# Quantum invariants of framed links from ternary self-distributive cohomology

Emanuele Zappala  Institute of Mathematics and Statistics, University of Tartu
Narva mnt 18, 51009 Tartu, Estonia<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract.

The ribbon cocycle invariant is defined by means of a partition function using
ternary cohomology of self-distributive structures (TSD) and colorings of
ribbon diagrams of a framed link, following the same paradigm introduced by
Carter, Jelsovsky, Kamada, Langfor and Saito in Transactions of the American
Mathematical Society 2003;355(10):3947-89, for the quandle cocycle invariant.
In this article we show that the ribbon cocycle invariant is a quantum
invariant. We do so by constructing a ribbon category from a TSD set whose
twisting and braiding morphisms entail a given TSD $2$-cocycle. Then we show
that the quantum invariant naturally associated to this braided category
coincides with the cocycle invariant. We generalize this construction to
symmetric monoidal categories and provide classes of examples obtained from
Hopf monoids and Lie algebras. We further introduce examples from Hopf-
Frobenius algebras, objects studied in quantum computing.

###### Contents

  1. 1 Introduction
    1. 1.1 Main results
    2. 1.2 Organization of the article
  2. 2 Preliminaries
    1. 2.1 Racks, quandles and cocycle invariants
    2. 2.2 Framed links and their diagrams
    3. 2.3 Ternary racks and their self-distributive cohomology
    4. 2.4 Braided monoidal categories and ribbon categories
  3. 3 Ribbon cocycle invariants
  4. 4 Ribbon categories from self-distributive ternary operations
  5. 5 The Ribbon cocycle invariant is a quantum invariant
  6. 6 Examples and Computations
    1. 6.1 Examples from compositions of binary quandles
    2. 6.2 Examples from heap structures
    3. 6.3 Computations of quantum invariants
  7. 7 Generalized construction
    1. 7.1 TSD objects in symmetric monoidal cateogries
    2. 7.2 Examples of TSD objects
    3. 7.3 Categorical 2-cocycle condition
    4. 7.4 Examples of categorical 2-cocycles
    5. 7.5 Construction of ribbon categories
  8. 8 Invariants of framed links in symmetric monoidal categories
  9. A Examples from ternary augmented racks

##  1\. Introduction

Self-distributivity of binary operations is well known to be an algebraic
formulation of the Reidemeister move III in knot theory. Sets with self-
distributive operations (i.e. shelves) satisfying extra conditions encoding
Reidemeister moves I and II have been used, starting in the 1980’s, to
construct invariants of knots and links. For instance, Joyce and Matveev
independently defined what is now known as the fundamental quandle of a knot
[Joy, Mat], whose construction is given as a presentation where the generators
correspond to the arcs of a knot diagram, while the relations formally
correspond to the conjugation operation in a group. Shelves satisfying the
algebraic Reidemeister move II condition are called racks, while those
satisfying also the algebraic counterpart of Reidemeister move I are called
quandles.

More recently, the notion of (co)homology of quandle has been introduced, and
a state-sum invariant of links that utilizes quandle cohomology has been
constructed in [CJKLS]. The resulting “cocycle invariant” is obtined as a sum
over all the colorings of a knot diagram, the states, of all the products of
Boltzmann weights, determined by quandle $2$-cocycles. Although computing
cocycle invariants introduces a new problem, that of obtaining nontrivial
quandle second cohomology classes, it is in general easier to compare two
cocycle invariants rather than comparing the fundamental quandle of two knots.

Moreover, it is known that quandles induce solutions to the set-theoretic
Yang-Baxter equation and therefore, upon linearizing the corresponding set-
theoretic map, they produce Yang-Baxter operators [Eis]. In fact, given a
quandle and a $2$-cocycle $\alpha$, one can construct a Yetter-Drinfel’d
module (i.e. a particular instance of a ribbon category) [Gra] and,
consequently, one can obtain quantum link invariants associated to the ribbon
category following a standard procedure as in [Tur]. It naturally arises the
question of whether the two types of invariant are somehow related. A positive
answer has been given in [Gra], where it is shown that the invariants coincide
in a suitable sense.

Ternary self-distributive structures are generalizations of binary shelves to
the setting of ternary operations. A suitable diagrammatic interpretation of
crossing of ribbons in terms of ternary operations translates the fundamental
moves for the isotopy equivalence of framed links into a ternary analogue of
rack. A corresponding state-sum invariant that uses cohomology of ternary
racks and colorings of ribbon diagrams associated to framed links is then
constructed [EZ] following the same reasoning as in the binary case. This
invariant, called ribbon cocycle invarinat, has been studied for a fundamental
class of ternary racks, called group heaps, and it has been seen to detect
nontrivial framing of links [SZ].

On the other hand, group heaps can be generalized to certain structures, named
quantum heaps, that naturally arise from involutory Hopf algebras, i.e. having
antipode that squares to the identity map. A corresponding construction for
Hopf monoids in symmetric monoidal categories exists [heap], providing a large
class of examples for ternary self-distributive objects in symmetric monoidal
categories, in the sense of [ESZ] Section 8. It is therefore possible to
develop an analogue of the ternary set-theoretic theory in symmetric monoidal
categories.

The scope of this article is that of using ternary self-distributive (TSD)
structures and their ternary cocycles to construct ribbon categories whose
twisting morphisms are nontrivial, and study the corresponding link
invariants. The starting point of this study follows the paradigm that has
been used in [Gra] to prove that the cocycle invariants are indeed quantum
invariants. We prove, in fact, that set-theoretic TSD structures and a choice
of a ternary $2$-cocycle are linearized to obtain a braiding in a suitably
constructed symmetric monoidal category. The construction is similar to that
of the braid category [FY, Kas], but braiding and twisting are induced by the
TSD structure following the doubling functorial procedure in [ESZ], and using
TSD cocycles to twist the morphisms obtained. Analogously to the fact that the
ribbon cocycle invariant detects nontrivial framings [SZ], we obtain that the
twisting defined in this category is nontrivial, as opposed to the case of
Yetter-Drinfel’d modules associated to (binary) quandle operations.

On the one hand, there is no strict need of defining a ribbon category out of
the data of a TSD and a ternary $2$-cocycle, in the sense that we can obtain a
representation of the framed braid group in a similar fashion as in [Tur],
from which a corresponding quantum invariant would naturally arise. On the
other hand, though, this construction easily generalises to multiple objects
where “self-distributive” ternary actions are defined. These produce a more
general family of ribbon categories where the twists are obtained by TSD
operations as in the previous case, while the braidings are obtained from
ternary actions. Moreover, the braiding and twisting morphisms can be deformed
by cohomological classes that twist the weights and entail the operations and
mutual actions of the underlying structure. Among the examples that we present
in this paper, we find mutually distributive structures and their labeled
cohomology, whose algebraic properties were studied in [ESZ], and $G$-families
of quandles and their cohomology theory, extensively studied in connection
with knotted handlebody invariants [IIJO, Nos].

The approach mentioned above, in addition, is particularly suitable to be
generalized to the case of TSD objects in symmetric monoidal categories. As
observed above, in fact, the notion of heap has a counterpart obtained from
involutory Hopf monoids in symmetric monoidal categories, therefore providing
a fertile ground for a general theory that associates a ribbon category to a
symmetric monoidal category along with a TSD object in it. Using the TSD
morphism we obtain, in fact, a Yang-Baxter operator in the tensor product of
the TSD object we start with, and use this to define the braiding of the
ribbon category. The twist is obtained via the same procuedure by interpreting
twists as self-intersections of a ribbon. In other words, Reidemeister move I
does not hold when we consider framed links, but it is replaced by a twisting
which can be defined using a variation of the braiding.

We have mentioned that we utilize TSD cohomology classes to deform the
braiding and twisting in the case of linearized TSD operations. When working
in a symmetric monoidal category, we can introduce a categorical version of
the $2$-cocycle condition. The setting, here, generalizes the set-theoretic
one in two fundamental ways. Recall the set-theoretic $2$-cocycle condition,
which reads

$\psi(x,y,z)-\psi([x,u,v],[y,u,v],[z,u,v])-\psi(x,u,v)+\psi([x,y,z],u,v)=0$

for all $x,y,z,u,v\in X$, where $(X,[-,-,-])$ is a TSD set. Firstly, observe
that certain elements appear in more than one term, and therefore are
repeated. This is no particular concern when dealing with set-theoretic
structures, but in a general symmetric monoidal category, it is required that
each instance of a repreated element is replaced by an instance of a
comultiplication morphism. In fact, the definition of TSD object, see for
instace Section 8 in [ESZ] for $n$-ary case, implements this perspective
already, and it is somehow natural to expect that it carries on to the
$2$-cocycle condition. Secondly, in the set-theoretic case coefficients of
cohomology are taken in a group, and linearization naturally requires the
coefficients to be represented in the ground field. In an arbitrary symmetric
monoidal category, we interpret this situation as an equality holding in the
unit object of the category. The object of coefficients naturally acts on the
TSD object allowing the “cocycle” to perturb the Yang-Baxter operator
associated to the TSD morphism. If one thinks of the group algebra associated
to a group as being a Hopf algebra where the comultiplication is simply the
splitting of an element in two identical copies, then the categorical
interpretation of the $2$-cocycle condition seems to be on the same footing as
the $2$-cocycle condition in the ground field of the linearization of a set-
theoretic operation.

In the general situation, one further assumption is necessary, in order to
apply the same construction as in the category of vector spaces. Namely, one
needs to assume that the category is $\mathbb{I}$-linear, where $\mathbb{I}$
is the unit object. Then the $2$-cocycles are assumed to take values in the
ground object $\mathbb{I}$ and, moreover, they are supposed to satisfy a
convolution inversion formula, in order to allow the definition of inverses.
This is naturally satisfied in the linearized case, since comultiplication is
simply diagonal, and coefficients in a group are automatically invertible.

Naturally, as in the set-theoretic case one can obtain ribbon categories from
multiple TSD sets having suitable ternary actions and families of ternary
$2$-cocycles, we can generalize the previous construction in a symmetric
monoidal category where multiple TSD objects along with certain ternary
morphisms are defined. An interesting class of examples arises from ternary
augmented racks, where the axioms of augmentation can be easily translated
from the case of vector spaces and Hopf algebras to that of Hopf monoids in a
symmetric monoidal category.

###  1.1. Main results

We proceed now to concisely summarize the main results of the present article.

The first result (Theorem 4.3) is that starting from a TSD set $(X,T)$ and a
given ternary $2$-cocycle $\alpha\in Z^{2}(X,A)$ with coefficients in an
abelian group $A$, we construct a ribbon category
$\mathcal{R}^{*}_{\alpha}(X)$ whose braidings are constructed out of a Yang-
Baxter operator arising from $(X,T)$ and deformed by the cocycle $\alpha$.
Moreover, it is shown that the ribbon category is well-defined, up to
equivalence of braided categories, with respect to the cohomology class of
$\alpha$, in the sense that if $\beta$ represents the same cohomology class,
then there exists a braided functor that gives an equivalence of categories
between $\mathcal{R}^{*}_{\alpha}(X)$ and $\mathcal{R}^{*}_{\beta}(X)$.
Moreover, a similar construction is shown to hold when starting with a family
of TSD sets $\\{X_{i}\\}_{i\in I}$ along with maps $T_{ij}:X_{i}\times
X_{j}\times X_{j}\longrightarrow X_{i}$ satisfying a generalized version of
TSD condition (Theorem 4.12). In this situation, we define the notion of
ternary $2$-cocycles for the family $\\{X_{i}\\}_{i\in I}$ and use them to
deform the Yang-Baxter operator associated to it. We therefore construct a
braiding and a twisting in order to obtain a ribbon category whose families of
objects and morphisms are larger than those of $\mathcal{R}^{*}_{\alpha}(X)$.
In Theorem 5.1 we show that the quantum invariants associated to the ribbon
category $\mathcal{R}^{*}_{\alpha}(X)$ coincide with the (state-sum) ribbon
cocycle invariant in a suitable sense, i.e. when we take a representation of
the coefficient abelian group of cohomology in the ground field. This
construction is generalized to the setting of TSD objects in symmetric
monoidal categories, of which TSD sets in the category of sets are a
particular instance. It is shown that in this setup, from a TSD object we
obtain a Yang-Baxter operator which is deformed by means of what is hereby
called a categorical $2$-cocycle. The new Yang-Baxter operator is used then to
construct the braiding in what is shown to be a ribbon category (Theorem
7.16).

###  1.2. Organization of the article

This article is structured as follows. We review some preliminary material in
Section 2, where we recall binary and ternary self-distributive structures,
the cocycle invariant and some basic notions regarding symmetric monoidal
categories. In Section 3 we give a detailed account of the construction of the
ribbon invariant of framed links as well as a proof of its being well-posed.
In Section 4 we show that starting from the data of a TSD set and a ternary
$2$-cocycle, there exists a ribbon category determined up to equivalence of
categories with respect to cohomology class of the $2$-cocycle. Moreover, it
is shown that a similar construct exists starting from a family of TSD
structures with some extra compatibility conditions and an analogue of the
notion of ternary $2$-cocycle. The corresponding ribbon cateogry has a wider
class of objects and morpshims with respect to the previous one. We then
proceed to show, in Section 5, that the (state-sum) ribbon cocycle invariant
coincides with the quantum invariant associated to the ribbon category arising
in Section 4. Section 6 presents various examples to elucidate the
construction in practice. Section 7 is devoted to generalizing the theory
developed in the previous sections in the context of symmetric monoidal
cateories and TSD objects. The notion of categorical $2$-cocycle condition is
introduced in order to deform the braidings obtained from TSD objects, in a
fashion that follows the paradigm of Section 4. Quantum invariants associated
to this class of ribbon categories are discussed in Section 8. Finally,
further examples arising from ternary racks are given in the Appendix.

Acknowledgements. This research has been funded by the Estonian Research
Council under the grant: MOBJD679. The author is grateful to M. Elhamdadi and
M. Saito for useful conversations.

##  2\. Preliminaries

In this section we provide preliminary material that is used throughout the
article.

###  2.1. Racks, quandles and cocycle invariants

Racks are (non-associative) magmas satisfying the self-distributive property
given by $(x*y)*z=(x*z)*(y*z)$ for all $x,y,z$, such that the right
multiplication maps are bijections. Self-distributivity is an “algebraization”
of the topological notion of Reidemeister move III, while the requirement that
right multiplications be bujective corresponds to imposing Reidemeister move
II. Idempotent racks are called quandles, where idempotence corresponds to the
remaining Reidemeister move I. It is well known that knot and link isotopy
classes in $\mathbb{R}^{3}$, or $\mathbb{S}^{3}$, can be characterized
combinatorially via their diagrams, i.e. projections on the plane satisfying
certain regularity properties, and Reidemeister moves I, II and III.
Consequently, quandles have been used in [CJKLS] to construct state-sum
invariants of links, named cocycle invariants. Fundamental roles in the
definition and validity of the cocycle invariant are played by the notion of
quandle coloring of a knot/link diagram, and a cohomology theory associated to
racks and quandles. In fact, loosely speaking, the invariant is defined by
considering all possible colorings of a fixed given diagram of a link
$\mathcal{L}$, and multiplying the weights of each crossing of the diagram,
each of which defined by applying a pre-determined $2$-cocycle to the colors
meeting at the crossing. When applying any of the Reidemeister moves to pass
from one diagram of $\mathcal{L}$ to the other, i.e. when performing an
isotopy on $\mathcal{L}$, the colors of the diagrams correspond bijectively by
virtue of the axioms defining a quandle, and the weights remain unchanged
because of the definition of quandle cohomology.

We proceed to briefly review the notion of quandle coloring of a link diagram,
and the definition of cohomology associated to a quandle $Q$. A reference for
both definitions is the article [CJKLS], where the cohomology utilizes abelian
coefficients, while the case with non-abelian coefficients is treated in
[CEGS]. Let $\mathcal{L}$ be an oriented link, let $\mathcal{D}$ indicate an
oriented diagram of $\mathcal{L}$, and let $Q$ be a quandle, with operation
denoted by the symbol $*$. A coloring of $\mathcal{D}$ by $Q$ is a map
$\mathcal{C}:R\longrightarrow Q$, where $R$ denotes the set of arcs of the
diagram $\mathcal{D}$, satisfying the conditions given in Figure 1, for
positive and negative crossings.

$x$$x*y$$y$$x$$y$$x*y$ Figure 1. Coloring condition for positive crossings
(left) and negative crossings (right).

Let $Q$ be a quandle and define chain groups $C_{n}(Q)$ to be the free abelian
group generated by the elements of $Q^{n}$ for each $n$. Then, we define the
$n^{\rm th}$-differential $\partial_{n}$ on generators according to the
assignment

$\displaystyle\partial_{n}(x_{1},\ldots,x_{n})$ $\displaystyle=$
$\displaystyle\sum_{i=2}^{n}(-1)^{n}[(x_{1},\ldots,x_{i-1},\hat{x}_{i},x_{i+1},\dots,x_{n})$
$\displaystyle\hskip
14.22636pt-(x_{1}*x_{i},\ldots,x_{i-1}*x_{i},\hat{x}_{i},x_{i+1},\ldots,x_{n})]$

where we have used $\hat{}$ to indicate omission of an element. Observe that
the first term in the sum is the “usual” simplicial term, while the second
term contains the information associated to the operation $*$, determining the
quandle structure. One proves directly that the maps $\partial_{n}$ satisfy
the pre-simplicial conditions and it follows automatically that
$\partial_{n-1}\circ\partial_{n}=0$, from which we have a well defined chain
complex and an associated homology theory called rack homology. Quandle
homology is obtained by quotienting out the sub-complex $C^{\rm q}_{n}(Q)$
generated as a free abelian group by $n$-tuples of $Q^{n}$ where
$x_{i}=x_{i+1}$ for some $i$. In fact, it is the indempotency condition that
induces well defined maps $\partial_{n}$ when restricting on the subgroups
$C^{\rm q}_{n}(Q)$. Taking $A$ to be an abelian group and dualizing the rack
and quandle chain complexes, one obtains associated cohomology theories which
we denote by $H^{n}(Q;A)$ and $H^{n}_{\rm q}(Q;A)$, respectively. A
representative $\phi$ of a second cohomology class $[\phi]\in H^{2}_{\rm
q}(Q;A)$ satisfies the $2$-cocycle condition, which takes the form

$\phi(x,y)-\phi(x*z,y*z)-\phi(x,z)+\phi(x*y,z)=0,$

for all $x,y,z\in Q$. The $2$-cocycle condition is related via a diagrammatic
interpretation to Reidemeister move III, as depicted in Figure 1 in [CJKLS],
while the $3$-cocycle condition, which we do not explicitly consider herein,
is related to the tetrahedron move and shadow colorings.

Fix now a coloring $\mathcal{C}$ by a quandle $Q$, defining at each crossing
$\tau_{i}$ of Figure 1 a Boltzmann sum, $\mathcal{B}(\tau_{i},\mathcal{C})$,
as $\psi(x,y)$, for positive crossing (left panel), and $\psi(x,y)^{-1}$ for
negative crossing (right panel), where $\psi\in Z^{2}(Q,A)$ is a quandle
$2$-cocycle, one defines the state-sum (or partition function)

$\sum_{\mathcal{C}}\prod_{i}\mathcal{B}(\tau_{i},\mathcal{C})$

for any given diagram of a knot, and where the sum runs over all the colorings
$\mathcal{C}$ of the fixed diagram, and the product runs over all the
crossings. This state-sum, called cocycle invariant, is shown to be an
invariant of knots in [CJKLS], where it has been firstly introduced. When
dealing with a link, one proceeds analogously for each component and defines
an invariant that is a vector with as many entries as the components of the
given link.

###  2.2. Framed links and their diagrams

Framed links are embeddings of finitely many copies of $\mathbb{S}^{1}\times
D^{2}$, i.e. solid tori, in the three dimensional space $\mathbb{R}^{3}$, or
its compactification $\mathbb{S}^{3}$. Alternatively, framed links can be
defined as links along with a choice of a section of their normal bundle.
Diagrammatically, a frmed link $\mathcal{L}$ is represented by a link diagram
whose arcs are thickened to be ribbons. This thickening is obtained in a
standard way by doubling each arc so to obtain a second copy of the link
diagram, parallel to the first one. The one lets the second copy mimick the
over/under passing information of the first diagram. Such a thickened diagram
is called blackboard framing. A crossing of a diagram whose arcs have been
thickened into a ribbon is represented in Figure 2. From the figure is clear
that the coloring paradigm corresponding to that of quandles changes. We can
think of each crossing as two arcs, each of them underpassing two arcs. The
coloring rule is suitably defined by means of ternary racks. This concept,
introduced to the author by M. Saito, is formalized in Section 3, where it is
also given a construction of the ribbon cocycle invariant.

Figure 2. Crossing of a blackboard framing of a framed link. $\cong$$+1$
Figure 3. Self-crossing of a ribbon introduces twists.

Reidemeister moves (R moves for short) of type II and III translate directly
into analogously defined moves where each arc is thickened to a ribbon, while
R move I does not hold in the context of framed links, since it introduces a
twist, i.e. a change in the framing. This is depicted in Figure 3. Throughout
this article we will depict positive, resp. negative, twists by a rectangle
inserted in a ribbon with a positive, resp. negative, integer indicating the
number of twists and their orientations. Isotopy equivalence of framed links
is characterized by moves RII, RIII and the cancellation of twists depicted in
Figure 4, where each twist is thought of as a self-crossing as in Figure 3
(with negative twists obtained by kinks in the opposite direction).

$-1$$+1$$\cong$ Figure 4. Twists with opposite signs annihilate each other.

###  2.3. Ternary racks and their self-distributive cohomology

Ternary racks are generalizations of racks to sets with ternary operations.
Specifically, a set $X$ endowed with a ternary operation $T:X\times X\times
X\longrightarrow X$ satisfying the conditions

$T(T(x,y,z),u,v)=T(T(x,u,v),T(y,u,v),T(z,u,v)),$

is said to be a ternary self-distributive (TSD) set. A TSD set such that the
map $X\longrightarrow X$ defined by $T(\bullet,x,y)$ is a bijection for all
$x,y\in X$ is said to be a ternary rack. A notable example of TSD structure is
the heap of a group, defined as $(x,y,z)\mapsto xy^{-1}z$. Heap operations
have been considered and studied in [SZ], in relation with their ribbon
cocycle invariants (see Section 3 below). TSD operations naturally arise also
by composing binary self-distributive operations. For instance, if $(Q,*)$ is
a rack, or quandle, then the operation $T(x,y,z):=(x*y)*z$ can be seen to
endow $Q$ with a TSD structure [ESZ].

We recall the notion of TSD (co)homology of ternary racks [Green] and, more
specifically, the TSD $2$-cocycle condition, since the ribbon cocycle
invariant utilizes $2$-cocycles as weights, in a fashion similar to the
original cocycle invariant introduced in [CJKLS]. Let $(X,T)$ be a TSD set,
and define $C_{n}(X)$ to be the free abelian group generated by
$(2n+1)$-tuples of elements of $X$. Define maps
$\partial_{n}:C_{n}(X)\longrightarrow C_{n-1}(X)$ by

$\displaystyle\partial_{n}(x_{1},\ldots,x_{2n+1})$
$\displaystyle\sum_{i=1}^{n}(-1)^{i}[(x_{1},\ldots,x_{2i-1},\widehat{x_{2i},x_{2i+1}},x_{2i+2},\ldots,x_{2n+1})$
$\displaystyle-(T(x_{1},x_{2i},x_{2i+1}),\ldots,T(x_{2i-1},x_{2i},x_{2i+1}),\widehat{x_{2i}x_{2i+1}},x_{2i+2},\ldots,x_{2n+1})],$

and extended by ${\mathbb{Z}}$-linearity. A (long) direct computation shows
that the maps $\partial_{n}$ are obtained as the alternating sum
$\partial_{n}=\sum_{i=1}^{n}(-1)^{i}\partial_{n}^{i}$, where the maps
$\partial_{n}^{i}$ satisfy the usual pre-simplicial module axioms and,
consequently, $(C_{n}(X),\partial_{n})$ defines a chain complex whose
associated homology, written $H_{n}(X)$, is called TSD homology. Given an
abelian group $A$ we obtain, upon dualizing the previous chain complex, TSD
cochain groups and associated cohomology. We indicate these groups with the
symbols $C^{n}(X;A)$ and $H^{n}(X;A)$, respectively. The $2$-cocycle
condition, for a $2$-cochain $\psi:X^{3}\longrightarrow A$, takes the form

$\displaystyle\psi(x,y,z)-\psi(T(x,u,v),T(y,u,v),T(z,u,v))$ $\displaystyle=$
$\displaystyle\psi(x,u,v)-\psi(T(x,y,z),u,v)$

for all $x,y,z,u,v\in X$. As it will be seen in Section 3, the ternary
$2$-cocycle condition, along with an appropriate interpretation of colorings
of blackboard framings by ternary racks, is invariant under moves RII, RIII
and cancellation move. It is therefore possible to define Boltzmann weights by
means of ternary $2$-cocycles and introduce a state-sum invariant of framed
links. Given a $1$-cochain $f:X\longrightarrow A$, the first cohomology
differential $\delta^{1}$ maps it to the function

$(x,y,z)\mapsto\delta^{1}f(x,y,z):=f(x)-f(T(x,y,z)).$

Therefore two $2$-cocycles $\psi$ and $\phi$ are in the same second cohomology
class if they differ by a term $\delta^{1}f$ as above, for some $1$-cochain
$f$. As it will be proved in Section 3, changing the representative of a
second cohomology class changes the ribbon cocycle invariant by a well
understood term, so that the invariant is a well defined function, up to a
known equivalence relation, of the cohomology group $H^{2}(X;A)$. This
observation did not appear in the original construction in [EZ], and has been
proven when $T$ is the heap operation in [SZ].

###  2.4. Braided monoidal categories and ribbon categories

Recall that given a monoidal category $(\mathcal{C},\otimes)$, a braiding in
$\mathcal{C}$ is a natural family of isomorphisms $c_{X,Y}:X\otimes
Y\longrightarrow Y\otimes X$ such that the Hexagon Axiom is satisfied [Kas],
Chapter XIII. Specifically, it is required that the diagram

${X\otimes(Y\otimes Z)}$${(Y\otimes Z)\otimes X}$${(X\otimes Y)\otimes
Z}$${Y\otimes(Z\otimes X)}$${(Y\otimes X)\otimes Z}$${Y\otimes(X\otimes
Z)}$$\scriptstyle{c_{X,Y\otimes
Z}}$$\scriptstyle{\alpha_{Y,Z,X}}$$\scriptstyle{\alpha_{X,Y,Z}}$$\scriptstyle{c_{X,Y}\otimes\mathbbm{1}}$$\scriptstyle{\alpha_{Y,X,Z}}$$\scriptstyle{\mathbbm{1}\otimes
c_{X,Z}}$

where $\alpha$ indicates the associativity constraint of the category
$\mathcal{C}$, and we have omitted the subscript of the identity morphism, as
no confusion arises. A similar diagram for the inverse of $c_{X,Y}$ is
required to commute, but this can be obtained from the commutativity of the
previous one. Therefore, it is not an independent axiom, see comment in [Kas]
right above Definition XIII.1.1. A monoidal category endowed with a braiding
is said to be a braided monoidal category. Observe that for any object
$X\in\mathcal{C}$, the braiding $c_{X,X}$ is a solution to the braid (Yang-
Baxter) equation. In fact, the diagrammatic interpretation coincides with the
RIII move for knot/link diagrams.

In this article we will consider our monoidal categories to be strict, so that
the associativity constraints will not be written now on. This assumption is
not particularly restrictive, as any monoidal category can be seen to be
equivalent to a struct monoidal category.

Typical examples of braided monoidal categories arise from braided bialgebras
(see Chapter VIII in [Kas]), where the category of $H$-modules, of a braided
bialgebra $H$, has a braided structure associated to the universal $R$-matrix
of $H$. Another important class of examples arises from crossed $G$-sets,
where the braiding is obtained using the crossed action. Linearizing these
structures produces bradings in some subcategory of vector spaces. More
generally, one can use a quandle operation, which generalizes the axioms of
crossed $G$-set.

A left dual of $X$ in a braided monoidal category is an object $X^{*}$ such
that there exist morphisms $coev:\mathbb{I}\longrightarrow X\otimes X^{*}$ and
$ev:X^{*}\otimes X\longrightarrow\mathbb{I}$ such that the equalities
$(\mathbbm{1}\otimes ev)\circ(coev\otimes\mathbbm{1})=\mathbbm{1}$ and
$(coev\otimes\mathbbm{1})\circ(\mathbbm{1}\otimes ev)=\mathbbm{1}$ hold. A
similar definition for right duals can be made. A category such that left and
right duals exist for all objects is said to be autonomous, and in this case
left and right duals coincide. In what follows, we will refer to left duality
simply as “duality”, if not otherwise specified.

In a braided monoidal category, the notion of dual introduces a diagrammatic
interpretation with different types of crossing orientations. The
corresponding RIII moves with new orientations are seen to be induced by the
original diagrammatic RIII, as in Figure 10 in [Tur]. See [Kas], Chapter XIV,
for the diagrammatic interpretation of duality.

Given a braided monoidal category with duals, a twist is a natural family of
isomorphisms $\theta_{X}$ such that $\theta_{X\otimes
Y}=(\theta_{X}\otimes\theta_{Y})\circ c_{Y,X}\circ c_{X,Y}$. Moreover,
$\theta$ is required to behave well with respect to dual objects, in the sense
that $\theta_{X^{*}}=(\theta_{X})^{*}$.

In the previous example of braided monoidal categories arising from
linearization of crossed $G$-sets, linear duals, and evaluation and
coevaluation maps in vector spaces give a ribbon cateogry structure along with
trivial twists. In fact, triviality of twists can be interpreted as a
consequence of the fact that quandles are idempotent following Figure 3. A
twist is introduced by means of a self-crossing, and the corresponding effect
of applying a quandle operation is trivial, due to idempotence. As it will be
seen below, using ternary operations and their diagrammatic interpretation
gives rise to ribbon categories, following a simialar paradigm, whose twisting
morphisms are nontrivial. Consequently the corresponding invariants detect the
framing of knot/link.

##  3\. Ribbon cocycle invariants

Following [CJKLS, CEGS], it is introduced in this section an invariant of
framed links, using colorings of ribbon diagrams by ternary quandles, and
ternary quandle $2$-cocycles. This invariant was originally introduced in
[EZ], and studied in [SZ] in the case of heap invariants. We give the details
of the construction, as they are relevant for the rest of the article.

Framed links are represented in the rest of the paper by their blackboard
framing. Therefore the arcs of a projection on the plane are represented by
ribbons bounded by two parallel arcs. Orientetions of the ribbons are
specified by orientations of the parallel arcs, which will be always assumed
to be concordant. The framing of a ribbon, which is an integer number, is
obtained by twisting the two arcs delimiting the ribbon. This is given by
consecutive self-intersections, and therefore a specified orientation of the
ribbon determines whether $n$ consecutive twists are positive or negative. A
diagram whose edges are specified by two parallel arcs, therefore defining a
ribbon, is called ribbon diagram. It follows from the definitions that the
blackboard framing of a framed link is a ribbon diagram.

Let $X$ be a tarnary quandle and let $\mathcal{D}$ be a diagram of a framed
link. Suppose for the moment that the link has a single component, in other
words it is a diagram of a framed knot. To each ribbon arc in $\mathcal{D}$,
associate a color by a pair of elements $(x_{1},x_{2})\in X\times X$,
corresponding to each side of the ribbon. At a positive crossing $\tau$ of
$\mathcal{D}$, where the arcs colored by $(x_{1}^{\tau},x_{2}^{\tau})$ and
$(y_{1}^{\tau},y_{2}^{\tau})$ meet, let the overpassing ribbon mantain the
same color, while change the color of the underpassing ribbon to
$(T(x_{1},y_{1},y_{2}),T(x_{2},y_{1},y_{2}))$. When a crossing $\tau$ is
negative, we change the color of the underpassing ribbon to $(z_{1},z_{2})$,
where $z_{i}$ is the unique element of $X$ such that
$T(y_{i},x_{1},x_{2})=z_{i}$, whose existence is guaranteed by the axioms of
ternary quandle. We now pose the following.

######  Definition 3.1.

Let $\mathcal{D}$ be a ribbon diagram whose set of ribbon arcs is denoted by
$\mathcal{R}$, and let $X$ be a ternary quandle. Then, a coloring of
$\mathcal{D}$ by $X$, is a (set-theoretic) map

$\mathcal{C}:\mathcal{R}\longrightarrow X\times X,$

that is consistent with the coloring rule above. The set of colorings of a
framed link is defined to be the set of colorings of a ribbon diagram of the
link.

######  Lemma 3.2.

Let $X$ be a ternary quandle and $\mathcal{D}$ a ribbon diagram of a framed
link. Then the set of colorings $\mathcal{C}$ of $\mathcal{D}$ by $X$ is
invariant under Reidemeister moves II and III, and moreover it respects
cancelling of kinks.

As a consequence, the notion of coloring of a framed link is well posed, and
the set of colorings of a framed link is an isotopy invariant.

Suppose $\phi$ is a ternary quandle $2$-cocycle of $X$, with coefficients in
$A$. For a given crossing $\tau$, define the Boltzmann weight at $\tau$,
depending on the coloring $\mathcal{C}$ and the $2$-cocycle $\phi$ by
$(\phi(x^{\tau}_{1},y^{\tau}_{1},y^{\tau}_{2})^{\epsilon(\tau)},\phi(x^{\tau}_{2},y^{\tau}_{1},y^{\tau}_{2})^{\epsilon(\tau)})\in
A\times A$, where the sign $\epsilon(\tau)$ is that of the crossing. Now we
can define the $2$-cocycle invariant of a link $L$ with respect to a ternary
quandle $X$ and a ternary $2$-cocycle $\psi\in Z^{2}(X,A)$ as follows. First
we give the definition in the case of framed knots, and then generalize it to
framed links.

######  Definition 3.3.

Let $\mathcal{D}$ be a ribbon diagram of a framed knot $K$ having $k$
crossings $\tau_{1},\cdots,\tau_{k}$, and let $\psi\in Z^{2}(X,A)$ be a
ternary $2$-cocycle. Define the cocycle invariant of $K$, by the $2$-cocycle
$\psi$ as

$\Theta_{\mathcal{D}}(X,T,\psi)=\sum_{\mathcal{C}}(\prod_{i=1}^{k}\psi(x^{\tau_{i}}_{1},y^{\tau_{i}}_{1},y^{\tau_{i}}_{2})^{\epsilon(\tau_{i})},\prod_{i=1}^{k}\psi(x^{\tau_{i}}_{2},y^{\tau_{i}}_{1},y^{\tau_{i}}_{2})^{\epsilon(\tau_{i})}),$

where $\epsilon(\tau_{i})$ is the sign of the $i^{th}$ crossing, and the sum
runs over all colorings $\mathcal{C}$ of the diagram $D$. We sometimes shorten
notation by writing $\Theta_{\psi}$ instead of
$\Theta_{\mathcal{D}}(X,T,\psi)$, and name $\Theta_{\mathcal{D}}(X,T,\psi)$
the ribbon cocycle invariant. Each term
$\psi(x^{\tau_{i}}_{j},y^{\tau_{i}}_{1},y^{\tau_{i}}_{2})^{\epsilon(\tau_{i})}$,
$j=1,2$, is also called the Boltzmann weight at $\tau$, relative to the
coloring $\mathcal{C}$ and the $2$-cocycle $\psi$. It is denoted by the symbol
$\mathcal{B}_{j}(\mathcal{C},\tau)$.

It remains to prove that the $2$-cocycle invariant does not depend on the
choice of ribbon diagram $\mathcal{D}$ used in Definition 3.3.

######  Theorem 3.4.

The cocycle invariant does not depend on the equivalence class of the ribbon
diagram $\mathcal{D}$. Therefore, it is well defined and it is an invariant of
framed knots. Moreover, changing a $2$-cocycle $\psi$ to another
representative of the same second cohomology group $[\psi]\in H^{2}(X,A)$,
changes the invariant to an integer multiple of the unit $(e,e)\in A\times A$.

###### Proof.

The Theorem is proved by showing that the state sum is invariant with respect
to Reidemeister moves II and III, and with respect to kink cancellation.
Compare with the moves T1-T5 and T6f given in [FY], page 166, and with
rel1-rel8 in [RT], page 14. As seen in Lemma 3.2, applying a Reidemeister move
II or III, or applying the kink cancellation move transforms one coloring of a
diagram to another coloring. Assuming that the colors at the top strands, hen
changing colorings between Reidemeister move II, are $(x,y)\times(z,w)$, we
see that the consecutive crossings $\tau_{1}$ and $\tau_{2}$ contribute with a
cocycle weight of $(\psi(x,z,w),\psi(y,z,w)$ and
$(\psi(x,z,w)^{-1},\psi(y,z,w)^{-1})$ respectively. So the contributions
cancel. Reidemeister mover III coincides with with the ternary $2$-cocycle
condition, and the corresponding invariance is guaranteed. Cancellation of
kinks is similar to the case of Reidemeister move II, and the sign in
Definition 3.3 ensures that the state sum does not change. Suppose $\psi$ is a
coboundary, i.e. $\psi=\delta f$, for some $f:X\longrightarrow A$. Let
$\mathcal{C}$ be a fixed coloring and suppose we order the crossings
$\tau_{1},\ldots,\tau_{n}$ starting from an arbitrarily chosen arc, and
numbering the crossings as we encounter them following the arc along the knot.
Consider $\tau_{i}$ and $\tau_{i+1}$, and assume without loss of generality
that $\tau_{i}$ is a positive crossing. Then the Boltzmann sum contribution on
the first entry of the state sum is given by $f(x)f(T(x,z,w))^{-1}$, assuming
that the colorings on top of $\tau_{i}$ are $(x,y)\times(z,w)$. Likewise, the
contribution on the second entry is $f(y)f(T(y,z,w))^{-1}$. Now, suppose
$\tau_{i+1}$ is positive, and let $(z^{\prime},w^{\prime})$ overpass at
$\tau_{i+1}$. Then the Boltzmann sum is given by
$f(T(x,z,w))f(T(T(x,z,w),z^{\prime},w^{\prime}))^{-1}$, which shows that the
two terms $f(T(x,z,w))$ and $f(T(x,z,w))^{-1}$ cancel. Similarly for the
second entry of the Boltzmann weight. If $\tau_{i+1}$ is negative, and we let
$(z^{\prime},w^{\prime})$ be underpassing, we see that the terms $f(T(x,z,w))$
and $f(T(y,z,w))$ again appear in both crossings with opposite signs, and
cancel out again. Proceeding in this fashion we see that when we have
$\tau_{n}$ and $\tau_{1}$ the remaining terms cancel out, since have assumed
that $\mathcal{C}$ is a colorng, hence it is well defined. Each coloring in
the state sum contributes with a trivial term, so the invariant of a
coboundary simply counts the number of colorings. Consequently, it is an
integer multiple of $(e,e)$. This fact implies that if $\psi$ and $\phi$
differ by a coboundary, their invariants differ by an integer multiple of
$(e,e)$. A more concise argument, based on an interpretation of the state-sum
invariant in terms of Kronecker pairing first described in [CJKLS], is given
in Theorem 5.8 of [SZ] for the case when $T$ is the heap of a group. The
reader can verify that it is applicable also for general TSD operations. ∎

######  Remark 3.5.

The invariant $\Theta_{\mathcal{D}}(X,T,\psi)$ is, by construction, an element
of the group ring ${\mathbb{Z}}[A\times A]$. Since there is an isomorphism
${\mathbb{Z}}[A\times A]\cong{\mathbb{Z}}[A]\otimes{\mathbb{Z}}[A]$, we can
identify $\Theta_{\mathcal{D}}(X,T,\psi)$ with a sum of tensor products of
elements of $A$.

We now generalize Definition 3.3 to framed links with multiple components.
First, if a link $\mathcal{L}$ has $t$ components
$\mathcal{L}=\mathcal{K}_{1}\cup\cdots\cup\mathcal{K}_{t}$, we label the
crossings $\tau$ of $\mathcal{L}$ with the number of the component the
underpassing ribbon belongs to. Therefore, for example, if at the crossing
$\tau$ the underpassing ribbon belongs to the component $i$, $\tau$ is denoted
by $\tau_{i}$. For $j=1,2$ we define the Boltzmann weight
$B_{j}^{(i)}(\mathcal{C},\tau_{i})$ relative to the crossing $\tau_{i}$, in
the $i^{\rm th}$ component, as $\psi(x_{j},y_{1},y_{2})^{\epsilon(\tau_{i})}$,
where $(y_{1},y_{2})$ is the coloring of the overpassing ribbon (not
necessarily in the component $K_{i}$) and $(x_{1},x_{2})$ is the coloring of
the underpassing ribbon (in the component $\mathcal{K}_{i}$ by assumption). In
the following definition, we denote a vector with multiple entries being pairs
with the notation $(a,b)\times\cdots\times(a^{\prime},b^{\prime})$.

######  Definition 3.6.

Let the notation be as in the previous paragraph. Then the (vector) ribbon
cocycle invariant of $\mathcal{L}$, relative to the ternary quandle $X$ and
the ternary $2$-cocycle $\psi$ is defined as

$\vec{\Theta}_{\mathcal{D}}(X,T,\psi)=\sum_{\mathcal{C}}\times_{i=1}^{t}(\prod_{s=1}^{k(i)}\mathcal{B}_{1}^{i}(\mathcal{C},\tau_{i}(s)),\prod_{s=1}^{k(i)}B_{2}^{i}(\mathcal{C},\tau_{i}(s))),$

where $k(i)$ is the number of crossings in the $i^{\rm th}$ component,
$\tau_{i}(s)$ indicates the $s^{\rm th}$ crossing in the $i^{\rm th}$
component, and the sum indicates that in each component of the vector we are
summing over all possible colorings $\mathcal{C}$.

An argument similar to that of Theorem 3.4, applied to each component, shows
that the (vector) ribbon cocycle invariant does not depend on the isotopy
class of the framed link $\mathcal{L}$ and it is therefore well defined.
Moreover, changing $\psi$ to another $2$-cocycle in the same second cohomology
class changes the invariant by an integer multiple of the vector
$(e,e)\times\cdots\times(e,e)$, where $e$ is the neutral element of the
coefficient group used for cohomology.

##  4\. Ribbon categories from self-distributive ternary operations

In this section it is given a generalization of known constructions that allow
to define a Yetter-Drinfeld module from a set-theoretic quandle [Gra]. More
specifically, the aim of this section is to define a ribbon category given a
ternary self-distributive object in the symmetric monoidal category of vector
spaces. As opposed to the case of a quandle, the ribbon category that is
obtained in this procedure admits nontrivial twisting morphisms. These are
defined by means of a self-crossing, similar to the Reidemeister move I, which
in this case is not the trivial map. In this section, along with sections 5
and 6 we focus on the case of ternary self-distributive objects obtained via
linearization of set-theoretic structures. We use ternary cohomology in the
usual sense [ESZ]. It will be shown in Section 7 that this construction can be
generalized to the case of ternary self-distributive objects in symmetric
monoidal categories endowed with duals, using a generalized version of
$2$-cocycle condition with coefficients in a group object of the given
category.

Let $Q:=\\{x_{1},\ldots,x_{n}\\}$ be a finite ternary quandle with operation
$T:Q\times Q\times Q\longrightarrow Q$. Then linearizing the operation over a
ground field $\mathbbm{k}$ gives a ternary self-distributive object in the
category of $\mathbbm{k}$-vector spaces, $(X,T)$, where $X:=\mathbbm{k}\langle
x_{1},\ldots,x_{n}\rangle$ and the linearized operation is indicated with the
same symbol $T$.

Let $H^{2}(Q,A)$ indicate the second ternary self-distributive cohomology
group of $Q$, with coefficients in the multiplicative (abelian) group $A$. Fix
a nontrivial 2-cocycle $\alpha:Q\times Q\times Q\longrightarrow A$. In the
examples treated below we have the case when $A\subset\mathbbm{k}^{\times}$
or, more generally, when it is given a group character of $A$ in the group of
units of $\mathbbm{k}$. The group $A$ therefore acts on the vector space $X$
and its tensor products via scalar multiplication of $\mathbbm{k}$. We will
use the symbol$\cdot$ to indicate this action, for clarity, and we use
juxtaposition to denote the multiplication in $A$.

Using this data, we define the brading $c^{\alpha}:X^{\otimes 2}\otimes
X^{\otimes 2}\longrightarrow X^{\otimes 2}\otimes X^{\otimes 2}$ by the
assignment

$(x\otimes y)\otimes(z\otimes
w)\mapsto\alpha(x,z,w)\alpha(y,z,w)\cdot(z\otimes w)\otimes(T(x,z,w)\otimes
T(y,z,w)),$

having used a comma to separate the entries of $T$ instead of the symbol
$\otimes$ to shorten notation. As it is proved below, the morphism
$c^{\alpha}$ satisfies the braid equation if and only if $\alpha$ is a
2-cocycle, i.e. $[\alpha]\in H^{2}(Q,A)$ (cf. with Proposition 3.3 in [Gra]).
The twisting morpshism $\theta^{\alpha}_{2}:X^{\otimes 2}\longrightarrow
X^{\otimes 2}$ is defined by extending the assignement

$x\otimes y\mapsto\alpha(x,x,y)\alpha(y,x,y)\cdot T(x,x,y)\otimes T(y,x,y).$

This definition is motivated by introducing a complete twist in a ribbon by
self crossing, Figure 3.

We introduce now the ribbon category whose objects are all even tensor powers
of the vector space $X$ and generated by braiding and twisting given above.
See[Kas] Chapter XII, Section XII.1, for the general definition of
presentation of a category. We give a very explicit definition below, to
describe the morphisms in detail.

######  Definition 4.1.

Let $(X,T)$ be a ternary self-distributive object in
$\mathcal{V}_{\mathbbm{k}}$, as above, and $[\alpha]\in H^{2}(Q,A)$. Define
the category $\mathcal{R}_{\alpha}(X)$ as follows. The objects are even powers
$X^{\otimes 2n}$ of $X$ in the category $\mathcal{V}_{\mathbbm{k}}$. The
tensor product of two objects $Y$ and $Z$, written $Y\boxtimes Z$, is defined
to be the tensor product $\otimes$ in $\mathcal{V}_{\mathbbm{k}}$. The trivial
power $X^{0}$ is set to be $\mathbbm{k}$ by definition. The morphisms of this
category are defined as follows. The set ${\rm Hom}(X^{\otimes 2},X^{\otimes
2})$ consists of the identity map and twists $(\theta^{\alpha})^{\circ m}$,
where $m\in{\mathbb{Z}}$ and $\circ$ indicates composition. The set ${\rm
Hom}(Y,Z)$ is empty if $Y\neq Z$. The morphism set ${\rm Hom}(X^{\otimes
4},X^{\otimes 4})$ is the free monoid generated under composition by twofold
tensor products of $f,g\in{\rm Hom}(X^{\otimes 2},X^{\otimes 2})$, and the
braiding $c^{\alpha}:=c^{\alpha}_{2,2}$. The set ${\rm Hom}(X^{\otimes
2n},X^{\otimes 2n})$ is defined inductively as the free monoid generated under
composition by tensor products $f\in{\rm Hom}(X^{\otimes 2m_{1}},X^{\otimes
2m_{1}})$ and $g\in{\rm Hom}(X^{\otimes 2m_{2}},X^{\otimes 2m_{2}})$ with
$m_{1}+m_{2}=n$. The braidings $c^{\alpha}_{n,m}:X^{\otimes 2n}\boxtimes
X^{\otimes 2m}\longrightarrow X^{\otimes 2m}\boxtimes X^{\otimes 2n}$ are the
morphisms $X^{2(n+m)}\longrightarrow X^{2(n+m)}$ corresponding to block
permutation switching $X^{\otimes 2n}$ and $X^{\otimes 2m}$, obtained by
subsequent applications of $c^{\alpha}_{2,2}$.

Endow the category $\mathcal{R}_{\alpha}(X)$ with duals by setting
$(X^{\otimes 2n})^{*}:=(X^{*})^{\otimes 2n}$, where $X^{*}$ is the linear dual
of $X$. The evaluation map $ev$ is determined by $x\otimes y\boxtimes f\otimes
g\mapsto f(y)g(x)$, and the coevaluation map $coev$ by $1\mapsto x_{j}\otimes
x_{i}\boxtimes x^{i}\otimes x^{j}$, where the Einstein summation convention is
used. Let $\mathcal{R}^{*}_{\alpha}(X)$ indicate the category
$\mathcal{R}_{\alpha}(X)$ with the addition of duals.

######  Remark 4.2.

Direct inspection shows that the twisting morphism $\theta^{\alpha}$ defined
above coincides with the self-intersection morphism obtained by means of $ev$
and $coev$ maps as composition: $(ev\otimes\mathbbm{1}^{\otimes
2})\circ(\mathbbm{1}^{\otimes 2}\otimes
c^{\alpha})\circ(coev\otimes\mathbbm{1}^{\otimes 2})$, cf. Figure 3.

######  Theorem 4.3.

With the same notation as above, the category $\mathcal{R}^{*}_{\alpha}(X)$
with braiding induced by $c^{\alpha}$ and twisting morphisms induced by
$\theta^{\alpha}$ is a ribbon category. Moreover, if $[\alpha]=[\beta]$, then
the two categories $\mathcal{R}^{*}_{\alpha}(X)$ and
$\mathcal{R}^{*}_{\beta}(X)$ are equivalent.

###### Proof.

The fact that $\mathcal{R}_{\alpha}(X)$ is a tensor category is a consequence
of the fact that $\mathcal{V}_{\mathbbm{k}}$ is a tensor category, and the
definitions. To verify naturality of the braiding, it is enough to verify the
commutativity of the square

${X^{\otimes 2n}\boxtimes X^{\otimes 2m}}$${X^{\otimes 2m}\boxtimes X^{\otimes
2n}}$${X^{\otimes 2n}\boxtimes X^{\otimes 2m}}$${X^{\otimes 2m}\boxtimes
X^{\otimes 2n}}$$\scriptstyle{c^{\alpha}_{2n,2m}}$$\scriptstyle{f\boxtimes
g}$$\scriptstyle{g\boxtimes f}$$\scriptstyle{c^{\alpha}_{2n,2m}}$

for all morphisms $f$ and $g$. This follows from the fact that the morphisms
are defined as block braidings induced by $c^{\alpha}$. So the naturality is a
direct consequence of the definition of morphisms, and the fact that
$c^{\alpha}$ satisfies the braid equation (proved below) in the same way it is
proved for the braid category. The fact that $c^{\alpha}$ is a family of
isomorphisms is a consequence of the invertibility of $T$ axiom, holding for
ternary racks. To verify that $c^{\alpha}$ is indeed a braiding, it has to be
shown that it satisfies the braid equation. To this objective, observe first
that it is enough to prove that $c^{\alpha}_{2,2}$ satisfies the braid
equation

$(c^{\alpha}_{2,2}\boxtimes\mathbbm{1})\circ(\mathbbm{1}\boxtimes
c^{\alpha}_{2,2})\circ(c^{\alpha}_{2,2}\boxtimes\mathbbm{1})=(\mathbbm{1}\boxtimes
c^{\alpha}_{2,2})\circ(c^{\alpha}_{2,2}\boxtimes\mathbbm{1})\circ(\mathbbm{1}\boxtimes
c^{\alpha}_{2,2}),$

since the general case would follow from iterations of this specific case.
Using the shorthand notation $T_{x}^{y,z}=T(x,y,z)$, on a general basis vector
$x\otimes y\boxtimes z\otimes w\boxtimes u\otimes v$, the left hand side
equals

$\displaystyle(c^{\alpha}_{2,2}\boxtimes\mathbbm{1})\circ(\mathbbm{1}\boxtimes
c^{\alpha}_{2,2})\circ(c^{\alpha}_{2,2}\boxtimes\mathbbm{1})x\otimes
y\boxtimes z\otimes w\boxtimes u\otimes v$ $\displaystyle=$
$\displaystyle\alpha(x,z,w)\alpha(y,z,w)\alpha(T_{x}^{z,w},u,v)\alpha(T_{y}^{z,w},u,v)\alpha(z,u,v)\alpha(w,u,v)$
$\displaystyle\cdot u\otimes v\boxtimes T_{z}^{u,v}\otimes
T_{w}^{u,v}\boxtimes T_{T_{x}^{z,w}}^{u,v}\otimes T_{T_{y}^{z,w}}^{u,v},$

while the right hand side equals

$\displaystyle(\mathbbm{1}\boxtimes
c^{\alpha}_{2,2})\circ(c^{\alpha}_{2,2}\boxtimes\mathbbm{1})\circ(\mathbbm{1}\boxtimes
c^{\alpha}_{2,2})x\otimes y\boxtimes z\otimes w\boxtimes u\otimes v$
$\displaystyle=$
$\displaystyle\alpha(z,u,v)\alpha(w,u,v)\alpha(x,u,v)\alpha(y,u,v)\alpha(T_{x}^{uv},T_{z}^{uv},T_{w}^{uv})\alpha(T_{y}^{uv},T_{z}^{uv},T_{w}^{uv})$
$\displaystyle\cdot u\otimes v\boxtimes T_{z}^{u,v}\boxtimes
T_{w}^{u,v}\boxtimes T_{T_{x}^{u,v}}^{T_{z}^{u,v},T_{w}^{u,v}}\otimes
T_{T_{y}^{u,v}}^{T_{z}^{u,v},T_{w}^{u,v}}.$

The two terms are seen to coincide by applying the $2$-cocycle condition to
$x,z,w,u,v$ and $y,z,w,u,v$ separately, and the definition of self-
distributivity of $T$. The fact that the duals turn $\mathcal{R}_{\alpha}(X)$
into a rigid category is standard. It is left to prove that the twisting
morphisms are naturural with respect to the braiding, i.e. that

$(\theta^{\alpha}\boxtimes\theta^{\alpha})\circ
c^{\alpha}=c^{\alpha}\circ(\theta^{\alpha}\boxtimes\theta^{\alpha}),$

and that

$\theta^{\alpha}_{Y\boxtimes Z}=c^{\alpha}_{Z,Y}\circ
c^{\alpha}_{Y,Z}\circ(\theta^{\alpha}_{Y}\boxtimes\theta^{\alpha}_{Z}).$

The latter follows immediately from the definition of $\theta^{\alpha}$ as
extension of $\theta^{\alpha}_{2}$. To prove naturality, observe that it is
enough to show that

$(\theta^{\alpha}\boxtimes\mathbbm{1})\circ
c^{\alpha}_{2,2}=c^{\alpha}_{2,2}\circ(\mathbbm{1}\boxtimes\theta^{\alpha})$

and

$(\mathbbm{1}\boxtimes\theta^{\alpha})\circ
c^{\alpha}_{2,2}=c^{\alpha}_{2,2}\circ(\theta^{\alpha}\boxtimes\mathbbm{1}),$

since the general naturality with respect of the braiding is obtained
diagrammatically by sliding twists below and above a crossing, as in Figure 5.
The case of a twist being slid below a crossing is similarly depicted.

$\cong$ Figure 5. A twist can be slid over a crossing.

On simple tensors $x\otimes y\boxtimes z\otimes w$, the left hand side of the
first equality becomes

$\displaystyle(\theta^{\alpha}\boxtimes\mathbbm{1})\circ
c^{\alpha}_{2,2}x\otimes y\boxtimes z\otimes w$ $\displaystyle=$
$\displaystyle\alpha(x,x,y)\alpha(y,x,y)\alpha(T_{x}^{z,w},z,w)\alpha(T_{y}^{z,w},z,w)$
$\displaystyle\cdot z\otimes w\boxtimes T_{T_{x}^{z,w}}^{z,w}\otimes
T_{T_{y}^{z,w}}^{z,w},$

while the right hand side is

$\displaystyle
c^{\alpha}_{2,2}\circ(\mathbbm{1}\boxtimes\theta^{\alpha})x\otimes y\boxtimes
z\otimes w$ $\displaystyle=$
$\displaystyle\alpha(x,z,w)\alpha(y,z,w)\alpha(T_{x}^{z,w},T_{x}^{z,w},T_{y}^{z,w})\alpha(T_{y}^{z,w},T_{x}^{z,w},T_{y}^{z,w})$
$\displaystyle\cdot T_{z}^{z,w}\otimes T_{w}^{z,w}\boxtimes
T_{T_{x}^{z,w}}^{T_{x}^{z,w},T_{y}^{z,w}}\otimes
T_{T_{y}^{z,w}}^{T_{x}^{z,w},T_{y}^{z,w}}.$

These are seen to coincide upon applying the $2$-cocycle condition twice to
$x,x,y,z,w$ and $y,x,y,z,w$, and using self-distributivity of $T$. Similarly,
for the second equality to be verified one has on the left hand side

$\displaystyle(\mathbbm{1}\boxtimes\theta^{\alpha})\circ
c^{\alpha}_{2,2}x\otimes y\boxtimes z\otimes w$ $\displaystyle=$
$\displaystyle\alpha(z,z,w)\alpha(w,z,w)\alpha(x,T_{z}^{z,w},T_{w}^{z,w})\alpha(y,T_{z}^{z,w},T_{w}^{z,w})$
$\displaystyle\cdot T_{z}^{z,w}\otimes T_{w}^{z,w}\boxtimes
T_{x}^{T_{z}^{z,w},T_{w}^{z,w}}\boxtimes T_{x}^{T_{z}^{z,w},T_{w}^{z,w}}.$

To verify that they are the same, apply the $2$-cocycle condition to
$T_{x}^{z,w},z,w,z,w$ and observe that

$\displaystyle T(x,z,w)$ $\displaystyle=$ $\displaystyle
T(T(T(x,z,w),z,w),z,w)$ $\displaystyle=$ $\displaystyle
T(T(T(x,z,w),z,w)T(z,z,w),T(w,z,w))$ $\displaystyle=$ $\displaystyle
T(x,T(z,z,w),T(w,z,w)),$

and similarly $T(y,z,w)=T(y,T(z,z,w),T(w,z,w))$. This sequence of equalities
is motivated diagrammatically by sliding a ribbon beneath a self crossing.

Suppose now that $[\alpha]=[\beta]$, i.e. there exists $f:X\longrightarrow A$
such that $\alpha(x,y,z)=\beta(x,y,z)f(x)f(T(x,y,z))^{-1}$ for all $x,y,z$.
Set $\tilde{f}:X^{\otimes 2}\longrightarrow X^{\otimes 2}$ as
$\tilde{f}(x\otimes y):=f(x)f(y)\cdot x\otimes y$, and extended by linearity.
The map $\tilde{f}$ has an inverse given by $\tilde{f}^{-1}(x\otimes
y):=f(x)^{-1}f(y)^{-1}\cdot x\otimes y$. The definition of $\tilde{f}$ and its
inverse clearly extends to objects of $\mathcal{R}^{*}_{\alpha}(X)$ and
$\mathcal{R}^{*}_{\beta}(X)$. Define a functor
$F:\mathcal{R}^{*}_{\beta}(X)\longrightarrow\mathcal{R}^{*}_{\alpha}(X)$ as
follows. On objects $F$ is defined to be the identity. On morphisms
$\tau\in{\rm Hom}(X^{\otimes 2n},X^{\otimes 2n})$ define
$F(\tau)=(\tilde{f}^{-1})^{\otimes n}\circ\tau\circ(\tilde{f}^{\otimes n})$.
This assignment is functorial, and moreover maps $\theta^{\beta}_{2}$ to
$\theta^{\alpha}_{2}$ and $c^{\beta}_{2,2}$ to $c^{\alpha}_{2,2}$, since

$\displaystyle(\tilde{f}^{-1}\theta^{\beta}_{2}\tilde{f})(x\otimes y)$
$\displaystyle=$
$\displaystyle(f(x)f(y)\beta(x,x,y)\beta(y,x,y)f(T(x,x,y))^{-1}f(T(y,x,y))^{-1})$
$\displaystyle\cdot x\otimes y$ $\displaystyle=$
$\displaystyle\alpha(x,x,y)\alpha(y,x,y)\cdot x\otimes y$ $\displaystyle=$
$\displaystyle\theta^{\alpha}_{2}(x\otimes y)$
$\displaystyle(\tilde{f}^{-1}c^{\beta}_{2,2}\tilde{f})(x\otimes y\boxtimes
z\otimes w)$ $\displaystyle=$
$\displaystyle(f(x)f(y)\beta(x,z,w)\beta(y,z,w)f(T((x,z,w))^{-1}f(T((y,z,w))^{-1})$
$\displaystyle\cdot z\otimes w\boxtimes T((x,z,w)\otimes T((y,z,w)$
$\displaystyle=$ $\displaystyle\alpha(x,z,w)\alpha(y,z,w)\cdot z\otimes
w\boxtimes T((x,z,w)\otimes T((y,z,w)$ $\displaystyle=$
$\displaystyle\theta^{\alpha}_{2,2}(x\otimes y\boxtimes z\otimes w).$

The definition of $F$ clearly respects tensor products and due to the
inductive definition of twists and braiding in the categories
$\mathcal{R}^{*}_{\alpha}(X)$ and $\mathcal{R}^{*}_{\beta}(X)$, it follows
that $F$ is a braided tensor functor. Since $F$ is essentially surjective on
objects (it is the identity), and fully faithful on morphisms (due to
invertibility of $\tilde{f}$), $F$ is an equivalence of braided tensor
categories that respects the twisting structure. This completes the proof. ∎

######  Remark 4.4.

Observe that the twisting morphisms $\theta^{\alpha}$ are in general
nontrivial, even for a trivial $\alpha$.

The main purpose of the construction of Theorem 4.3, is to show that the
natural invariants associated to the ribbon category coincide with the cocycle
invariants defined in Section 3. It is in fact possible to bypass this
construction and prove that the twisting morphsism $\theta^{\alpha}$ and the
braiding morphism $c^{\alpha}_{2,2}$ induce a representation of the infinite
framed braid group (the inductive limit of the framed braid groups $FB_{n}$ of
[KS]) similarly to [Tur], using the linear map $\Phi_{b}$ of Section 5 (given
below). It is convenient, though, to define a ribbon category from a ternary
self-distributive structure and a ternary $2$-cocycle since this is suitable
for a generalization to multiple compatible self-distributive structures.

######  Definition 4.5.

A compatible system of ternary self-distributive structures is a finite family
$\\{(Q_{i},T_{i})\\}_{i\in I}$ of ternary self-distributive sets along with
actions

$T_{ij}:Q_{i}\times Q_{j}\times Q_{j}\longrightarrow Q_{i},$

of $Q_{j}\times Q_{j}$ on $Q_{i}$ for all $i,j=1,\ldots n$, satisfying the
compatibility condition

$T_{ik}\circ(T_{ij}\times\mathbbm{1}\times\mathbbm{1})=T_{ij}\circ(T_{ik}\times
T_{jk}\times T_{jk})\circ\shuffle\circ(\mathbbm{1}^{\times
3}\times\Delta_{3}\times\Delta_{3}),$

where $\Delta_{3}=(\Delta\times\mathbbm{1})\circ\Delta$ and $\shuffle$ is the
permutation map corresponding to ternary self-distributivity. Such a system is
denoted $\\{Q_{i},T_{ij}\\}$, where it is implicitly assumed that
$T_{ii}:=T_{i}$.

Linearizing such a system of compatible structues gives a multi-object
analogue of ternary self-distributive object in the category of vector spaces.
We will sometime call the linearized object a compatible system of self-
distributive structures, since no confusion will arise.

######  Remark 4.6.

There are infinitely many examples of the structure given in Definition 4.5
arising from mutually distributive structures as in [ESZ]. In fact let
$(X,T_{0},T_{1})$ be a mutually self-distributive set (with $X$ finite).
Linearizing $T_{0}$ and $T_{1}$ over a field $\mathbbm{k}$ and defining
$T_{10}=T_{0}=T_{00}$ and $T_{01}=T_{1}=T_{11}$ over the vector space, it is
seen by direct inspection that the structure $\\{\mathbbm{k}\langle
X\rangle,T_{ij}\\}_{i,j=0,1}$ is a compatible system of ternary self-
distributive structures.

In Section 6 it will be seen that compositions of $G$-families of quandles
provide other natural examples of these structures. Furthermore, in the
Appendix, more examples from augmented Hopf modules will be introduced. In
particular it will be seen that there are compatible systems with multiple
base spaces.

To the notion of compatible system of ternary self-distributive structures,
there corresponds the notion of compatible system of ternary $2$-cocycles as
follows.

######  Definition 4.7.

Let $\\{(Q_{i},T_{ij})\\}$ be a compatible system of ternary self-distributive
structures. A compatible system of $2$-cocycles with coefficients in $A$
(abelian group with multiplicative notation) is a family of maps
$\alpha_{ij}:X_{i}\times X_{j}\times X_{j}\longrightarrow A$ such that

$\displaystyle\alpha_{ij}(x,y,z)\cdot\alpha_{ik}(T_{ij}(x,y,z),u,v)$
$\displaystyle=$
$\displaystyle\alpha_{ik}(x,u,v)\cdot\alpha_{ij}(T_{ik}(x,u,v),T_{jk}(y,u,v),T_{jk}(z,u,v)),$

for all $x\in X_{i}$, $y,z\in X_{j}$ and $u,v\in X_{k}$ and all $i,j,k$. Such
a family of maps is denoted by the symbol $\\{\alpha_{ij}\\}$, where
parentheses can be omitted to shorten notation.

######  Definition 4.8.

A compatible system of $2$-cocycles $\\{\alpha_{ij}\\}$ is said to be trivial,
or cobounded, if there exists a family of maps $f_{i}:X_{i}\longrightarrow A$
such that

$\delta f_{i}(x\times y_{1}\times y_{2}):=f_{i}(x)f_{i}(T_{ij}(x\times
y_{1}\times y_{2}))^{-1}=\alpha_{ij}(x\times y_{1}\times y_{2})$

for all $i,j$ and all $x\in X_{i}$, $y_{1},y_{2}\in X_{j}$.

Two systems of $2$-cocycles, $\\{\alpha_{ij}\\}$ and $\\{\beta_{ij}\\}$ are
said to be equivalent if the system $\\{\alpha_{ij}\beta^{-1}_{i,j}\\}$ is
trivial.

######  Remark 4.9.

Observe that when $i=j=k$ it follows that each $\alpha_{ii}$ is a ternary
$2$-cocycle for the ternary self-distributive structure $T_{i}$, and moreover
the triviality condition gives that $\alpha_{ii}$ represents the trivial class
in second ternary self-distributive cohomology group. It follows that if
$\alpha_{ii}$ is not cobounded in the ternary self-distributive cochain, then
a compatible system of $2$-cocycles is nontrivial in the sense of Definition
4.8.

######  Remark 4.10.

Definitions 4.7 and 4.8 are clearly reminiscent of a cohomology theory. It is
natural to ask whether such a theory derives from a deformation theory of
compatible systems. The answer is no, in that infinitesimal deformations of
compatible systems require more conditions to be satisfied, than just the
$2$-cocycle condition of Definition 4.7.

######  Remark 4.11.

In the case of Remark 4.6, when a system of compatible structures is defined
on the same base space, a compatible system of $2$-cocycles is the same as a
$2$-cocycle in the labeled cohomology of [ESZ].

Let $\\{(Q_{i},T_{ij})\\}$ be a compatible system of ternary self-distributive
structures and let $\\{\alpha_{ij}\\}$ be a compatible system of $2$-cocycles.
Let $X_{i}=\mathbbm{k}\langle Q_{i}\rangle$ and denote the linearized maps
$T_{ij}$ by the same symbols. To these data it is possible to associate
twisting morphisms and braidings as follows. For each $X_{i}$ there is a
twisting $\theta^{\alpha_{ij}}:X_{i}\otimes X_{i}\longrightarrow X_{i}\otimes
X_{i}$ defined by extending the assignment

$x\otimes y\mapsto\alpha_{ii}(x,x,y)\alpha_{ii}(y,x,y)\cdot
T_{i}(x,x,y)\otimes T_{i}(y,x,y).$

For each pair of objects $X_{i}$ and $X_{j}$ there is a braiding
$c^{\alpha_{ij}}:X_{i}^{\otimes 2}\otimes X_{j}^{\otimes 2}\longrightarrow
X_{j}^{\otimes 2}\otimes X_{i}^{\otimes 2}$ induced by

$(x\otimes y)\otimes(z\otimes
w)\mapsto\alpha_{ij}(x,z,w)\alpha_{ij}(y,z,w)\cdot(z\otimes
w)\otimes(T_{ij}(x,z,w)\otimes T_{ij}(y,z,w)).$

The construction of Definition 4.1 is adapted to this case and the
corresponding category is denoted by
$\mathcal{R}^{*}_{\\{\alpha_{ij}\\}}(\\{X_{i}\\})$.

######  Theorem 4.12.

The category $\mathcal{R}^{*}_{\\{\alpha_{ij}\\}}(\\{X_{i}\\})$ with braiding
and twisting morpshisms defined above is a ribbon category. Moreover, if two
systems $\\{\alpha_{ij}\\}$ and $\\{\beta_{i,j}\\}$ are equivalent, then
$\mathcal{R}^{*}_{\\{\alpha_{ij}\\}}(\\{X_{i}\\})$ and
$\mathcal{R}^{*}_{\\{\beta_{ij}\\}}(\\{X_{i}\\})$ are equivalent.

###### Proof.

The proof is substantially the same as that of Theorem 4.3, where the
$2$-cocycle condition of $\alpha$ is replaced by compatibility condition of
the family $\\{\alpha_{ij}\\}$ and self-distributivity of $T$ is replaced by
compatibility of the system $\\{T_{ij}\\}$. When $\\{\alpha_{ij}\\}$ and
$\\{\beta_{ij}\\}$ are equivalent, one can construct maps $\tilde{f}_{i}$ as
in Theorem 4.3 and show that these induce an equivalence of ribbon categories.
∎

Observe that $\mathcal{R}^{*}_{\\{\alpha_{ij}\\}}(\\{X_{i}\\})$ is a ribbon
category with “distinguished” objects $X_{i}\otimes X_{i}$, and other objects
given by tensor products obtained from the distinguished ones.

##  5\. The Ribbon cocycle invariant is a quantum invariant

The ribbon category $\mathcal{R}^{*}_{\alpha}(X)$ allows to define an
invariant of framed links from any ternary self-distributive operation and a
fixed ternary $2$-cocycle, following standard procedures [Tur]. Let $(X,T)$ be
a ternary self-distributive object arising from a set-theoretic ternary
quandle $Q$ as above, and $[\alpha]\in H^{2}(Q,A)$ be fixed. A framed link is
represented by the closure of an element $b\in FB_{n}$ of the framed braid
group on $n$ ribbons [KS] where, since twisting of the ribbon and crossings
commute, it is assumed that the twists are on top of the braid. Using the same
notation of [KS],

$b=t_{1}^{r_{1}}\cdots t_{n}^{r_{n}}\cdot\tau,$

where $r_{i}$ are integers indicating the number of twists of the $i^{\rm th}$
ribbon and $\tau$ is an element of the braid group $B_{n}$.

Then, the quantum invariant associated to $(X,T)$ and $\alpha$, is obtained by
considering the object $X^{2}\boxtimes\cdots\boxtimes X^{2}$ ($n$-fold
product) in $\mathcal{R}^{*}_{\alpha}(X)$ and taking the trace of the morphism
$\Phi_{b}:X^{2}\boxtimes\cdots\boxtimes X^{2}\longrightarrow
X^{2}\boxtimes\cdots\boxtimes X^{2}$ corresponding to $b$ as follows. Each
generator $t_{i}^{r_{i}}$ corresponds to $(\theta^{\alpha})^{r_{i}}$, and each
crossing $\sigma_{i}^{\pm 1}$ in the product defining $\tau$ corresponds to
$(c^{\alpha}_{2,2})^{\pm 1}$. This invariant is denoted by the symbol

$\Psi_{\mathcal{D}}(X,T,\alpha),$

where $\mathcal{D}$ is the ribbon diagram representing $b$.

The following theorem establishes that two procedures described in Section 3
and this section coincide.

######  Theorem 5.1.

Let $L$ be a framed link, with presentation given by a framed braid $b$ as
above, $(X,T)$ be a ternary self-distributive structure that is linearized
over $\mathbbm{k}$, and $\alpha$ a $2$-cocycle of $(X,T)$ with coefficitents
in $A$. Suppose that $\chi:A\longrightarrow\mathbbm{k}^{\times}$ is a group
character. Fix a diagram $\mathcal{D}$ of $L$. Then the ribbon cocycle
invariant $\chi\Theta_{\mathcal{D}}(X,T,\alpha)$ and the quantum invariant
$\Psi_{\mathcal{D}}(X,T,\chi\circ\alpha)$ coincide.

###### Proof.

The proof is very similar to that of Theorem 3.5 in [Gra]. Observe that in
order to compute $\Psi_{\mathcal{D}}(X,T,\alpha)$, one has to consider all
combinations of basis vectors $x_{i_{1}}^{\epsilon(i_{1})}\otimes\cdots\otimes
x_{i_{n}}^{\epsilon(i_{n})}$ apply the endomorphism $\Phi_{b}$ and therefore
apply it to all the possible combinations
$x_{j_{1}}^{\epsilon(j_{1})}\otimes\cdots\otimes x_{j_{n}}^{\epsilon(j_{n})}$.
It follows that the only nontrivial contributions to
$\Psi_{\mathcal{D}}(X,T,\alpha)$ are obtained when
$x_{j_{1}}^{\epsilon(j_{1})}\otimes\cdots\otimes
x_{j_{n}}^{\epsilon(j_{n})}=x_{i_{1}}^{-\epsilon(i_{1})}\otimes\cdots\otimes
x_{i_{n}}^{-\epsilon(i_{n})}$, by definition of $ev$ and $coev$ maps in
$\mathcal{R}^{*}_{\alpha}(X)$. At each crossing, whether corresponding to a
twisting $t^{r_{i}}$ or one of the factors of the braid $\tau$, $\Phi_{b}$
contributes with a scalar given by either

$\alpha(z_{1},z_{1},z_{2})\alpha(z_{2},z_{1},z_{2}),$

or

$\alpha(z_{1},w_{1},w_{2})\alpha(w_{2},w_{1},w_{2}),$

where it has been assumed that $\theta^{\alpha}$ is applied to the vector
$z_{1}\otimes z_{2}$ and $c^{\alpha}$ is applied to the vector $z_{1}\otimes
z_{2}\boxtimes w_{1}\otimes w_{2}$. The vector output is given by either

$T(z_{1},z_{1},z_{2})\otimes T(z_{2},z_{1},z_{2}),$

or

$w_{1}\otimes w_{2}\boxtimes T(z_{1},w_{1},w_{2})\otimes
T(w_{2},w_{1},w_{2}).$

It therefore follows that the only nontrivial contribution to
$\Psi_{\mathcal{D}}(X,T,\alpha)$ corresponds to the colorings of $\mathcal{D}$
by $X$ and each contribution equals one of the summands that define
$\Theta_{\mathcal{D}}(X,T,\alpha)$. The proof is complete. ∎

The quantum invariant $\Psi_{\mathcal{D}}(X,T,\alpha)$ does not only provide a
differennt interpretation of the ribbon cocycle invariant
$\Theta_{\mathcal{D}}(X,T,\alpha)$, but it is also suitable for a
generalization to framed braids (not necessarily closed).

######  Theorem 5.2.

Let $F$ be a diagram of a framed braid $b\in FB_{n}$. Then $\Phi_{b}$ defined
as above is an invariant of $F$.

##  6\. Examples and Computations

###  6.1. Examples from compositions of binary quandles

In this subsection it is showed that the notions of $G$-family of quandles and
$G$-family of $2$-cocycles [IIJO, Nos] provide interesting examples of
compatible systems of TSD structures. An explicit example using Nosaka
cocycles is also shown in detail.

Recall first, that a $G$-family of quandles is a set $X$ along with a family
of quandle operations $*^{g}$ indexed by a group $G$ (i.e. $g\in G$ for all
$g$), and satisfying the compatibility conditions

$\displaystyle(x*^{g}y)*^{h}y$ $\displaystyle=$ $\displaystyle x*^{gh}y,$
$\displaystyle(x*^{g}y)*^{h}z$ $\displaystyle=$
$\displaystyle(x*^{h}z)*^{h^{-1}gh}(y*^{h}z).$

Given a $G$-family of quandles one can construct a compatible system of
ternary self-distributive operations as follows. Linearize the base set $X$ to
obtain $V=\mathbbm{k}\langle X\rangle$. Then define maps $T_{gh}:V\otimes
V\otimes V\longrightarrow V$ by the assignment $x\otimes y\otimes
z\mapsto(x*^{h}y)*^{h^{-1}}z$. A direct computation shows that the system so
defined is indeed compatible. Observe that $T_{gg}$ being self-distributive is
an instance of the fact that composing mutually distributive binary operations
produces a ternary self-distributive operation, as in [ESZ].

######  Definition 6.1.

The compatible system constructed above from the $G$-family of quandles
$(X,*^{g})_{g\in G}$ is called the compatible system associated to a
$G$-family.

Since the main construction of Section 4 associate a ribbon category to a
compatible system of distributive structures by means of a system of
$2$-cocycles, it is fundamental to obtain such objects for compatible system
associated to $G$-families. As shown in the next result, it is possible to do
so using the notion of $G$-family $2$-cocycles, see [IIJO, Nos] for the
definition of $G$-family (co)homology. In what follows it is assumed that the
$X$-set $Y$ appearing in $G$-family cohomology is a singleton endowed with the
trivial $G$-family action. The set $Y$ is therefore omitted without further
notice, but this should not cause any difficulties.

######  Proposition 6.2.

Let $(X,*^{g})_{g\in G}$ and $(V,T_{gh})_{g,h\in G}$ be a $G$-family of
quandles and the associated compatible system of self-distributive structures,
respectively. Let $\theta$ be a $G$-family $2$-cocycle. Then there is an
associated compatible system of $2$-cocycles $\theta_{gh}$ defined as follows

$\theta_{gh}(x\times y\times
z):=\theta((x,e)\times(y,g))+\theta((x*^{g}y,e)\times(z,h)).$

######  Remark 6.3.

A couple of observations are due. Firstly, notice that since $X$ acts
trivially on $Y$, the two terms corresponding to deleting the first entry of
$\theta$, according to the definition of $G$-family $2$-cocycle cancel each
other, so that it is reasonable to arbitrarily choose on element of $G$ to
label all the first entries $x$, where the obvious choice falls upon the
neutral element $e$ of $G$. Secondly, there is a parallel between labels
assignment in the definition of chain maps in “labeled cohomology” of Theorem
5.3 in [ESZ], particularly clear from Remark 5.7 in the same article, and the
group element enriching $y$ or $z$ to a pair $(y,g)$ or $(z,h)$, respectively.

###### Proof of 6.2.

Using Remark 6.3, the proof is almost immediate. In fact, the condition that
$\theta_{gh}$ has to satisfy is (in additive notation)

$\displaystyle\theta_{fg}(x\times y\times z)+\theta_{fh}(T_{ij}(x\times
y\times z)\times u\times v)$ $\displaystyle=$
$\displaystyle\theta_{fh}(x\times u\times v)+\theta_{fg}(T_{fh}(x\times
u\times v),T_{gh}(y\times u\times v)\times T_{gh}(z\times u\times v)).$

Using the definition of $\theta_{gh}$, the $G$-family $2$-cocycle condition
becomes equivalent to labeled cohomology $2$-cocycle condition, since terms
obtained by deleting $x$ cancel. Now the same proof as in Remark 5.7 of [ESZ]
can be applied, mutatis mutandis, to complete. ∎

Let now $G={\rm SL}(2;{\mathbb{Z}}_{3})$ and
$X={\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}$, with operations $\\{*^{g}\\}_{g\in
G}$ defined as follows [IIJO], $x*^{g}y:=gx+(\mathbbm{1}-g)y$ for all $x,y\in
X$ and $g\in G$, where $G$ acts on $X$ by matrix multiplication on column
vectors, and $\mathbbm{1}$ is the identity of $G$. A direct computation shows
that these operations define a $G$-family structure on $(X,G)$ (Proposition
2.3 in [IIJO]). From this data, Nosaka has constructed [Nos] a $G$-family
$2$-cocycle, that has been employed in [IIJO] to compute cocycle invariants of
certain handlebody knots and distinguish them from their mirror images. As
pointed out above, it is not restrictive to omit the singleton set
$Y:=\\{y\\}$ in the original construction. Define $\alpha:(X\times G)^{\times
2}\longrightarrow{\mathbb{Z}}_{3}$ by

$(x,g)\times(y,h)\mapsto\lambda(g){\rm det}(x-y,(1-h)^{-1}y),$

where $\lambda$ is the abelianization function, defined by
$\lambda(A)=(a+d)(b-c)(1-bc)$, for a matrix
$A:=\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)$. Then $\alpha$ is a
$G$-family $2$-cocycle and it follows that $\\{\alpha_{gh}\\}$ defined as in
Proposition 6.2 is a compatible system of $2$-cocycles.

######  Example 6.4.

Applying Theorem 4.12 to the compatible system of $2$-cocycles
$\\{\theta_{gh}\\}_{g,h\in G}$ associated to Nosaka’s $G$-family $2$-cocycle
$\alpha$ via Proposition 6.2 one obtains a ribbon category. The braiding at
level $2$ is given explicitly by the maps

$\displaystyle c^{\alpha_{gh}}_{2,2}(x\otimes y\boxtimes z\otimes w)$
$\displaystyle=$ $\displaystyle\alpha((x,e),(z,g)\alpha((x*^{g}z,e),(w,h))$
$\displaystyle\hskip 28.45274pt\alpha((y,e),(z,g)\alpha((x*^{g}z,e),(w,h))$
$\displaystyle\cdot z\otimes
w\boxtimes(x+(h^{-1}-\mathbbm{1})z+(\mathbbm{1}-h)w)\otimes$
$\displaystyle\hskip 28.45274pt(y+(h^{-1}-\mathbbm{1})z+(\mathbbm{1}-h)w),$

while twisting morphisms are given by

$\displaystyle\theta^{\alpha_{gh}}(x\otimes y)$ $\displaystyle=$
$\displaystyle\alpha((x,e),(x,g)\alpha((x,e),(y,h))$ $\displaystyle\hskip
28.45274pt\alpha((y,e),(x,g)\alpha((y*^{g}x,e),(w,h))$
$\displaystyle\cdot(h^{-1}x+(\mathbbm{1}-h^{-1})y)\otimes((h^{-1}-\mathbbm{1})x-h^{-1}y).$

###  6.2. Examples from heap structures

Recall that given a group $G$, the heap operation $G\times G\times
G\longrightarrow G$ defined by $x\times y\times z\mapsto xy^{-1}z$, defines a
ternary quandle structure on $G$. Linearizing this assignment over a field
$\mathbbm{k}$ produces a ternary quandle object in the category of vector
spaces, where the diagonal map is induced by $x\mapsto x\otimes x\otimes x$.
This definition in fact coincides with the quantum heap of the Hopf algebra
structure on the group ring $\mathbbm{k}[G]$, as it can be seen directly.

######  Example 6.5.

Let ${\mathbb{Z}}_{2}$ be the cyclic group of order $2$ and let
$\mathbb{C}[\mathbb{Z}_{2}]$ be the structure defined above, obtained by
linearizing the heap operation of ${\mathbb{Z}}_{2}$. The elements of
$\mathbb{Z}_{2}$ are identified with the symbols $e_{x}$, with
$x\in\mathbb{Z}_{2}$, generating the two dimensional vector space. A direct
computation shows that
$H^{2}({\mathbb{Z}}_{2},{\mathbb{Z}}_{2})={\mathbb{Z}}_{2}\oplus{\mathbb{Z}}_{2}$
with generators corresponding to the equivalence classes of characteristic
functions $\chi_{(0,0,0)}$ and $\chi_{(0,1,1)}$. Fix the cocycle $(0,1)\in
H^{2}({\mathbb{Z}}_{2},{\mathbb{Z}}_{2})$, i.e. the map $\phi(x,y,z)=1$ if
$(x,y,z)=(0,1,1)$ and $\phi(x,y,z)=0$ otherwise. Identifying
${\mathbb{Z}}_{2}$ with $(-1)^{i}\in{\mathbb{C}}^{\times}$ one gets a
nontrivial cohomology class in
$H^{2}({\mathbb{Z}}_{2},{\mathbb{C}}^{\times})$, still denoted by $\phi$. The
corresponding twisting morphisms and braiding morphism are as follows

$\displaystyle\theta^{\phi}(e_{x}\otimes e_{y})$ $\displaystyle=$
$\displaystyle(-1)^{\phi(x,x,y)+\phi(y,x,y)}\cdot e_{x+x+y}\otimes e_{y+x+y}$
$\displaystyle=$ $\displaystyle e_{y}\otimes e_{x},$

that is, the twisting morphism is given by transposition, and

$\displaystyle c^{\phi}(e_{x}\otimes e_{y}\boxtimes e_{z}\otimes e_{w})$
$\displaystyle=$ $\displaystyle(-1)^{\phi(x,z,w)+\phi(y,z,w)}\cdot
e_{z}\otimes e_{w}\boxtimes e_{x+z+w}\otimes e_{y+z+w}$ $\displaystyle=$
$\displaystyle\begin{cases}e_{z}\otimes e_{w}\boxtimes e_{x}\otimes e_{y}\ \
z=w\\\ e_{z}\otimes e_{w}\boxtimes e_{x+1}\otimes e_{y+1}\ \ {\rm
otherwise}\end{cases}$

######  Example 6.6.

Let $X$ be the abelian group ${\mathbb{Z}}_{n}$ with group heap structure, and
set $A={\mathbb{Z}}$ taken with multiplicative notation, with generator $g$.
Suppose $\rho$ is a given group character mapping $A$ in the group of units of
$\mathbbm{k}$. In [SZ], Lemma 3.7, it is shown that the $2$-cochain
$\phi_{i}:X^{\times 3}\longrightarrow{\mathbb{Z}}$ defined by the formula

$\phi_{i}:=\sum_{x\in{\mathbb{Z}}_{n}}[\sum_{j=0}^{n-1}\chi_{(x,j,j+i)}],$

where $\chi_{(x,y,z)}$ is the characteristic function at the triple
$(x,y,z)\in X^{\times 3}$ is a nontrivial $2$-cocycle for any choice of
$i=1,\ldots,n-1$. It is in fact thereby proved that $[\phi_{i}]\neq[\phi_{k}]$
in the second cohomology group, whenever $i\neq k$ in ${\mathbb{Z}}_{n}$. The
ribbon category corresponding to $\phi$, for some choice of $n\in{\mathbb{N}}$
and $0\neq i\in{\mathbb{Z}}_{n}$ is determined by braiding and twisting
morphisms obtained as follows. For all
$e_{x},e_{y},e_{z},e_{w}\in\mathbbm{k}[X]$, the linearization of $X$ over
$\mathbbm{k}$ coinciding with the group algebra of ${\mathbb{Z}}_{n}$,
$c_{2,2}^{\phi_{i}}$ maps simple tensors according to the assignment

$\displaystyle c_{2,2}^{\phi_{i}}(x\otimes y\boxtimes z\otimes w)$
$\displaystyle=$
$\displaystyle\rho(g^{\phi_{i}(x,z,w)})\rho(g^{\phi_{i}(y,z,w)})e_{z}\otimes
e_{w}\boxtimes e_{x-z+w}\otimes e_{y-z+w}$ $\displaystyle=$
$\displaystyle\rho(g^{2\phi(x,z,w)})e_{z}\otimes e_{w}\boxtimes
e_{x-z+w}\otimes e_{y-z+w}$ $\displaystyle=$
$\displaystyle\begin{cases}\rho(g^{2})e_{z}\otimes e_{w}\boxtimes
e_{x+i}\otimes e_{y+i}\ \ {\rm if}\ w-z=i\\\ e_{z}\otimes e_{w}\boxtimes
e_{x+k}\otimes e_{y+k}\ \ {\rm if}\ w-z=k\neq i\end{cases}$

The twisting morphism $\theta_{2}^{\phi_{i}}$ maps simple tensors as

$\displaystyle\theta_{2}^{\phi_{i}}(e_{x}\otimes e_{y})$ $\displaystyle=$
$\displaystyle\rho(g^{\phi_{i}(x,x,y)})\rho(g^{\phi_{i}(y,x,y)})e_{y}\otimes
e_{2y-x}$ $\displaystyle=$
$\displaystyle\rho(g^{2\phi_{i}(x,x,y)})e_{y}\otimes e_{2y-x}$
$\displaystyle=$ $\displaystyle\begin{cases}\rho(g^{2})e_{y}\otimes e_{y+i}\ \
{\rm if}\ y-x=i\\\ e_{y}\otimes e_{y+k}\ \ {\rm if}\ y-x=k\neq i\end{cases}$

We note that the twisting morphism $\theta_{2}^{\phi_{i}}$ is determined, up
to scalar multiplication, by the Takasaki quandle operation $x*y=2y-x$
associated with the abelian group ${\mathbb{Z}}_{n}$. This is in fact a
general feature of the twisting morphism of an abelian heap. It is also easy
to see that for non-abelian heaps one obtains the core of a quandle, instead
of the Takasaki structure. Generalization of the preceding braiding and
twisting structure to the case of linear combinations of $\phi_{i}$’s is
easily obtained from the previous equations.

######  Example 6.7.

Let $X=D_{3}$ be the dihedral group on $6$ elements, with presentation
$\langle s,r\ |\ s^{2}=r^{3}=1,\ srs=r^{-1}\rangle$. Once again consider the
group heap strucutre on $X$ and linearize it to obtain a quantum heap on the
group ring $\mathbb{C}[X]$. Denote characteristic functions $\chi_{(x,y,z)}$,
those cochains with coefficients in ${\mathbb{Z}}_{3}$ defined by
$\chi_{(x,y,z)}(x^{\prime},y^{\prime},z^{\prime})=1$ if
$(x^{\prime},y^{\prime},z^{\prime})=(x,y,z)$, and $0$ otherwise. Define the
$2$-cochain

$\psi=\sum_{x}[\chi_{(x,1,r)}+\chi_{(x,r,r^{2})}+\chi_{(x,r^{2},1)}]+\sum_{x}[\chi_{(x,s,sr)}+\chi_{(x,sr,sr^{2})}+\chi_{(x,sr^{2},s)}].$

A direct computation shows that $\psi$ satisfies the $2$-cocycle condition and
it is therefore a $2$-cocycle. moreover $\psi$ is nontrivial [SZ], Example
5.13 and Proposition 5.14, so that $[\psi]\neq 0$. Mapping ${\mathbb{Z}}_{3}$
to the $3^{\rm rd}$ roots of unity $G_{3}$ we obtain $\psi\in
Z^{2}({\mathbb{Z}}_{3},G_{3})$, where $G_{3}$ acts on
${\mathbb{C}}[{\mathbb{Z}}_{3}]$, and therefore on
${\mathbb{C}}[{\mathbb{Z}}_{3}]\otimes{\mathbb{C}}[{\mathbb{Z}}_{3}]$, by
scalar multiplication. Twisting morphisms are obtained as

$\displaystyle\theta^{\psi}(x\otimes y)$ $\displaystyle=$ $\displaystyle
e^{\frac{2\pi i}{3}(\psi(x,x,y)+\psi(y,x,y))}\cdot y\otimes yx^{-1}y,$

where the scalar $e^{\frac{2\pi i}{3}(\psi(x,x,y)+\psi(y,x,y))}$ is nontrivial
if and only if $x^{-1}y=r$, in which case we obtain $e^{\frac{2\pi
i}{3}(\psi(x,x,y)+\psi(y,x,y))}=e^{\frac{4\pi i}{3}}$. Observe that $yx^{-1}y$
is the core quandle operation. In fact, $\theta^{\psi}$ corresponds (up to the
multiplying scalar) to the $R$ matrix obtained by linearizing the set-
theoretic solution of Yang-Baxter equation corresponding to the core of group
$D_{3}$. The braiding $c_{2,2}^{\psi}$ is given explicitly by

$\displaystyle c^{\psi}_{2,2}(x\otimes y\boxtimes z\otimes w)$
$\displaystyle=$ $\displaystyle e^{\frac{2\pi i}{3}(\psi(x,z,w)+\psi(y,z,w))}$
$\displaystyle\hskip 28.45274pt\cdot z\otimes w\boxtimes xz^{-1}w\otimes
yz^{-1}w,$

where the scalar multiple is nontrivial if and only if $z^{-1}w=r$, in which
case we have $e^{\frac{2\pi i}{3}(\psi(x,z,w)+\psi(y,z,w))}=e^{\frac{4\pi
i}{3}}$, similarly to the case of twisting.

###  6.3. Computations of quantum invariants

In this subsection we illustrate the theory by computing some invariants of a
few basic framed links corresponding to linearized ternary self-distributive
structures and $2$-cocycles given above. We compare with the invariants
computed in [SZ]. In fact it will be apparent how the two types of invariants
encode the same information, as proved in Theorem 5.1.

######  Example 6.8.

Let us consider the unknot with frame $n\in{\mathbb{N}}$. Let
$H={\mathbb{Z}}_{m}$ for some arbitrary $m$, considered with abelian heap
structure, i.e. $T(x,y,z)=x-y+z$. As already argued in Example 6.6, for each
$i=1,\ldots,m-1$ there exists a nontrivial cocycle with coefficients in
${\mathbb{Z}}$ given by
$\phi_{i}:=\sum_{x\in{\mathbb{Z}}_{m}}[\sum_{j=0}^{m-1}\chi_{(x,j,j+i)}]$. Let
us fix an arbitrary $i$ and let $\zeta\in{\mathbb{C}}$ denote a primitive
$m^{\rm th}$-root of unity. We assume ${\mathbb{Z}}$ to be generated (in
multiplicative notation) by $g$, and define the character
$\tilde{\chi}:{\mathbb{Z}}\longrightarrow{\mathbb{C}}^{\times}$ by reducing
the exponents of $g$ modulo $m$ and identifying it with the corresponding
power of $\zeta$. The map that is associated to the $n$-framed unknot is
$\theta^{n}$, where we set $\theta:=\theta^{\phi_{i}}$ for simplicity. We
denote the basis vectors of the vector space $X=\mathbbm{k}\langle H\rangle$
generated by $H$ by the symbols $e_{x}$, for $x\in H$. Then $\theta^{n}$ is
easily seen to be given by the map $e_{x}\otimes e_{y}\mapsto q(x,y)\cdot
e_{ny-(n-1)x}\otimes e_{(n+1)y-nx}$, where the unit
$q(x,y)\in{\mathbb{C}}^{\times}$ is determined below. For a vector to
contribute to the trace of $\theta^{n}$ we need $e_{x}=e_{ny-(n-1)x}$ and
$e_{y}=e_{(n+1)y-nx}$, which gives $n(y-x)=0$. If $(n,m)=1$ then there are
$|X|$ vectors that satisfy this condition. We have that
$tr_{q}(\theta^{n})=\mid X\mid$, since $i\neq 0$. Observe that the condition
of $e_{x}\otimes e_{y}$ contributing to the trace of $\theta^{n}$ can be
rephrased as $x,y$ giving a coloring of the diagram of the $n$-framed unknot.
So, when choosing $m$ coprime with $n$ the invariant is simply counting
colorings. If $(n,m)=d\neq 1$, then we have $d$ elements $\alpha$ divisible by
$m/d$. Each of these solutions gives $|H|=m$ vectors that contribute
nontrivially to the trace of $\theta^{n}$. Moreover, whenever $i=\alpha$ for
one of the previous solutions, we have a contribution of $\zeta^{2n}$. We have

$tr_{q}(\theta^{n})=\begin{cases}dm\ \ \ (i,m/d)\neq m/d\\\
(d-1)m+\zeta^{2n}m\ \ \ (i,m/d)=m/d\end{cases}$

The cocycle invariant $\Psi$, by direct computation, is seen to be equal to
$me\otimes e$ when $m$ and $n$ are coprime, and

$\Psi=\begin{cases}md\cdot e\otimes e\ \ \ (i,m/d)\neq m/d\\\ m(d-1)\cdot
e\otimes e+mg^{n}\otimes g^{n}\ \ \ (i,m/d)=m/d\end{cases}$

when $(m,n)=d$. We see that applying $\chi$ to $\Psi$ we obtain
$tr_{q}(\theta^{n})$, as required. Moreover, we can choose $m$ and $i$ such
that $tr_{q}(\theta^{n})=dm+\zeta^{2n}m$ and $\zeta^{2n}\neq 1$, so the
invariant detects twisting.

######  Example 6.9.

Let us now consider the torus link $T(2,2n)$ on two strings, with even number
of crossings. We compute the quantum invariant corresponding to the cocycle
$\phi_{i}\in Z^{2}({\mathbb{Z}}_{m},{\mathbb{Z}})$, for some $i=1,\ldots,n-1$.
Set $X={\mathbb{C}}\langle{\mathbb{Z}}_{m}\rangle$. The framed braid whose
closure gives $T(2,2n)$ corresponds to the endomorphism $c^{2n}:X^{\otimes
2}\boxtimes X^{\otimes 2}\longrightarrow X^{\otimes 2}\boxtimes X^{\otimes
2}$, obtained by composing the braiding of two ribbons $2n$ times. We use the
symbol $\boxtimes$ to distinguish pairs corresponding to the two edges of a
ribbon, following previous conventions. We choose, as before, an integer $m$
and a primitive $m^{\rm th}$-root of unitiy $\zeta$ and we use again the map
$\tilde{\chi}$ that sends a generator, say $g$, of ${\mathbb{Z}}$ in
multiplicative notation to $\zeta$. On basis vectors we have, by direct
computation, $c^{2n}(e_{x}\otimes e_{y}\boxtimes e_{z}\otimes
e_{w})=q(x,y,z,w)\cdot e_{x+n(w-z)}\otimes e_{y+n(w-z)}\boxtimes
e_{z+n(y-x)}\otimes e_{w+n(y-x)}=q(x,y,z,w)\cdot e_{x}\otimes e_{y}\boxtimes
e_{z}\otimes e_{w}$, where $q(x,y,z,w)\in{\mathbb{C}}^{\times}$ is determined
as follows. We shorten notation by omitting the variables, therefore writing
just $q$. When $y-x=w-z=i$ we have $q=\zeta^{4n}$, when just one of $w-z$ or
$y-x$ equals $i$, $q=\zeta^{2n}$, while $q=1$ whenever both $w-z$ and $y-x$
are different from $i$. Then $c^{2n}$ is diagonal and each vector contributes
to the trace so, to complete the computation, one needs just to count how many
vectors contribute with either of the weights $q$ given above. We have
$tr_{q}(c^{2n})=n^{2}\zeta^{4n}+2n(n-1)\zeta^{2n}+n^{4}+n$. Observe that the
computation parallels perfectly that of Example 5.12 in [SZ], where the fact
that each initial arc coloring defines a full coloring of $T(2,2n)$ translates
into the statement that $c^{2n}$ is diagonal, and where in the cocycle
invariant the weights contribute to the entries of the tensor product of the
group algebra ${\mathbb{Z}}_{n}[{\mathbb{Z}}]$ depending on the components of
the link.

##  7\. Generalized construction

In this section, we develop a generalized version of the theory described
above, where we consider symmetric monoidal categories. To this objective, we
first need to provide a framework for ternary self-distributivity in symmetric
monoidal categories and introduce a categorical version of $2$-cocycle
condition. The first construction has been introduced by the author, along
with M. Elhamdadi and M. Saito, in [ESZ], while the second construction will
be introduced herein.

Our main interest in generalizing the construction from the linearized case
described above to more general objects in symmetric monoidal categories lies
in the following discussion. When computing the quantum invariant associated
to a TSD set we see that a vector contributes to the invariant, i.e. to the
(quantum) trace of the linear map $X^{\otimes 2n}\longrightarrow X^{\otimes
2n}$ associated to a framed braid diagram with n doubled-strings, if the
coloring condition on the edges of the diagram is satisfied. This is due to
the fact that the comultiplication of $X$ is simply given by producing two
copies of an element of $X$. For a general TSD object, say in the category of
vector spaces, the comultiplication is usually not the same as $x\mapsto
x\otimes x$, so that there can be contributions to the quantum trace that do
not correspond to colorings of the framed diagram by $X$. We expect that this
phenomenon produces stronger invariants than the ribbon cocycle invariant. In
fact, while a trivial cocycle does not produce new non-trivial invariants in
the linearized case (it simply counts the colorings of a diagram, which is a
known invariant), with a general comultiplication we can obtain nontrivial
invariants corresponding to the YB operator associated to the TSD object even
when this is not deformed by a nontrivial cocycle.

Let $\mathcal{C}$ be a symmetric monoidal category, with tensor functor
denoted by $\otimes$ and let $X$ be a fixed object in $\mathcal{C}$. Then,
associating the switching morphism $\tau:X^{\otimes 2}\longrightarrow
X^{\otimes 2}$ to the transposition $(12)\in\mathbb{S}_{2}$ we obtain a
“representation” of the infinite symmetric group $\mathbb{S}_{\infty}$ as
follows. Let $\sigma\in\mathbb{S}_{n}$ for some $n\in\mathbb{N}$, we decompose
the permutation in a product of transpositions
$\sigma=\sigma_{1}\cdots\sigma_{k}$ for some $k$. Then we define the
corresponding automorphism of $X^{\otimes n}$ to be
$\tau_{1}\circ\cdots\circ\tau_{k}$, where $\tau_{i}:X^{\otimes
n}\longrightarrow X^{\otimes n}$ is the automorphism given by
$\mathbbm{1}^{i-1}\otimes\tau_{X,X}\otimes\mathbbm{1}^{n-i+1}$. Then it is
verified that this assignment does not depend on the choice of decomposition
of a permutation into transpositions, and it is therefore well defined. We
obtain a correspondence
$\shuffle:\mathbb{S}_{\infty}\longrightarrow\cup_{n}{\rm Hom}(X^{\otimes
n},X^{\otimes n})$, and $\shuffle_{\sigma}$ is the automorphism of $X^{\otimes
n}$ that $\shuffle$ associates to the permutation on $n$ elements $\sigma$. In
the rest of the section we will say that $\shuffle_{\sigma}$ is the morphism
corresponding to the permutation $\sigma$.

###  7.1. TSD objects in symmetric monoidal cateogries

Let $(\mathcal{C},\otimes,\tau)$ be a symmetric monoidal category and let $X$
be a ternary self-distributive (TSD) object in $\mathcal{C}$, i.e. a comonoid
object with a morphism $T:X\otimes X\otimes X\longrightarrow X$ satisfying
categorical self-distributivity and commuting with the diagonal morphism
$\Delta_{3}:=\Delta\circ(\Delta\otimes\mathbbm{1})$. Specifically, categorical
self-distributivity means that $X$ is endowed with a morphism
$\Delta:X\longrightarrow X\otimes X$ which makes the same diagrams of
coassociativity commute, a morphism $\epsilon:X\longrightarrow\mathbbm{k}$
which satisfies the same diagrams of counit in a coalgebra, and the morphism
$T$ makes the following diagram commute (case $n=3$ in [ESZ])

${X^{\otimes 9}}$${X^{\otimes 5}}$${X^{\otimes 9}}$${X^{\otimes
n}}$${X^{\otimes
3}}$${X}$$\scriptstyle{\shuffle_{t}}$$\scriptstyle{\mathbbm{1}^{\otimes
3}\otimes\Delta_{3}^{\otimes 2}}$$\scriptstyle{T\otimes\mathbbm{1}^{\otimes
2}}$$\scriptstyle{T\otimes T\otimes T}$$\scriptstyle{T}$$\scriptstyle{T}$

where
$\Delta_{3}:=(\Delta\otimes\mathbbm{1})\circ\Delta=(\mathbbm{1}\otimes\Delta)\circ\Delta$,
and $\shuffle_{t}$ is the shuffle corresponding to ternary self-distributivity

$t=(2,4)(3,5)(6,8)(3,7).$

In the rest of the article we will denote
$\Delta_{n}:=(\Delta\otimes\mathbbm{1}^{n-1})\circ\cdots\circ(\Delta\otimes\mathbbm{1})\circ\Delta:X\longrightarrow
X^{\otimes n}$, to indicate the $n$-fold diagonal of the comonoid object $X$.

######  Remark 7.1.

In what follows we will assume our TSD objects to be cocommutative (as
comonoids), as the main proof of the section will make use of this assumption.
We point out that the preliminary definitions and results make sense without
this further assumption, unless otherwise specified.

The final objective of this section is to generalize the construction of
ribbon category from a TSD operation to the case of TSD objects in symmetric
monoidal categories. When we linearize a TSD operation $T$ over a field (or
ring), as we have seen in Section 4, the rack axiom of $T$ which states that
$T(x,y,z)=d$ has a solution for $x$ for every $d$, once we fix two elements
$y,z$, automatically provides a way of defining an inverse to the braiding
morphisms. In the setting of symmetric monoidal categories, the definition of
TSD object given above does not provide any condition guaranteeing the
possibility of introducing an inverse to the braiding morphism. The next
definition provides an answer to this issue.

######  Definition 7.2.

Let $X$ be a TSD object in a symmetric monoidal category $\mathcal{C}$. Then a
ternary rack in $\mathcal{C}$ is a comonoid object $X$ in $\mathcal{C}$
together with a pair of morphisms $T,T^{-1}:X^{\otimes 3}\longrightarrow X$
satisfying the TSD condition given above, along with the equation

$T^{-1}\circ[T\otimes\mathbbm{1}^{\otimes
2}]\circ\mathbbm{1}\otimes\Delta\otimes\Delta=\mathbbm{1}\otimes\epsilon\otimes\epsilon,$

where equality is meant as an equality of morphisms $X^{\otimes
3}\longrightarrow X$ in $\mathcal{C}$. We also require an analogous equation
to be satisfied, where the roles of $T$ and $T^{-1}$ are exchanged.

######  Example 7.3.

The fundamental example of TSD object in the symmetric monoidal category of
vector spaces is that of an involutory Hopf algebra with quantum heap
operation. This is a “categorical” version of the notion of heap of a group
whose cocycle invariants have been introduced and studied in [SZ]. The quantum
heap operation is given by extending the operation $x\otimes y\otimes z\mapsto
xS(y)z$ by linearity. A similar construction holds replacing a Hopf algebra by
a Hopf monoid in a symmetric category $\mathcal{C}$. This means that $H$ is a
bimonoid, i.e. an object that is both monoid and comonoid, and it is endowed
with a morphism $s:H\longrightarrow H$ that satisfies the same commuative
diagrams for the antipode as in the usual definition of Hopf algebra. It has
been proved in [heap] that an involutory Hopf monoid gives rise to a TSD
object by a generalization of the quantum heap construction (see Theorem 7.12
therein). Both vector space and symmetric monoidal category versions of the
proofs utilize the fact that

$T\circ[T\otimes\mathbbm{1}^{\otimes
2}]\circ[\mathbbm{1}\otimes\hat{\tau}]\circ\mathbbm{1}\otimes\Delta\otimes\Delta=\mathbbm{1}\otimes\epsilon\otimes\epsilon,$

where $\hat{\tau}$ is the morphism $X^{\otimes 4}\longrightarrow X^{\otimes
4}$ corresponding to the permutation $\bigl{(}\begin{smallmatrix}1&2&3&4\\\
1&3&4&2\end{smallmatrix}\bigr{)}$. See for instance the first part of the
proof of Proposition 7.10, and Lemma 7.11 in [heap]. If $X$ is the heap object
corresponding to an involutory Hopf monoid in $\mathcal{C}$, i.e. a TSD
object, we have a ternary rack object in $\mathcal{C}$ by taking $T^{-1}$ to
be $T\circ[\mathbbm{1}\otimes\tau]$, where $\tau$ denotes the switching
morphism of $\mathcal{C}$. Hopf algebras (or monoids) naturally give rise to
TSD objects as well as ternary racks. More generally, one can replace the
notion of involutory Hopf monoid by the more general one of (categorical)
heap, of which Hopf monoids provide a fundamental example. See Definition 7.1
in [heap].

###  7.2. Examples of TSD objects

There is a very natural situation in which TSD objects arise in a symmetric
monoidal category, as described in the following example, which generalizes
Example 7.3

######  Example 7.4.

We describe explicitly the quantum heap TSD morphism in symmetric monoidal
categories mentioned above. Let $X$ be an involutory Hopf monoid in
$\mathcal{C}$. Then it has been shown in [heap] that $X$ can be endowed with a
morphism $T:X^{\otimes 3}\longrightarrow X$ that turns it into a TSD object in
$\mathcal{C}$ as follows. Set
$T:=\mu\circ(\mu\otimes\mathbbm{1})\circ(\mathbbm{1}\otimes
s\otimes\mathbbm{1})$, where $\mu$ is the multiplication morphism and $s$ is
the (involutive) antipode in $X$. Observe that when $X$ is a Hopf algebra in
the category of vector spaces, we have that $(X,T)$ coincides with the quantum
heap of Example 7.3. In fact, if in addition $X$ is the group algebra of some
group $G$, then $T$ is the linearization of set-theoretic heap structure of
$G$, as in Subsection 6.2, so that this is a natural generalization of objects
already encountered. These structures will also be referred to as quantum
heaps, as in the vector space case.

######  Example 7.5.

Let $X$ denote a Hopf monoid in $\mathcal{C}$. Then, we can define an
operation akin to conjugation in a group, by means of the antipode $S$ of $X$.
We define $q=\mu\circ(\mathbbm{1}\otimes\mu)\circ(S\otimes\mathbbm{1}^{\otimes
2})\shuffle\circ(\mathbbm{1}\otimes\Delta)$, where $\shuffle$ corresponds to
the transposition $(12)\in\mathbb{S}_{3}$. In the category of vector spaces or
modules, i.e. when dealing with a Hopf algebra in the usual sense, this
operation takes the form $x\otimes y\mapsto S(y^{(1)})xy^{(2)}$, where
juxtaposition denotes multiplication in $X$. We refer to this operation as
quantum conjugation. Iterating the operation $q$, i.e. defining
$T=q\circ(q\otimes\mathbbm{1})$ we obtain a ternary operation, called double
quantum conjugation. In [ESZ] Section 8, it is seen that $q$ is binary self-
distributive, and it is also proven that composing binary self-distributive
operations yields a TSD operation.

######  Example 7.6.

Let $\mathcal{C}$ denote the category of vector spaces over a field
$\mathcal{k}$, and let $L$ denote a Lie algebra. Set $X=\mathbbm{k}\oplus L$
and, denoting its elements by pairs $(a,x)$, we can define a coproduct on $X$
by the assingment

$(a,x)\mapsto(a,x)\otimes(1,0)+(1,0)\otimes(0,x),$

and counit $\epsilon(a,x)=a$. It is easy to see that this structure defines a
coalgebra in $\mathcal{C}$. Then, we apply Example 8.8 in [ESZ] defining $T$
by iteration of Lie bracket structure on basis vectors $(a,x)$ by the
assignment

$(a,x)\otimes(b,y)\otimes(c,z)\mapsto(abc,bcx+c[x,y]+b[x,z]+[[x,y],z]).$

Then $T$ turns $X$ into a TSD object in the category of vector spaces (proof
in [ESZ], Theorem 8.6 or Appendix A).

In fact, it can be proved that the TSD structures of Example 7.6 are
invertible, and therefore are ternary racks. We will not consider these
objects in detail, leaving the proof of the previous claim to subsequent work,
as they are not cocommutative.

###  7.3. Categorical 2-cocycle condition

Let us now consider an $\mathbb{I}$-linear symmetric monoidal category
$\mathcal{C}$, where $\mathbb{I}$ denotes the unit object of $\mathcal{C}$.
Suppose $X$ is a unitary comonoid object in $\mathcal{C}$. This means that $X$
is endowed with morphisms $\Delta:X\longrightarrow X\otimes X$,
$\epsilon:X\longrightarrow\mathbb{I}$ and $\eta:\mathbb{I}\longrightarrow X$
that make commute the diagrams defining a unitary coalgebra in the category of
vector spaces. In this situation, we say that a morphism
$\alpha:X\longrightarrow\mathbb{I}$ is convolution invertible if there exists
a morphism $\alpha^{-1}$ making the following diagram commute

${X^{\otimes 3}}$${X^{\otimes 3}\otimes X^{\otimes
3}}$${\mathbb{I}}$${\mathbb{I}\otimes\mathbb{I}}$$\scriptstyle{\shuffle\Delta^{\otimes
3}}$$\scriptstyle{\epsilon^{\otimes
3}}$$\scriptstyle{\alpha\otimes\alpha^{-1}}$

where $\shuffle$ reorders the outputs of $\Delta^{\otimes 3}$ as in the
comultiplication of a tensor coalgebra structure.

######  Definition 7.7.

Let $X$ be a TSD object in a symmetric monoidal category. Then a convolution
invertible morphism $\alpha$ is called categorical 2-cocycle with coefficients
in $\mathbb{I}$ if the diagram

${X^{\otimes 5}}$${X^{\otimes 8}}$${\mathbb{I}^{\otimes 2}}$${X^{\otimes
12}}$${\mathbb{I}^{\otimes
2}}$${\mathbb{I}}$$\scriptstyle{\shuffle_{1}\circ(\Delta^{3}\mathbbm{1}^{2})}$$\scriptstyle{\shuffle_{2}\circ(\Delta\mathbbm{1}^{2}\Delta_{2}^{2})}$$\scriptstyle{\alpha\alpha\circ(\mathbbm{1}^{3}T\mathbbm{1}^{2})}$$\scriptstyle{\alpha\alpha\circ(\mathbbm{1}^{3}T^{3})}$

commutes, where $\shuffle_{1}=(2,4)(3,5)$ and
$\shuffle_{2}=(2,4)(3,5)(6,8)(3,7)$, and we have shortened notation by
omitting tensor product of morphisms, which has been indicated as
juxtaposition. We also assume that $2$-cocycles are normalized, in the sense
that $\alpha\circ(\eta\otimes\mathbbm{1}^{\otimes
2})=\alpha\circ(\mathbbm{1}\otimes\eta\otimes\mathbbm{1})=\alpha\circ(\mathbbm{1}^{\otimes
2}\otimes\eta)=\epsilon\otimes\epsilon$.

We now illustrate the prevoius definitions in the category of vector spaces,
using Sweedler notation to denote comultiplication. We will give examples of
the above structures later in the section, while we just assume in the
following example that such objects exist in the category of vector spaces.

######  Example 7.8.

Suppose that $\mathcal{C}$ is the category of vector spaces over some ground
field $\mathbbm{k}$, and $X$ is as above. We want to show how the
commutativity of categorical $2$-cocycle diagram translates in $\mathcal{C}$.
Using Sweedler notation, on a basis vector $x\otimes y\otimes z\otimes
u\otimes v$ we get

$\displaystyle\alpha(x^{(1)}\otimes y^{(1)}\otimes
z^{(1)})\cdot\alpha(T(x^{(2)}\otimes y^{(2)}\otimes z^{(2)})\otimes u\otimes
v)$ $\displaystyle=$ $\displaystyle\alpha(x^{(1)}\otimes u^{(1)}\otimes
v^{(1)})$ $\displaystyle\hskip 28.45274pt\cdot\alpha(T(x^{(2)}\otimes
u^{(2)}\otimes v^{(2)})\otimes T(y\otimes u^{(3)}\otimes v^{(3)})\otimes
T(z\otimes u^{(4)}\otimes v^{(4)}))$

where $\cdot$ indicates multiplication in $\mathbbm{k}$.

The fact that $\alpha$ is convolution invertible in the “coefficients” object
$\mathbb{I}$ plays a fundamental role in constructing inverses in the general
construction, as it will be seen below.

Lastly, we define an equivalence relation between categorical $2$-cocycles.

######  Definition 7.9.

Let $\alpha$ and $\beta$ denote two categorical $2$-cocycles. We say that
$\alpha$ and $\beta$ are equivalent if there exists a convolution invertible
morphism $f:X\longrightarrow\mathbb{I}$ such that the following diagram

${X\otimes X\otimes X}$${X^{\otimes 3}\otimes X^{\otimes 3}}$${X^{\otimes
3}\otimes X}$${\mathbb{I}}$$\scriptstyle{\shuffle\circ\Delta^{\otimes
2}}$$\scriptstyle{(234)\circ(\Delta\otimes\mathbbm{1}^{\otimes
2})}$$\scriptstyle{\alpha\otimes(f\circ T)}$$\scriptstyle{\beta\otimes f}$

commutes.

We show the previous definition in the category of vector spaces.

######  Example 7.10.

Let $(X,T)$ be a TSD object in the category of vector spaces. Then two
categorical $2$-cocycles $\alpha$ and $\beta$ are equivalent, by definition,
if it holds $\alpha(x^{(1)}\otimes y^{(1)}\otimes z^{(1)})f(T(x^{(2)}\otimes
y^{(2)}\otimes z^{(2)}))=\beta(x^{(1)}\otimes y\otimes z)f(x^{(2)})$, for some
$f$, for all $x,y,z\in X$.

###  7.4. Examples of categorical 2-cocycles

We still need to provide examples of categorical $2$-cocycles, as per
Definition 7.7. We begin by observing that the setting of linearized
operations and set-theoretic ternary $2$-cocycles of Section 4 provides first
examples of such morphisms.

######  Example 7.11.

Let $Q$ be a ternary (set-theoretic) quandle and let $G$ be a (multiplicative)
group. Suppose that $\alpha$ is a $2$-cocycle, i.e. $\alpha\in Z^{2}(Q,G)$. As
in Section 4 we let $X$ denote the linear space generated by the elements of
$Q$ and define $T$ to be the linearized operation defined from the set-
theoretic one of $Q$. Let $\chi:G\longrightarrow\mathbbm{k}^{\times}$ denote a
group character. We define a linear map $\tilde{\alpha}:X\otimes X\otimes
X\longrightarrow\mathbbm{k}$ by the assignment on simple vectors $x\otimes
y\otimes z\mapsto\chi\alpha(x,y,z)$. Let us verify that $\tilde{\alpha}$ is
indeed a categorical $2$-cocycle. Since the diagonal in $X$ is induced by the
set-theoretic diagonal $x\mapsto x\times x$, applying Example 7.8 it is enough
to verify

$\displaystyle\tilde{\alpha}(x\otimes y\otimes
z)\cdot\tilde{\alpha}(T(x\otimes y\otimes z)\otimes u\otimes v)$
$\displaystyle=$ $\displaystyle\tilde{\alpha}(x\otimes u\otimes
v)\cdot\tilde{\alpha}(T(x\otimes u\otimes v)\otimes T(y\otimes u\otimes
v)\otimes T(z\otimes u\otimes v))$

which holds true using the definition of $\tilde{\alpha}$ and $T$ as
linearizations of set-theoretic structures.

We consider now, temporarily, the binary version of the previous
constructions. Let us set $\mathcal{C}$ to be the category of vector spaces
over a ground field $\mathbbm{k}$. Recall, in this case, that a unital
coalgebra $(X,\Delta)$ of $\mathcal{C}$ along with a morphism of coalgebras
$q:X\otimes X\longrightarrow X$ is said to be (binary) self-distributive if
the equality

$q(q(x\otimes y)\otimes z)=q(q(x\otimes z^{(1)})\otimes q(y\otimes z^{(2)})),$

holds for all $x,y,z\in X$. The (binary) categorical $2$-cocycle condition for
a convolution invertible morphism $\sigma:X\otimes
X\longrightarrow\mathbbm{k}$ is readily introduced as follows

$\alpha(x^{(1)}\otimes y^{(1)})\alpha(q(x^{(2)}\otimes y^{(2)})\otimes
z)=\alpha(x^{(1)}\otimes z^{(1)})\alpha(q(x^{(2)}\otimes z^{(2)})\otimes
q(y\otimes z^{(3)})).$

As previously observed, in [ESZ] Section 8, it has been proven that composing
binary self-distributive operations yields TSD operations. In the set-
theoretic case, moreover, it is shown that binary $2$-cocycles can be used to
construct ternary $2$-cocycles. We want to use a vector space version of the
set-theoretic result to obtain examples of (ternary) categorical $2$-cocycles.
We consider the cocommutative case for simplicity.

######  Lemma 7.12.

In the setting above, suppose that $\alpha$ is a binary $2$-cocycle. Then,
defining $\psi(x\otimes y\otimes z):=\alpha(x^{(1)}\otimes
y^{(1)})\alpha(q(x^{(2)}\otimes y^{(2)})\otimes z)$, it follows that $\psi$ is
a ternary $2$-cocycle for the doubled TSD operation
$T=q\circ(q\otimes\mathbbm{1})$.

###### Proof.

The proof of this fact follows the same lines of Remark 5.7 in [ESZ], where
the cocycles introduced in Remark 5.6 are assumed to coincide, and the binary
operations are taken both to be $q$. The geometric intepretation of the
various applications of the $2$-cocycle conditions are depicted as in Figure 3
in [ESZ], where extra care is to be taken to utilize the correct superscript
corresponding to Sweedler notation for comultiplication. To prove that $\psi$
is convolution invertible, one needs the fact that $q$ is a morphism of
coalgebra, which is assumed by hypothesis. ∎

######  Example 7.13.

Let $X$ be an involutory Hopf algebra over a ground field $\mathbbm{k}$, and
let $\alpha$ denote a $2$-cocycle with coefficients in $\mathbbm{k}$. Recall,
from the introduction of [AS], or [KSq] Section 10.2.3, that this means that
$\sigma:X\otimes X\longrightarrow\mathbbm{k}$ is convolution invertible,
satisfies the equation

$\sigma(x^{(1)}\otimes y^{(1)})\sigma(x^{(2)}y^{(2)}\otimes z)=\sigma(x\otimes
y^{(1)}z^{(1)})\sigma(y^{(2)}\otimes z^{(2)}),$

for all $x,y,z\in X$, and it is normalized $\sigma(1\otimes x)=\sigma(x\otimes
1)=\epsilon(x)$. Let us consider now the (double) quantum conjugation
operation of Example 7.5. Then, assuming that the underlying coalgebra
structure is cocommutative for simplicity, one can verify that the morphism
$\alpha:X\otimes X\longrightarrow\mathbbm{k}$ defined as $\alpha(x\otimes
y):=\sigma(x^{(1)}\otimes y^{(1)})\sigma^{-1}(y^{(2)}\otimes
S(y^{(3)})x^{(2)}y^{(4)})$ satisfies the (binary) self-distributive cocycle
condition. Applying Lemma 7.12 we obtain a (ternary) categorical $2$-cocycle.
In fact, the correspondence $\sigma\mapsto\alpha$ is a quantum version of the
map given in [CJKLS], Theorem 7.1. This was originally motivated by
diagrammatic computations.

Cocycles of Hopf algebras are used to twist the product structure and the
antipode, to obtain a new Hopf algebra [AS, KSq]. It is known, and easily
verified, that when the underlying coalgebra structure is cocommutative, Hopf
$2$-cocycles generate the same product of the Hopf algebra one starts with.
Example 7.13 considers, in fact, a subclass of a family of $2$-cocycles that
are referred to as invariant cocycles, or lazy cocycles [EG]. These are
defined as those $2$-cocycles whose corresponding twisted Hopf algebra
structure is unchanged. Invariant cocycles are defined up to an equivalence
relation, and their equivalence classes constitute a group, as for the
cohomology of groups, or algebras etc. It seems a relevant question to study
equivalence classes of categorical $2$-cocycles obtained from invariant
cocycles. Moreover, as seen in the next section, categorical $2$-cocycles are
used to twist the Yang-Baxter operator associated to a TSD object, and the
ribbon category constructed below is well defined within an equivalence class.
It might be of interest to relate the equivalence classes of invariant
$2$-cocycles to those of ribbon categories associated to TSD cocycles. We
point out that we do not know whether it is possible to construct categorical
TSD $2$-cocycles from Hopf $2$-cocycles that are not invariant.

###  7.5. Construction of ribbon categories

We define a general version of ribbon categories from TSD objects in symmetric
monoidal categories. This construction gives the linearized definition in
Section 4 when the symmetric monoidal category is that of vector spaces over a
field, the TSD object is defined by linearizing a set-theoretic ternary
rack/quandle, and the coefficients are taken to be a subgroup of the units of
the ground field, via a group character.

Let $\mathcal{C}$ be an $\mathbb{I}$-linear symmetric monoidal category, and
let $X$ be a ternary self-distributive object in $\mathcal{C}$. Let $\alpha$
be a categorical $2$-cocycle in the sense of Definition 7.7. Then we define a
braiding morphism $c_{2,2}^{\alpha}$ following the case of linearized set-
theoretic operations as follows

$c_{2,2}^{\alpha}=(\mathbbm{1}^{\otimes}2\otimes([\alpha\otimes\alpha]\otimes
T\otimes T))\shuffle_{c}(\Delta^{\otimes 2}\Delta_{4}^{\otimes 2}),$

where $\shuffle_{c}$ is the morphism corresponding to the permutation

$\sigma=\bigl{(}\setcounter{MaxMatrixCols}{14}\begin{smallmatrix}1&2&3&4&5&6&7&8&9&10&11&12&13&14\\\
3&9&6&12&1&4&7&10&13&2&5&8&11&14\end{smallmatrix}\bigr{)}.$

Similarly we define a twisting morphism as follows

$\theta^{\alpha}_{2}=([\alpha\otimes\alpha]\otimes T\otimes
T)\shuffle_{\theta}(\Delta_{6}^{\otimes 2}),$

where $\shuffle_{\theta}$ is the morphism corrsponding to the permutation

$\sigma=\bigl{(}\setcounter{MaxMatrixCols}{12}\begin{smallmatrix}1&2&3&4&5&6&7&8&9&10&11&12\\\
1&2&5&7&8&11&4&3&6&10&9&12\end{smallmatrix}\bigr{)}.$

######  Example 7.14.

We want to illustrate braiding and twisting morphisms, as given above, in the
category of vector spaces. Let $\alpha$ be a categorical $2$-cocycle as in
Example 7.8. We now explicitly give the form of switching and twisting
morphisms on simple tensors. We have for $c_{2,2}^{\alpha}$

$\displaystyle c_{2,2}^{\alpha}(x\otimes y\otimes\otimes z\otimes w)$
$\displaystyle=$ $\displaystyle z^{(1)}\otimes w^{(1)}\otimes$
$\displaystyle[\alpha(x^{(1)}\otimes z^{(2)}\otimes
w^{(2)})\cdot\alpha(y^{(1)}\otimes z^{(3)}\otimes w^{(3)})]\cdot$
$\displaystyle T(x^{(2)}\otimes z^{(4)}\otimes w^{(4)})\otimes
T(y^{(2)}\otimes z^{(5)}\otimes w^{(5)}),$

and for twisting morphism $\theta^{\alpha}_{2}$

$\displaystyle\theta^{\alpha}_{2}(x\otimes y)$ $\displaystyle=$
$\displaystyle[\alpha(x^{(1)}\otimes x^{(2)}\otimes
y^{(2)})\cdot\alpha(y^{(1)}\otimes x^{(3)}\otimes y^{(3)})]\cdot$
$\displaystyle T(x^{(4)}\otimes x^{(5)}\otimes y^{(5)})\otimes
T(y^{(4)}\otimes x^{(6)}\otimes y^{(6)}).$

We suppose now, for simplicity, that the comultiplication of $X$ is
cocommutative. Although, in principle this condition can be weakened to some
classes of object whose comultiplication satisfies some special symmetries in
the next results, the proofs are significantly easier when dealing with a
cocommutative object. For instance, the comultiplication of Example 7.6 is not
cocommutative. It is possible to show nontheless that the following
constructions hold. We consider this class of objects in a subsequent article.

######  Definition 7.15.

Let $\mathcal{C}$ be an $\mathbb{I}$-linear symmetric monoidal category with
duals, $(X,T)$ a TSD object in $\mathcal{C}$, and $\alpha$ a categorical
$2$-cocycle of $X$ with coefficients in $\mathbb{I}$. Let $c_{2,2}^{\alpha}$
and $\theta_{2}^{\alpha}$ be braiding a twisting morphisms, respectively, as
previously defined in this section. Then we define
$\mathcal{R}^{*}_{\alpha}(X)$ to be the monoidal category freely generated by
$X\otimes X$ where morphisms are defined as in Definition 4.1 from
$c_{2,2}^{\alpha}$ and $\theta_{2}^{\alpha}$.

######  Theorem 7.16.

$\mathcal{R}^{*}_{\alpha}(X)$ is a ribbon category. Moreover, if $\alpha$ and
$\beta$ are equivalent, then
$\mathcal{R}^{*}_{\alpha}(X)\cong\mathcal{R}^{*}_{\beta}(X)$ as ribbon
categories.

Before proceeding with the actual proof, we give some key steps of the proof
in the setting of the category of vector spaces. This special case is rather
illuminating when considering the actual proof, and functions as prototype for
the general case. We want to show that the braid equation holds, by computing
the LHS and RHS on a simple tensor $x\otimes y\boxtimes z\otimes w\boxtimes
u\otimes v\in X^{\otimes 2}\boxtimes X^{\otimes 2}\boxtimes X^{\otimes 2}$. We
obtain, using Sweedler’s notation, and applying one of the maps whose
composition is the LHS of the braid equation

$\displaystyle x\otimes y\boxtimes z\otimes w\boxtimes u\otimes v$
$\displaystyle\mapsto$ $\displaystyle z^{(1)}\otimes w^{(1)}\boxtimes$
$\displaystyle\hskip 14.22636pt[\alpha(x^{(1)}\otimes z^{(2)}\otimes
w^{(2)})\alpha(y^{(1)}\otimes z^{(3)}\otimes w^{(3)})]\cdot$
$\displaystyle\hskip 14.22636ptT(x^{(2)}\otimes z^{(4)}\otimes w^{(4)})\otimes
T(y^{(2)}\otimes z^{(5)}\otimes w^{(5)})\boxtimes u\otimes v$
$\displaystyle\mapsto$ $\displaystyle z^{(1)}\otimes w^{(1)}\boxtimes
u^{(1)}\otimes v^{(1)}\boxtimes[\alpha(T(x^{(21)}\otimes z^{(41)})\otimes
w^{(41)}\otimes u^{(2)}\otimes v^{(2)})$ $\displaystyle\hskip
14.22636pt\alpha(T(y^{(21)}\otimes z^{(51)})\otimes w^{(51)}\otimes
u^{(3)}\otimes v^{(3)})\alpha(x^{(1)}\otimes z^{(2)}\otimes w^{(2)})$
$\displaystyle\hskip 14.22636pt\alpha(y^{(1)}\otimes z^{(3)}\otimes
w^{(3)})]\cdot T(T(x^{(22)}\otimes z^{(42)}\otimes w^{(42)}\otimes
u^{(4)}\otimes v^{(4)})$ $\displaystyle\hskip 14.22636pt\otimes
T(T(y^{(22)}\otimes z^{(52)}\otimes w^{(52)}\otimes u^{(5)}\otimes v^{(5)})$
$\displaystyle\mapsto$ $\displaystyle u^{(11)}\otimes
v^{(11)}\boxtimes[\alpha(z^{(11)}\otimes u^{(12)}\otimes
v^{(12)})\alpha(w^{(11)}\otimes u^{(13)}\otimes v^{(13)})]$
$\displaystyle\hskip 14.22636pt\cdot T(z^{(12)}\otimes u^{(14)}\otimes
v^{(14)})\otimes T(w^{(12)}\otimes u^{(15)}\otimes v^{(15)})\boxtimes$
$\displaystyle\hskip 14.22636pt[\alpha(T(x^{(21)}\otimes z^{(41)})\otimes
w^{(41)}\otimes u^{(2)}\otimes v^{(2)})$ $\displaystyle\hskip
14.22636pt\alpha(T(y^{(21)}\otimes z^{(51)})\otimes w^{(51)}\otimes
u^{(3)}\otimes v^{(3)})\alpha(x^{(1)}\otimes z^{(2)}\otimes w^{(2)})$
$\displaystyle\hskip 14.22636pt\alpha(y^{(1)}\otimes z^{(3)}\otimes
w^{(3)})]\cdot T(T(x^{(2)}\otimes z^{(42)}\otimes w^{(42)}\otimes
u^{(4)}\otimes v^{(4)})$ $\displaystyle\hskip 14.22636pt\otimes
T(T(y^{(22)}\otimes z^{(52)}\otimes w^{(52)}\otimes u^{(5)}\otimes v^{(5)}).$

Let us now compute the RHS of the braid equation when evaluated on $x\otimes
y\otimes z\otimes w\otimes u\otimes v$. We do not write each step, but rather
provide the final result

$\displaystyle(\mathbbm{1}\boxtimes
c_{2,2}^{\alpha})\circ(c_{2,2}^{\alpha}\boxtimes\mathbbm{1})\circ(\mathbbm{1}\boxtimes
c_{2,2}^{\alpha})(x\otimes y\boxtimes z\otimes w\boxtimes u\otimes v)$
$\displaystyle=$ $\displaystyle u^{(11)}\otimes
v^{(11)}\boxtimes[\alpha(z^{(1)}\otimes u^{(2)}\otimes
v^{(2)})\alpha(w^{(1)}\otimes u^{(3)}\otimes v^{(3)})]$ $\displaystyle\hskip
14.22636pt\cdot T(z^{(21)}\otimes u^{(41)}\otimes v^{(41)})\otimes
T(w^{(21)}\otimes u^{(51)}\otimes v^{(51)})\boxtimes$ $\displaystyle\hskip
14.22636pt[\alpha(T(x^{(21)}\otimes u^{(141)}\otimes v^{(141)})\otimes
T(z^{(22)}\otimes u^{(42)}\otimes v^{(42)})\otimes T(w^{(22)}\otimes$
$\displaystyle\hskip 56.9055ptu^{(52)}\otimes v^{(52)}))$ $\displaystyle\hskip
14.22636pt[\alpha(T(y^{(21)}\otimes u^{(151)}\otimes v^{(151)})\otimes
T(z^{(23)}\otimes u^{(43)}\otimes v^{(43)})\otimes T(w^{(23)}\otimes$
$\displaystyle\hskip 56.9055ptu^{(53)}\otimes v^{(53)}))\alpha(x^{(1)}\otimes
u^{(12)}\otimes v^{(12)})\alpha(y^{(1)}\otimes u^{(13)}\otimes v^{(13)})]$
$\displaystyle\hskip 14.22636pt\cdot T(T(x^{(12)}\otimes u^{(142)}\otimes
v^{(142)})\otimes T(z^{(24)}\otimes u^{(44)}\otimes v^{(44)})\otimes
T(w^{(24)}\otimes$ $\displaystyle\hskip 56.9055ptu^{(54)}\otimes
v^{(54)}))\otimes$ $\displaystyle\hskip 14.22636ptT(T(y^{(12)}\otimes
u^{(152)}\otimes v^{(152)})\otimes T(z^{(15)}\otimes u^{(45)}\otimes
v^{(45)})\otimes T(w^{(25)}\otimes$ $\displaystyle\hskip
56.9055ptu^{(55)}\otimes v^{(55)}))).$

Now, comparing the two expressions we see that we cannot apply directly
ternary self-distributivity of $T$ and the categorical $2$-cocycle condition
to $\alpha$, as the terms corresponding to comultiplications in Sweedler
notation are shuffled differently. We can rearrenge them by means of the
cocommutativity of $X$ and then conclude that they coincide applying
categorical $2$-cocycle condition and TSD property of $T$.

A similar direct reasoning is applied to show that braiding morphism and
twisting commute. We will not add the details of this computation.

The proof of Theorem 7.16 will consist of a similar reasoning, but replacing
equations by commutative diagrams. We compare the expressions by writing them
as compositions of “all comultiplications”, “shuffle”, “$T$’s and $\alpha$’s”
and finally multiplying the $\alpha$’s by identifying two copies of the unit
object $\mathbb{I}$ with $\mathbb{I}$ itself. Then using cocommutativity and
naturality of shuffle morphisms in a symmetric monoidal category, one draws
the conclusion that the two terms are equal.

###### Proof of Theorem 7.16.

As in the proof of Theorem 4.3, the crucial step is to show that
$c_{2,2}^{\alpha}$ satisfies the braid equation, and that
$\theta_{2}^{\alpha}$ satisfies the compatibility relation with respect to
$c_{2,2}^{\alpha}$. We proceed to prove the first assertion. We need to show
that the diagram

${X^{\otimes 2}\boxtimes X^{\otimes 2}\boxtimes X^{\otimes 2}}$${X^{\otimes
2}\boxtimes X^{\otimes 2}\boxtimes X^{\otimes 2}}$${X^{\otimes 2}\boxtimes
X^{\otimes 2}\boxtimes X^{\otimes 2}}$${X^{\otimes 2}\boxtimes X^{\otimes
2}\boxtimes X^{\otimes 2}}$${X^{\otimes 2}\boxtimes X^{\otimes 2}\boxtimes
X^{\otimes 2}}$${X^{\otimes 2}\boxtimes X^{\otimes 2}\boxtimes X^{\otimes
2}}$$\scriptstyle{c_{2,2}^{\alpha}\boxtimes\mathbbm{1}^{\otimes
2}}$$\scriptstyle{1^{\otimes 2}\boxtimes
c_{2,2}^{\alpha}}$$\scriptstyle{1^{\otimes 2}\boxtimes
c_{2,2}^{\alpha}}$$\scriptstyle{c_{2,2}^{\alpha}\boxtimes\mathbbm{1}^{\otimes
2}}$$\scriptstyle{c_{2,2}^{\alpha}\boxtimes\mathbbm{1}^{\otimes
2}}$$\scriptstyle{1^{\otimes 2}\boxtimes c_{2,2}^{\alpha}}$

is commutative. We will refer to the left perimeter, from top to bottom, of
the preceding hexagon as the “left-hand side” of the braid diagram, or simply
LHS, and similarly for the right perimeter we will say the “right-hand side”
or RHS of the braid diagram. In what follows we will keep using the same
shortened notation in which a tensor product of maps of such as
$\mathbbm{1}^{\otimes 2}\boxtimes T^{\otimes 2}$ will be denoted by
juxtaposition, $\mathbbm{1}^{2}T^{2}$, where powers of tensor products of type
$\otimes 2$ are simply indicated by a power $2$. A similar notation is also
used for objects, where we omit the tensor symbol $\otimes$ in the exponents,
although we keep the $\boxtimes$ symbol for clarity.

The proof will consists of rewriting the LHS and the RHS of the braid diagram
in a convenient way, by using the axioms of the symmetric monoida category
$\mathcal{C}$, and the left Frobenius module axiom. Then we will argue that
the two expressions coincide by means of the categorical $2$-cocycle
condition, and the rack axioms of $T$. We start with the LHS.

Using the definition of $c_{2,2}^{\alpha}$, the LHS of the braid diagram fits
in the commutative diagram of Figure 6.

${X^{2}\boxtimes X^{2}\boxtimes X^{2}}$${X^{4}\boxtimes X^{10}\boxtimes
X^{2}\cong X^{2}\boxtimes X^{12}\boxtimes X^{2}}$${X^{2}\boxtimes
X^{12}\boxtimes X^{2}}$${X^{2}\boxtimes X^{2}\otimes X^{2}\boxtimes
X^{2}}$${X^{2}\boxtimes X^{3}\boxtimes X^{2}}$${X^{2}\boxtimes X^{2}\boxtimes
X^{2}}$${X^{2}\boxtimes X^{2}\boxtimes X^{2}}$${X^{2}\boxtimes X^{2}\boxtimes
X^{2}}$$\scriptstyle{(c_{2,2}^{\alpha}\mathbbm{1}^{2})\circ(\mathbbm{1}^{2}c_{2,2}^{\alpha})\circ(c_{2,2}^{\alpha}\mathbbm{1}^{2})}$$\scriptstyle{c_{2,2}\mathbbm{1^{2}}}$$\scriptstyle{\Delta^{2}\Delta_{4}^{2}\mathbbm{1^{2}}}$$\scriptstyle{\shuffle_{c}1^{2}}$$\scriptstyle{\alpha^{2}T^{2}\mathbbm{1}^{2}}$$\scriptstyle{\mu\mathbbm{1}^{2}\mathbbm{1}^{2}}$$\scriptstyle{\mathbbm{1}^{2}\mathbbm{1}^{2}}$$\scriptstyle{\mathbbm{1}^{2}c_{2,2}^{\alpha}}$$\scriptstyle{c_{2,2}^{\alpha}\mathbbm{1}^{2}}$
Figure 6.

Now we rewrite the top-right perimeter of the previous diagram as in Figure 7,

${X^{2}\boxtimes X^{2}\boxtimes X^{2}}$${X^{4}\boxtimes X^{10}\boxtimes
X^{10}}$${X^{2}\boxtimes X^{12}\boxtimes X^{10}}$${X^{2}\boxtimes
X^{12}\boxtimes X^{2}}$${X^{2}\boxtimes X^{12}\boxtimes
X^{10}}$${X^{2}\boxtimes X^{12}\boxtimes X^{2}}$${X^{2}\boxtimes X^{6}\otimes
X^{12}\boxtimes X^{10}}$${X^{2}\boxtimes X^{2}\boxtimes
X^{26}}$${X^{2}\boxtimes\mathbb{I}^{2}\otimes X^{2}\boxtimes
X^{2}}$${X^{2}\boxtimes\mathbb{I}^{2}\otimes X^{4}\boxtimes
X^{10}}$${X^{2}\boxtimes X^{2}\boxtimes X^{6}\otimes\mathbb{I}^{2}\otimes
X^{6}}$${X^{2}\boxtimes\mathbb{I}\otimes X^{2}\boxtimes
X^{2}}$${X^{2}\boxtimes X^{4}\boxtimes X^{10}}$${X^{2}\boxtimes X^{2}\boxtimes
X^{12}}$${X^{2}\boxtimes X^{2}\boxtimes X^{2}}$${X^{2}\boxtimes X^{2}\boxtimes
X^{2}}$${X^{2}\boxtimes X^{2}\boxtimes
X^{2}}$$\scriptstyle{\Delta^{2}\Delta_{4}^{2}\mathbbm{1}^{2}}$$\scriptstyle{\Delta^{2}\Delta_{4}^{2}\Delta_{4}^{2}}$$\scriptstyle{(\Delta\mathbbm{1})^{2}(\Delta\mathbbm{1}^{4})^{2}\mathbbm{1}^{10}}$$\scriptstyle{\shuffle}$$\scriptstyle{\mathbbm{1}^{14}\Delta_{4}^{2}}$$\scriptstyle{\shuffle_{c}\mathbbm{1}^{2}}$$\scriptstyle{(\Delta\mathbbm{1})^{2}(\Delta\mathbbm{1}^{4})^{2}\mathbbm{1}^{10}}$$\scriptstyle{\mathbbm{1}^{8}\Delta^{6}\Delta_{4}^{2}}$$\scriptstyle{\mathbbm{1}^{2}\alpha^{2}T^{2}\mathbbm{1}^{2}}$$\scriptstyle{\mathbbm{1}^{2}\alpha^{2}T^{4}\mathbbm{1}^{10}}$$\scriptstyle{\shuffle}$$\scriptstyle{\mathbbm{1}^{4}(T\mathbbm{1}^{2})^{2}\alpha^{2}(T\mathbbm{1}^{2})^{2}}$$\scriptstyle{\mathbbm{1}^{4}\Delta^{2}\Delta_{4}^{2}}$$\scriptstyle{\mathbbm{1}^{2}\mu\mathbbm{1}^{4}}$$\scriptstyle{\mathbbm{1}^{2}(\mu\mathbbm{1}^{4})\mathbbm{1}^{10}}$$\scriptstyle{\shuffle}$$\scriptstyle{\mathbbm{1}^{10}\mu\mathbbm{1}^{6}}$$\scriptstyle{(\mathbbm{1}^{3}\Delta^{2}\Delta_{4}^{2})}$$\scriptstyle{\mathbbm{1}^{2}\mathbbm{1}^{3}}$$\scriptstyle{\mathbbm{1}^{2}\shuffle_{c}}$$\scriptstyle{(\mathbbm{1}^{4}(\mu\mathbbm{1})\mathbbm{1}^{2})\circ(\mathbbm{1}^{10}T^{2})}$$\scriptstyle{\mathbbm{1}^{2}\Delta^{2}\Delta_{4}^{2}}$$\scriptstyle{\mathbbm{1}^{2}c_{2,2}^{\alpha}}$$\scriptstyle{(c_{2,2}^{\alpha}\mathbbm{1}^{2})\circ(\mathbbm{1}^{2}c_{2,2}^{\alpha})}$$\scriptstyle{c_{2,2}^{\alpha}\mathbbm{1}^{2}}$
Figure 7.

where we have used naturality of the shuffle morphisms in the category
$\mathcal{C}$, coassociativity of the morphism $\Delta$ and, in the bottom-
left triangle, the fact that $\mathbb{I}$ commutes with objects of
$\mathcal{C}$. Similar considerations show that the diagram of Figure 8

${X^{2}\boxtimes X^{12}\boxtimes X^{10}}$${X^{2}\boxtimes X^{20}\boxtimes
X^{18}}$${X^{2}\boxtimes X^{2}\boxtimes X^{2}}$${X^{2}\boxtimes X^{2}\boxtimes
X^{26}}$${X^{2}\boxtimes X^{12}\boxtimes X^{26}}$${X^{2}\boxtimes
X^{2}\boxtimes X^{6}\otimes\mathbb{I}^{2}\otimes X^{6}}$${X^{4}\boxtimes
X^{10}\boxtimes X^{6}\otimes\mathbb{I}^{2}\otimes X^{6}}$${X^{2}\boxtimes
X^{2}\boxtimes X^{12}}$${X^{2}\boxtimes X^{2}\boxtimes\mathbb{I}^{2}\otimes
X^{2}}$${X^{2}\boxtimes X^{2}\boxtimes\mathbb{I}\otimes
X^{2}}$${X^{2}\boxtimes X^{2}\boxtimes
X^{2}}$$\scriptstyle{\shuffle}$$\scriptstyle{\mathbbm{1}^{6}(\Delta_{2}\mathbbm{1}^{5})^{2}(\Delta_{4}\mathbbm{1}^{4})^{2}}$$\scriptstyle{\shuffle}$$\scriptstyle{\mathbbm{1}^{4}(T\mathbbm{1})^{2}\alpha^{2}(T\mathbbm{1}^{2})^{2}}$$\scriptstyle{\mathbbm{1}^{4}(\mu\circ\mu^{2})\mathbbm{1}^{8}}$$\scriptstyle{\mathbbm{1}^{4}\alpha^{2}T^{2}}$$\scriptstyle{\mathbbm{1}^{4}\mu\mathbbm{1}^{2}}$$\scriptstyle{\mathbbm{1}^{4}\mathbbm{1}}$$\scriptstyle{\Delta^{2}\Delta_{4}^{2}\mathbbm{1}^{2}}$$\scriptstyle{c_{2,2}^{\alpha}\mathbbm{1}^{2}}$
Figure 8.

commutes, where the horizontal maps are obtained from (appropriate) products
of the comultiplication morphism, and similarly for the hypotenuse of the
central triangle. The morphism $X^{2}\boxtimes X^{12}\boxtimes
X^{26}\longrightarrow X^{2}\boxtimes X^{2}\boxtimes X^{2}$ is
$\mathbbm{1}^{2}\alpha^{2}T^{2}((\mu\circ\mu^{2})\circ(T\mathbbm{1}^{2})^{2})T\circ(T\mathbbm{1}^{2})^{2}$.
Pasting together the three diagrams along the corresponding edges shows that
the morphism $(c_{2,2}^{\alpha}\boxtimes\mathbbm{1})\circ(\mathbbm{1}\boxtimes
c_{2,2}^{\alpha})\circ(c_{2,2}^{\alpha}\boxtimes\mathbbm{1})$ can be rewritten
by first aplying the comultiplication morphisms $\Delta$ all together to
produce the right number of copies of $X$, shuffle them in the right position
and then apply all the morphisms $T$ and $\alpha$ at the end. This is in fact
better understood by thinking of the same proof for Hopf algebras. A (very
tedious) direct inspection, simplified by diagrammatic reasoning, can be
applied to determine step by step the shuffles used in the commutative
diagrams. Thus proceeding we see that we have obtained for the LHS of the
braid diagram
$(c_{2,2}^{\alpha}\boxtimes\mathbbm{1})\circ(\mathbbm{1}\boxtimes
c_{2,2}^{\alpha})\circ(c_{2,2}^{\alpha}\boxtimes\mathbbm{1})=(\mathbbm{1}^{\otimes
2}\boxtimes(\mathbbm{1}\circ\mu\otimes\mathbbm{1}^{\otimes
2})\boxtimes(\mu^{\otimes}))\circ(\mathbbm{1}^{2}\boxtimes\alpha^{\otimes
2}T^{\otimes 2}\boxtimes(\alpha\circ(T\otimes\mathbbm{1}^{\otimes
2}))^{\otimes 2}\otimes\alpha^{\otimes
2}\otimes(T\circ(T\otimes\mathbbm{1}^{\otimes 2})^{\otimes
2}))\circ\shuffle\circ(\Delta_{2}^{\otimes 2}\boxtimes\Delta_{7}^{\otimes
2}\boxtimes\Delta_{8}^{\otimes 2})$, where the shuffle morphism $\shuffle$
corresponds to the permutation that rearrenges the terms in the last
expression of the computation of
$(c_{2,2}^{\alpha}\boxtimes\mathbbm{1})\circ(\mathbbm{1}\boxtimes
c_{2,2}^{\alpha})\circ(c_{2,2}^{\alpha}\boxtimes\mathbbm{1})$. A similar
procedure shows that the RHS of the braid diagram is written as
$(\mathbbm{1}\boxtimes
c_{2,2}^{\alpha})\circ(c_{2,2}^{\alpha}\boxtimes\mathbbm{1})\circ(\mathbbm{1}\boxtimes
c_{2,2}^{\alpha})=(\mathbbm{1}^{\otimes 2}\boxtimes(\mu\circ\alpha^{\otimes
2})\circ T^{\otimes 2})\boxtimes(\mu\circ(\mu\circ\alpha^{\otimes
2}\otimes\mu\circ\alpha^{\otimes 2})\otimes(T\circ T^{\otimes 3})^{\otimes
2})\shuffle\circ(\Delta_{2}^{\otimes 2}\boxtimes\Delta_{5}^{\otimes