# Planar Equivalence of Knotoids and Quandle Invariants

MOHAMED ELHAMDADI  Department of Mathematics and Statistics, University of
South Florida, Tampa, FL 33620, USA<EMAIL_ADDRESS>,  WOUT MOLTMAKER
Korteweg-de Vries Institute, University of Amsterdam, Amsterdam, 1098XG,
Netherlands<EMAIL_ADDRESS>and  MASAHICO SAITO  Department of
Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA
<EMAIL_ADDRESS>
###### Abstract.

While knotoids on the sphere are well-understood by a variety of invariants,
knotoids on the plane have proven more subtle to classify due to their
multitude over knotoids on the sphere and a lack of invariants that detect a
diagram’s planar nature. In this paper, we investigate equivalence of planar
knotoids using quandle colorings and cocycle invariants. These quandle
invariants are able to detect planarity by considering quandle colorings that
are restricted at distinguished points in the diagram, namely the endpoints
and the point-at-infinity. After defining these invariants we consider their
applications to symmetry properties of planar knotoids such as invertibility
and chirality. Furthermore we introduce an invariant called the triangular
quandle cocycle invariant and show that it is a stronger invariant than the
end specified quandle colorings.

##  1\. Introduction

Knotoids were introduced by V. Turaev [Turaev] and are a generalization of
knot diagrams, modelled on the unit interval rather than the circle. This
means that a knotoid diagram is allowed to have two end-points; what
distinguishes knotoids from long knots is that the endpoints are allowed to
lie anywhere in the diagram, including interior regions. What makes them
nontrivial is that we impose an equivalence relation on knotoid diagrams that
prevents us from moving any arcs over or under these endpoints, so that we may
imagine them as fixed in place on the plane by two infinitely long rods or
‘rails’ [rail].

Knotoids were originally introduced to study knots [Turaev], but since have
also received considerable research interest in their own right. Several
constructions of strong invariants for knots have been extended to knotoids
[newinvariants, moltmaker, winding], and novel invariants of knotoids have
been constructed [barbensi, winding]. This work has culminated in a first
classification of knotoids for low crossing number [systematic].

Since this initial investigation, knotoids have mainly found applications to
protein topology, and more generally to studying the topology of physical
systems of open-ended filamentous structures. We model these using ‘open
curves’, i.e. smooth embeddings $[0,1]\hookrightarrow\mathbb{R}^{3}$. In
particular, the following approach was taken in [Eleni] to construct a Jones
polynomial of open curves: given an open curve $C$ in $\mathbb{R}^{3}$, for a
unit vector $v$ in $S^{2}$ define $C_{v}$ to be the knotoid diagram obtained
from projecting $C$ onto the plane normal to $v$. Then for all but a measure
zero subset of $S^{2}$, $C_{v}$ is a well-defined knotoid diagram. The idea is
then that we can ‘measure’ the entanglement of $C$ by considering the multiset
$\\{C_{v}\\}_{v\in S^{2}}$. For any knotoid type $K\in\\{C_{v}\\}_{v\in
S^{2}}$ we consider the area $A_{K}$ of the subset of $S^{2}$ that is sent to
$K$ by the map $C\mapsto[C_{v}]$. The Jones polynomial of an open curve is
then defined as the (a priori infinite) sum of Jones polynomials $V_{K}(t)$ of
all knotoid types $K\in\\{C_{v}\\}_{v\in S^{2}}$, weighted by the areas
$A_{K}$. This convergent sum can be written as an integral as follows, noting
that the integrand is well-defined on all but a measure zero subset of $S^{2}$
[Eleni]:

$V_{C}(t)=\frac{1}{4\pi^{2}}\int_{v\in S^{2}}V_{C_{v}}(t)dA.$

This Jones polynomial of open curves is not an isotopy invariant of open
curves, indeed all open curves are ambient isotopic to one another, but rather
it is a ‘measure’ for the open curve’s knottedness. Other measures of
entanglement are obtained from following the same procedure using a knotoid
invariant other than the Jones polynomial: as long as the invariant takes
values in some $\mathbb{Z}$-module $M$ we obtain a knot measure taking values
in $M\otimes\mathbb{R}$. For example see [Eleni2] where $\mathbb{R}$-valued
entanglement measures are derived from $\mathbb{Z}$-valued knotoid invariants.

In all applications of this kind considered so far, the knotoid diagrams are
thought of as lying on a sphere, rather than their plane projection. This can
of course be done by adding a point at infinity to the plane. However, unlike
for knots and links, for knotoids the equivalence classes on the plane differ
from those on the sphere. Clearly adding a point at infinity to the plane is a
surjection from knotoids on the plane to knotoids on the sphere, but this map
is not injective. In fact it is many-to-one: for example, there are 8
equivalence classes of knotoids with crossing number 4 on the sphere, but 154
on the plane [systematic].

This observation implies that if one employs an entanglement measure that
relies on an invariant of knotoids on the plane rather than on the sphere,
then we can obtain a measure with a ‘higher resolution’ for entanglement. This
is in the sense that the underlying invariant can distinguish more different
knotoids and hence different ‘kinds of knottedness’. The problem with this
approach is that currently, not very many invariants of knotoids on the plane
are known. Most straight-forward generalizations of knot and link invariants
to knotoids turn out to factor through the one-point compactification of the
plane, and hence yield invariants of knotoids on the sphere.

In this paper we define several new planar knotoid invariants based on
quandles. Quandles were introduced to knot theory independently by D. Joyce
[Joyce] and S. Matveev [Matveev], and axiomatize the notion of ‘coloring’ a
knot diagram, where the quandle acts as a pallet of colors to use. The
simplest quandle invariants count the number of valid colorings of a diagram,
yielding an $\mathbb{N}$-valued invariant. Extending these quandle-coloring
counting invariants to planar knotoids, we obtain an $\mathbb{R}$-valued
planar entanglement measure for open curves, which enjoys the increased
resolution of employing a planar invariant as well as the convenience for
applications of taking values in $\mathbb{R}$. Further extending these
invariants to quandle homology invariants yields planar knotoid measures
taking values in more complicated spaces.

Quandle invariants of knotoids were first studied in [GN], but the invariants
described there do not tend to detect the planar character of knotoids on the
plane, and are invariant under one of the ‘forbidden moves’ for knotoids;
namely that which pulls an arc over an endpoint. In fact, this invariance was
already noted in a different context in [CKS-Geom]. In this paper we give a
refinement of these invariants that does detect planarity. We note that a
recent article [GR] constructs similar refinements using fundamental and
pointed quandles; the work presented here is independent and as such we do not
refer to these constructions, although the colorings here can be interpreted
as quandle homomorphisms from pointed fundamental quandles.

The organization of this article is as follows. In Section 2 we briefly give
preliminary descriptions of knotoids and quandles. In Section 3 we construct
quandle invariants of planar knotoids, and discuss how well these invariants
can distinguish inequivalent planar knotoids that become equivalent on the
sphere. In Section 4 we consider our invariants in relation to planar
invertibility. We refine the basic quandle invariants to quandle cocycle
invariants in Section 5, and discuss these in relation to planar chirality in
Section 6. Finally in Section 7 we describe another quandle cocycle invariant
for planar knotoids, called the ‘triangular’ quandle 2-cocycle invariant.

##  2\. Preliminaries of knotoids and quandles

In this section we review materials used in this paper.

###  2.1. Knotoids

######  Definition 2.1.

Let $\Sigma$ be a surface. A knotoid diagram on $\Sigma$ is a smooth immersion
$K:[0,1]\to\Sigma$ whose only singularities are transversal self-intersections
that are decorated with over/under-crossing data. The images of $0$ and $1$
are called the initial point (or tail) and terminal point (or head) of the
knotoid, respectively. Knotoid diagrams are oriented from tail to head. Two
knotoids are said to be equivalent if they can be related by a sequence of
isotopies of $\Sigma$ and Reidemeister moves; see Figure 1. A knotoid on
$\Sigma$ is an equivalence class of knotoid diagrams on $\Sigma$.

Figure 1.

Note that the Reidemeister moves are not allowed to involve any endpoints of a
knotoid, and therefore the forbidden moves on a knotoid diagram, depicted in
Figure 2 are explicitly not allowed. Note that if these forbidden moves were
allowed, all knotoids on any surface $\Sigma$ would be equivalent to the
trivial knotoid that has no crossings.

Figure 2.

In this work we restrict our attention to the cases $\Sigma=S^{2}$ and
$\Sigma=\mathbb{R}^{2}$. Knotoids on $S^{2}$ are also called spherical
knotoids, and knotoids on $\mathbb{R}^{2}$ are called planar knotoids.

Clearly there is a surjection from planar knotoids to spherical knotoids,
given by the one-point compactification of the plane. This surjection acts on
planar knotoid diagrams by adding a point at infinity; this operation
preserves the equivalence of diagrams. For knots the same map exists, but is a
bijection since an isotopy carrying an arc of a knot diagram across the added
point at infinity can also be achieved by moving the same arc in the other
direction, over the entire knot diagram. For knotoids this is not the case, as
the endpoints get in the way of an arc moving over the entire diagram. As a
result, for knotoids this surjection is not injective. In other words: there
are many inequivalent planar knotoids that become equivalent when drawn on the
sphere, the only difference between them being that they have different
regions as their exterior region.

Conversely, given a spherical knotoid we can turn it into a planar knotoid
diagram (non-canonically) by choosing a region in the diagram and removing a
point in that region from $S^{2}$, i.e. designating it as the point at
infinity. As such we now introduce the following notation: Let $K$ be a
spherical knotoid diagram. Let $R$ be a region in the diagram. The planar
knotoid of $K$ when $R$ is specified as the region at infinity is denoted by
$(K,R)$. The spherical and planar equivalence of knotoids are denoted by
$\cong_{{\mathbb{S}}^{2}}$ and $\cong_{{\mathbb{R}}^{2}}$, respectively.

Spherical knotoids have a natural product operation given by concatenation:
for any spherical knotoid diagrams $K_{1}$ and $K_{2}$ we may assume without
loss of generality that the head of $K_{1}$ and the tail of $K_{2}$ lie in the
exteriors of their respective diagrams. The product $K_{1}\cdot K_{2}$ is then
defined by identifying the head of $K_{1}$ with the tail of $K_{2}$ in the
disjoint union $K_{1}\sqcup K_{2}$.

For planar knotoids we define the product operation analogously, as long as
the exterior region of $K_{2}$ is appropriate:

######  Definition 2.2.

Let $(K_{1},R_{1})$ and $(K_{2},R_{2})$ be planar knotoids such that the tail
of $K_{2}$ lies in $R_{2}$. Then the product $K_{1}\cdot K_{2}$ is the planar
knotoid diagram obtained by placing a small copy of $K_{2}$ next to the head
of $K_{1}$ and identifying the tail of $K_{2}$ with the head of $K_{1}$.

###  2.2. Quandle invariants

We start by reviewing the definition of a quandle and give a few examples.

######  Definition 2.3.

[Joyce, Matveev] A quandle $(X,*)$ is a set with a binary operation
$(x,y)\mapsto x*y$ satisfying the following three axioms:

(I) For all $x\in X$, $x*x=x$.

(II) For all $y,z\in X$, there exists a unique $x\in X$ such that $x*y=z$.

(III) For all $x,y,z\in X$, we have $(x*y)*z=(x*z)*(y*z).$

Recall that these three axioms correspond respectively to Reidemeister moves
of type I, II, and III.

A function $\phi:(X,*)\rightarrow(Y,\triangleright)$ between quandles is
called a homomorphism if $\phi(x\ast y)=f(x)\triangleright f(y)$ for any
$x,y\in X$. The notions of isomorphisms and automorphisms are defined
naturally. The inner automorphism group of a quandle $X$ is the subgroup of
its automorphism group generated by the right action defined by the quandle
operation. A quandle is called connected if its inner automorphism group acts
transitively. Throughout this paper we assume that all quandles are connected,
as it is sufficient to focus on them in terms of colorings of knotoid
diagrams.

The following are a few examples of quandles.

  * •

Any set $X$ with the operation given by $x*y=x$ for any $x,y\in X$ is a
quandle called the trivial quandle.

  * •

The binary operation of conjugation $x*y=y^{-1}xy$ defines a quandle structure
on any group $G$. Furthermore, for the abelian cyclic group
$G=\mathbb{Z}_{n}$, the binary operation can be written as $x*y=-x+2y$. This
quandle is called the dihedral quandle and denoted ${\bf R}_{n}$.

  * •

Let $\Lambda$ be $\mathbb{Z}[t^{\pm 1}]$ or $\mathbb{Z}_{n}[t^{\pm 1}]$. Then
any $\Lambda$-module $M$ is a quandle with $x*y=tx+(1-t)y$, $x,y\in M$, called
an Alexander quandle.

A coloring of an oriented knot or knotoid diagram is a function
${\mathcal{C}}:R\rightarrow X$, where $X$ is a fixed quandle and $R$ is the
set of over-arcs in the diagram satisfying, at each crossing, the conditions
(A) and (B) of Figure 3. The set of colorings of a knotoid diagram $K$ by a
quandle $(X,*)$ is denoted by ${\rm Col}_{X}(K)$.

Let $D\subset\Sigma$ be a diagram of a knotoid. The connected components of
$\Sigma\setminus D$ are called the regions. A coloring of the regions is
defined as a map from the set of regions of a colored knotoid diagram to a
quandle that satisfy the conditions (C), (D) and (E) of Figure 3.

######  Lemma 2.4 ([CKS-Geom]).

Let $\cal C\in{\rm Col}_{X}(K)$ be a coloring of the set of arcs of a knot
diagram $K$ by a connected quandle $(X,*)$. For a given color $x$ of a
particular region $R$, there is a unique region coloring $\tilde{\cal C}$ that
extends $\cal C$.

The proof of this lemma is to define region colors along arcs that go under
the knot diagram and determine colors according to the rule. The uniqueness
follows from independence under homotopy of such arcs, as shown in Figure 3
(E), where the rightmost region color is independent of choice of arcs that
goes over and under the center crossing. This lemma can be extended to
knotoids in the following manner.

######  Lemma 2.5.

Let $\cal C\in{\rm Col}_{X}(K)$ be a coloring of the set of arcs of a
spherical knotoid diagram $K$ by a connected quandle $(X,*)$. Then there is a
unique region coloring $\tilde{\cal C}$ that extends $\cal C$.

###### Proof.

Let $p_{\iota}$ and $p_{\tau}$ be the initial and terminal points of $K$. Let
$\gamma$ be an arc from $p_{\iota}$ to $p_{\tau}$ that goes under the arcs of
$K$. Let $\hat{K}$ be the knot diagram given by the union $K\cup\gamma$,
called an under closure of $K$. Extend the coloring ${\cal C}$ of $K$ along
$\gamma$ as it goes under each arc of $K$, following the coloring rule. This
argument is also used in [Turaev]. When it reaches near $p_{\tau}$, the colors
of these points coincide. This is seen by an argument showing that the end
colors of (1-1)-tangles coincide, see [CDS, Jozef]. Independence of the choice
of $\gamma$ can be seen by the homotopy argument as in Lemma 2.4. ∎

Figure 3.

##  3\. End specified triangular colorings

Let $K$ be a spherical oriented knotoid diagram. Let $R_{\infty}$ be the
region specified as the region at infinity. Let $R_{\iota}$ and $R_{\tau}$ be
the regions at the initial and terminal end points, respectively. Let
$p_{\infty}$ be a point in $R_{\infty}$, and $p_{\iota}$, $p_{\tau}$ be the
initial and terminal end points, respectively.

Let $(X,*)$ be a quandle. Denote by ${\mathbb{Z}}[X]$ the free integral module
over $X$, consisting of elements of the formal finite sum $\sum a_{x}x$, where
$a_{x}\in{\mathbb{Z}}$ and $x\in X$. The addition is defined monomial wise and
extended linearly, $(\sum a_{x}x)+(\sum b_{y}y)=\sum(a_{z}+b_{z})z$. Since
${\mathbb{Z}}[X\times X]\cong{\mathbb{Z}}[X]\otimes{\mathbb{Z}}[X]$, we denote
their elements by $\sum a_{(x,y)}x\otimes y$, and similar for multiple
tensors.

######  Definition 3.1.

Let $(X,*)$ be a quandle and let $(K,R_{\infty})$ be a planar oriented
knotoid. An ordered quandle triple $(a,b,c)\in X^{3}$ is an ordered triple
such that $a,b,c$ belong to the same connected component of $X$.

An end specified coloring ${\cal C}:{\cal A}\cup{\cal R}\rightarrow X$ with
respect to a quandle triple $(a,b,c)$ is a coloring of arcs and regions of $K$
by $X$ such that $({\cal C}(p_{\infty}),{\cal C}(p_{\iota}),{\cal
C}(p_{\tau}))=(a,b,c)$. The set of all end specified colorings with respect to
a quandle triple $(a,b,c)$ is denoted by ${\rm
Col}^{(a,b,c)}_{X}(K,R_{\infty})$, and its cardinality is denoted by $|{\rm
Col}^{(a,b,c)}_{X}(K,R_{\infty})|$. The end specified coloring invariant of
$(K,R)$ with respect to $X$ and $(a,b,c)$ is a formal sum

${\cal Col}_{X}(K,R_{\infty}):=\sum_{(a,b,c)\in X^{3}}|{\rm
Col}_{X}^{(a,b,c)}(K,R_{\infty})|(a\otimes b\otimes
c)\in{\mathbb{Z}}[X]^{\otimes 3}.$

Similarly, for a spherical knotoid $K$, an end specified coloring ${\cal C}$
is defined by a coloring such that $({\cal C}(p_{\iota}),{\cal
C}(p_{\tau}))=(b,c)$ for $(b,c)\in X^{2}$. The set of all end specified
coloring with respect to a quandle pair $(b,c)\in X^{2}$ is denoted by ${\rm
Col}^{(b,c)}_{X}(K)$. Similar colorings are defined for planar knot diagrams
when no color is specified for the region at infinity.

######  Lemma 3.2.

Let $(K,R)$ be a planar knotoid and $X$ be a quandle. Then $\sum_{a\in X}|{\rm
Col}^{(a,b,c)}(K,R)|=|{\rm Col}^{(b,c)}(K)|$.

###### Proof.

For a given $(b,c)\in X^{2}$, let ${\cal C}\in{\rm Col}^{(b,c)}(K)$. Then by
Lemma 2.5, there is a unique $a\in X$ such that ${\cal C}\in{\rm
Col}^{(a,b,c)}(K,R)$. Hence the statement follows. ∎

Figure 4.

######  Example 3.3.

In Figure 4, a knotoid $T_{n}$ is depicted, where $n=2h$ is an even integer.
Arcs $A_{i}$ and regions $R_{j}$ ($i=0,1,\ldots,n$, $j=0,1,\ldots,n-1$) are
indicated, as well as the region at infinity $R$. In this example we show that
$(T_{n},R_{1})\not\equiv_{{\mathbb{R}}^{2}}(T_{n},R_{2})$ using colorings.

We examine possible colorings. Let ${\cal C}(p_{\iota})=x$ and ${\cal
C}(A_{n-1})=y$. Recall that an Alexander quandle $X$ has the operation
$u*v=tu+(1-t)v$ for $u,v\in X$. If $T_{n}$ is colored by an Alexander quandle
$X$ such that ${\cal C}(p_{\iota})=x$ and ${\cal C}(A_{n-1})=y$, then one
inductively computes that

${\cal C}(A_{k})=t\xi_{k-1}(t)x+\xi_{k}(t)y,\ {\rm where}\
\xi_{k}(t)=1-t+t^{2}-\cdots+(-t)^{k};\ (\xi_{0}(t)=1,\ \xi_{1}(t)=1-t).$

Similarly, we have

$\displaystyle{\cal C}(R_{1})$ $\displaystyle=$ $\displaystyle tx+(1-t)y,$
$\displaystyle{\cal C}(R_{2})$ $\displaystyle=$
$\displaystyle(2-t)tx+(1-t)^{2}y,$ $\displaystyle{\cal C}(R_{k})$
$\displaystyle=$ $\displaystyle[1+(1-t)\xi_{k-2}(t)]tx+(1-t)\xi_{k-1}(t)y\
{\rm for}\ k>2.$

For a dihedral quandle ${\bf R}_{\ell}={\mathbb{Z}}_{\ell}[t]/(t+1)$, we have
$\xi_{k}(-1)=k+1$, ${\cal C}(A_{k})=-kx+(k+1)y$ and ${\cal
C}(R_{k})=(1-2k)x+(2k)y$ for $k>0$ and ${\cal C}(R_{0})=-x+2y$. A pair $(x,y)$
gives rise to a coloring if and only if ${\cal C}(A_{n-1})=y$ and ${\cal
C}(R_{0})={\cal C}(A_{n})$, that is,

$-(n-1)x+ny=y\quad{\rm and}\quad-x+2y=-nx+(n+1)y,$

the same condition. This condition is satisfied if and only if $n=1$ mod
$\ell$. Thus we consider the case $n=\ell+1$, and take $X={\bf R}_{n-1}={\bf
R}_{2h-1}$. Then there is a coloring of $K$ for any $x,y\in X$. In this case
we also have ${\cal C}(R_{0})={\cal C}(A_{n})=2y-x$. Since $2$ is invertible,
the correspondence $(x,y)\mapsto(x,2y-x)$ is bijective. Since ${\cal
C}(R_{1})=2y-1$ as well, for $x=0$ and $y=1$, we have $|{\rm
Col}^{(2,0,2)}(T_{n},R_{1})|=1$. Since ${\cal C}(R_{2})=4y-3x=4$ for $x=0$ and
$y=1$, we have $|{\rm Col}^{(2,0,2)}(T_{n},R_{2})|=0$. Hence we obtain that
$(T_{n},R_{1})$ and $(T_{n},R_{2})$ are not planarily equivalent.

######  Proposition 3.4.

For any positive integer $m$, there exists a spherical knotoid $K$ that has
$m$ regions $R_{i}$, $i=1,\ldots,m$, such that the planar knotoids $(K,R_{i})$
are pairwise inequivalent as planar knotoids.

###### Proof.

We set $K$ to be $T_{n}$ in Example 3.3, where $n\geq m+2$. We use the
notations as in Figure 4, and computations in Example 3.3 with the dihedral
quandle ${\bf R}_{\ell}$, where $n=\ell+1$, $\ell=n-1=2h-1$.

For a fixed $k$, $1\leq k\leq m$, we consider the quandle triple $(2k,0,2)$,
where $2k$ is regarded as an element of ${\mathbb{Z}}_{n}$. Then as in the
example, we have

$|{\rm Col}^{(2k,0,2)}_{X}(K,R_{k})|=1,\ {\rm and}\ |{\rm
Col}^{(2k,0,2)}_{X}(K,R_{k^{\prime}})|=0$

for all $k^{\prime}\neq k$, $k^{\prime}=1,\ldots,m$. By varying
$k=1,\ldots,m$, we obtain that $(K,R_{i})$ are pairwise inequivalent. ∎

######  Example 3.5.

To compute the number of colorings over all possible quandle triples for
$(T_{n},R_{k})$, we use two colors $x,y$ in Figure 4 as parameters. To have
nontrivial colorings, we set $X={\bf R}_{\ell}$ with $\ell=n-1$ as computed in
Example 3.3. We have computed that for a given $(x,y)$, we have

${\cal C}(p_{\iota})=x,\ {\cal C}(p_{\tau})={\cal C}(A_{n})={\cal
C}(R_{0})=2y-x,\ {\cal C}(p_{\infty})={\cal C}(R_{k})=(1-2k)x+(2k)y.$

For a given $(x,y,k)$, we have

$|{\rm Col}^{((1-2k)x+(2k)y,x,2y-x)}_{X}(K,R_{k})|=1,$

and for other $(a,b,c)$ we have $|{\rm Col}^{(a,b,c)}_{X}(K,R_{k})|=0$.
Therefore we obtain

${\cal Col}_{X}(K,R_{k})=\sum_{(a,b,c)\in Qt(X)}|{\rm
Col}^{(a,b,c)}_{X}(K,R_{k})|(a\otimes b\otimes c)=\sum_{x,y\in
X}([(1-2k)x+(2k)y]\otimes x\otimes[2y-x]).$

######  Definition 3.6.

The matching product on ${\mathbb{Z}}[X]^{\otimes 3}$,

$\vartriangle:({\mathbb{Z}}[X]^{\otimes 3})\times({\mathbb{Z}}[X]^{\otimes
3})\rightarrow{\mathbb{Z}}[X]^{\otimes 3},$

is defined on monomials by

$a_{x,y,z}(x\otimes y\otimes z)\vartriangle b_{u,v,w}(u\otimes v\otimes
w)=\begin{cases}(a_{x,y,z}b_{u,v,w})(x\otimes y\otimes w)&{\text{if }}z=u=v\\\
0&{\text{otherwise}}\end{cases}$

and extended linearly.

######  Lemma 3.7.

Let $(X,*)$ be a quandle. Let $(K_{1},R_{1})$, $(K_{2},R_{2})$ be planar
knotoids, and $p_{\iota}^{i}$ and $p_{\tau}^{i}$ be the initial and terminal
points of $K_{i}$ for $i=1,2$. Suppose $K_{2}$ satisfies $p_{\iota}^{2}\in
R_{2}$ so that the product $K_{1}\cdot K_{2}$ is defined. Then we have

${\cal Col}_{X}(K_{1}\cdot K_{2},R_{1})={\cal
Col}_{X}(K_{1},R_{1})\vartriangle{\cal Col}_{X}(K_{2},R_{2}).$

###### Proof.

This is checked monomial wise. Then the initial and terminal points
$p_{\iota}$ and $p_{\tau}$ of $K_{1}\cdot K_{2}$ are $p_{\iota}=p_{\iota}^{1}$
and $p_{\tau}=p_{\tau}^{2}$, respectively. The other points $p_{\tau}^{1}$ and
$p_{\iota}^{2}$ are identified in $K_{1}\cdot K_{2}$. For $a,b,c,d\in X$,
${\cal C}_{1}\in{\rm Col}^{(a,b,c)}_{X}(K_{1})$ and ${\cal C}_{2}\in{\rm
Col}^{(c,c,d)}_{X}(K_{2})$, a coloring ${\cal C}={\cal C}_{1}\cdot{\cal
C}_{2}\in{\rm Col}^{(a,b,d)}_{X}(K_{1}\cdot K_{2})$, that restricts to ${\cal
C}_{1}$ and ${\cal C}_{2}$ on each factor, is well defined after identifying
$p_{\tau}^{1}$ and $p_{\iota}^{2}$, since ${\cal C}_{1}(p_{\tau}^{1})={\cal
C}_{2}(p_{\iota}^{2})$. Hence we obtain

$|{\rm Col}^{(a,a,d)}_{X}(K_{1}\cdot K_{2})|=|{\rm
Col}^{(a,b,c)}_{X}(K_{1})||{\rm Col}^{(c,c,d)}_{X}(K_{2})|.$

When ${\cal C}_{1}(p_{\tau}^{1})\neq{\cal C}_{2}(p_{\iota}^{2})$, the
colorings do not extend. Thus there is a bijection between the set of
colorings of $K_{1}$ and $K_{2}$ such that ${\cal C}_{1}(p_{\tau}^{1})={\cal
C}_{2}(p_{\iota}^{2})$ and the set of colorings of $K_{1}\cdot K_{2}$, and the
equality holds for monomials. ∎

######  Example 3.8.

Let $T_{n}$ be the knotoid in Example 3.3, where it was shown that ${\cal
Col}_{X}(T_{n},R)=\sum_{a,b\in{\bf R}_{2h-1}}a\otimes a\otimes b$ for a
positive integer $h$, $n=2h$. Lemma 3.7 implies that

${\cal Col}_{X}(T_{n}\cdot T_{n},R)={\cal Col}_{X}(T_{n},R)\vartriangle{\cal
Col}_{X}(T_{n},R)=(\sum a\otimes a\otimes b)\vartriangle(\sum c\otimes
c\otimes d).$

See Figure 10 for a diagram of $T_{n}\cdot T_{n}$. For any $(a,d)\in{\bf
R}_{2h-1}$, there are $2h-1=n-1$ elements $b=c\in{\bf R}_{2h-1}$ such that
$(a\otimes a\otimes b)\vartriangle(c\otimes c\otimes d)=(a,a,d)$, hence

$(\sum a\otimes a\otimes b)\vartriangle(\sum c\otimes c\otimes
d)=\sum(n-1)(a\otimes a\otimes d).$

Thus we have ${\cal Col}_{X}(T_{n}\cdot T_{n})=\sum_{a,d\in{\bf
R}_{n-1}}(n-1)(a\otimes a\otimes d)$.

######  Corollary 3.9.

For any positive integers $m,\ell$, there exist a knotoid $K$ and a quandle
$(X,*)$ such that ${\cal Col}_{X}(K)$ consists of more than $m$ monomials
whose coefficients are all larger than $\ell$.

###### Proof.

Repeated applications of the computations given in Example 3.8 show that
${\cal Col}_{{\bf R}_{n-1}}((T_{n})^{\cdot k})$, where the knotoid is the
$k$-fold knotoid product, is equal to $\sum_{a,b\in{\bf
R}_{n-1}}(n-1)^{k-1}a\otimes b$. Thus sufficiently large choices of $n$ and
$k$ for given $m$ and $\ell$ give the result. ∎

######  Question 3.10.

What are the possible values of the invariant? There are more specific
questions that can be raised, such as: Is it true that for any positive
integer $n$, there exist a knotoid $K$ and a quandle $(X,*)$ such that ${\cal
Col}_{X}(K)$ consists of $n$ nonzero monomials? Are there any relations among
coefficients of nonzero monomials?

##  4\. Planar equivalence and invertibility

In this section we investigate relations between planar equivalence and
invertibility. For a planar or spherical knotoid $K$, we denote by $rK$ the
knotoid obtained from $K$ by reversing its orientation. If a planar knotoid
$(K,R)$ is planarily equivalent to $(rK,R)$, then it is called (planarily)
invertible. Otherwise it is called (planarily) noninvertible. Spherical
invertibility is defined analogously, but without reference to a region at
infinity.

######  Example 4.1.

Let $T_{n}$ be a knotoid depicted in Figure 4, $n>0$. Let $R$ be the region at
infinity. Then $(rT_{n},R)$ is not equivalent to $(T_{n},R)$. This is because
for $(T_{n},R)$, $p_{\infty}$ and $p_{\iota}$ both belong to $R$, but
$p_{\tau}$ does not, while for $(rT_{n},R)$, $p_{\infty}$ and $p_{\tau}$ both
belong to $R$, but $p_{\iota}$ does not.

However, $T_{n}$ and $rT_{n}$ are spherically equivalent, as the outer arc can
be isotoped through infinity, so that $T_{n}$ is spherically invertible.

######  Remark 4.2.

The argument in the preceding example motivates to define the following
concept of distances. Let $\gamma_{\iota}$, $\gamma_{\tau}$ and
$\gamma_{\iota\tau}$ be oriented paths from $p_{\infty}$ to $p_{\iota}$,
$p_{\infty}$ to $p_{\tau}$ and $p_{\iota}$ to $p_{\tau}$, respectively, all
whose intersections with $K$ are transversal crossing points, away from the
crossings of $K$. Assume that these arcs intersect with $K$ in a finitely many
double crossings.

Then the crossing distance from the terminal point $p_{\infty}$ to the initial
point $p_{\iota}$ (resp. the terminal point $p_{\tau}$) is the minimum number
of crossings of $\gamma_{\iota}$ and $K$ among all equivalent diagrams, and
denoted by $d_{\iota}$ (resp. $d_{\tau}$). The distance between the initial
and terminal points, $d_{\iota\tau}$ was similarly defined as the height of
$K$ in [GK]. The distances $d_{\iota}$ and $d_{\tau}$ can be used to detect
invertibility.

A quandle is called involutive if its right action is involutive, i.e.
$(x*y)*y=x$ for all elements $x,y$. For example the dihedral quandles ${\bf
R}_{n}$ are involutive. As colorings by involutive quandles are well defined
for unoriented diagrams, there is a bijection (identity correspondence)
between the sets of colorings before and after the orientation reversal. Since
the reversal changes the initial and terminal end points, we have the
following.

######  Lemma 4.3.

Let $X$ be an involutive quandle. For planar knotoid $(K,R)$ and for any
quandle triple $(a,b,c)$, we have

$|{\rm Col}^{(a,b,c)}_{X}(K,R)|=|{\rm Col}^{(a,c,b)}_{X}(rK,R)|.$

######  Example 4.4.

In the proof of Proposition 3.4, using the same notation, we have $|{\rm
Col}^{(2k,0,2)}_{X}(T_{n},R_{k})|=1$. Using the computation in Example 3.5:

${\cal Col}_{X}(K,R_{k})=\sum_{x,y\in X}((1-2k)x+(2k)y)\otimes
x\otimes(2y-x).$

If $x=2$ and $2y-x=2$, then we obtain $y=2$, then $(1-2k)x+(2k)y=2$. Hence if
$k\neq 1$, then $|{\rm Col}^{(2k,0,2)}_{X}(rT_{n},R_{k})|=0$, so that
$(T_{n},R_{k})$ and $(rT_{n},R_{k})$ are not planarily equivalent, and
$(T_{n},R_{k})$ is noninvertible.

By varying values of $n$ and $k$ in the preceding example, we obtain the
following result.

######  Theorem 4.5.

For any positive integer $m$, there exists a spherically invertible knotoid
$K$ with regions $R_{i}$, $i=1,\ldots,m$, such that $(K,R_{k})$ are planarily
noninvertible for $i=1,\ldots,m$.

##  5\. Planar cocycle invariants with end specified colors

In this section we define quandle cocycle invariants for knotoids in a manner
similar to [CJKLS]. Such invariants have been defined in [Cazet], and the
difference here is to use end restricted colorings and obtain stronger
invariants, in particular of planar knotoids, that distinguish properties
between planar versus spherical equivalences. In the following section we use
these invariants to study planar chirality.

We recall [CJKLS] that a function $\phi:X\times X\rightarrow A$, where $X$ is
a quandle and $A$ is an abelian group, is called a quandle 2-cocycle if it
satisfies

$\phi(x,x)=0\qquad\text{and}\qquad\phi(x,y)-\phi(x,z)+\phi(x*y,z)-\phi(x*z,y*z)=0$

for all $x,y,z\in X$. The abelian group of all quandle 2-cocycles is denoted
by $Z^{2}_{\rm Q}(X,A)$. Further details of quandle (co)homology theory can be
found in [CKS].

######  Definition 5.1.

Let ${\cal C}:{\cal A}\cup{\cal R}\rightarrow X$ be an end specified coloring
with respect to a quandle triple $(a,b,c)$ of $K$ by $X$ such that $({\cal
C}(p_{\infty}),{\cal C}(p_{\iota}),{\cal C}(p_{\tau}))=(a,b,c)$.

Let $(X,*)$ be a quandle, $(K,R)$ a planar knotoid diagram. Let $\phi\in
Z^{2}_{Q}(X,A)$ be a quandle 2-cocycle with a finite abelian coefficient group
$A$. Let $B_{\phi}(K,{\cal C})$ denote the product of the cocycle values
$\phi(x_{j},y_{j})^{\epsilon(j)}$ over all crossings $j$ with respect to a
coloring ${\cal C}$. Here $\epsilon(j)=\pm 1$, depending on the sign of
crossing $j$. Recall that the product is with respect to the group operation
of $A$, which is abelian but denoted with multiplicative notation.

The planar quandle 2-cocycle invariant is defined by

$\Phi_{\phi}^{(a,b,c)}(K)=\sum_{C\in{\rm
Col}_{X}^{(a,b,c)}(K,R)}B_{\phi}(K,{\cal C}).$

Consider the quandle module ${\mathbb{Z}}A[X](=({\mathbb{Z}}[A])[X])$, whose
elements consists of finite sums $\sum a_{x}x$, $a_{x}\in{\mathbb{Z}}[A]$.
Then define

${\Phi}_{\phi}(K,R)=\sum\Phi_{\phi}^{(a,b,c)}(K,R)(a\otimes b\otimes
c)\in{\mathbb{Z}}A[X]^{\otimes 3}.$

Similarly, the 2-cocycle invariants $\Phi_{\phi}^{(b,c)}(K)$ and
$\Phi_{\phi}(K)$ are defined for spherical knotoids using ${\rm
Col}^{(b,c)}(K)$ instead of ${\rm Col}^{(a,b,c)}(K,R)$.

In a similar manner, end-specified 3-cocycle invariants can be also defined;
see [CJKLS] for details on 3-cocycle invariants of knots.

######  Definition 5.2.

Let $\psi\in Z^{3}_{Q}(X,A)$ be a quandle 3-cocycle of $X$ with a finite
abelian coefficient group $A$. For a region (and arc) coloring ${\cal C}$, let
$B_{\psi}(K,{\cal C})$ denote the product of the cocycle values
$\psi(u_{j},x_{j},y_{j})^{\epsilon(j)}$ over all crossings $j$ with respect to
a coloring ${\cal C}$, where $u_{j}$ is the source color at the crossing.

Define the planar quandle 3-cocycle invariant for knotoids  to be

$\Psi_{\psi}^{(a,b,c)}(K)=\sum_{C\in{\rm
Col}^{(a,b,c)}_{X}(K)}B_{\psi}(K,{\cal C}).$

Then define

${\Psi}_{\psi}(K)=\sum\Phi_{\psi}^{(a,b,c)}(K)(a\otimes b\otimes
c)\in{\mathbb{Z}}A[X]^{\otimes 3}.$

By routine check of Reidemeister moves as in [CJKLS], we have that these
cocycle invariants are well defined.

##  6\. Planar equivalence and chirality

In this section we discuss differences of chirality of knotoid between
spherical and planar equivalences. For a spherical or planar knotoid diagram
$K$, we denote by $mK$ the knotoid obtained from $K$ by reversing all its
crossings.

First we apply the quandle 3-cocycle invariant defined in the preceding
section to show that there exists a knotoid that is achiral spherically but
chiral planarily.

Let $K$ be a figure eight knotoid diagram depicted in solid arcs in Figure 5.
We show that $K^{\prime}$ is spherically amphicheiral but not planarily. Let
$K^{\prime}$ be a diagram obtained by replacing the outer solid arc by the
dotted arc. By rotating $K$ by 180 degrees about the midpoint of the two end
points, we see that $K^{\prime}$ is equivalent to $rmK$. Since this
replacement of the outer arc is a spherical move, we have that
$K^{\prime}=rmK$, hence $K\cong_{{\mathbb{S}}^{2}}rmK$. The diagram
$K^{\prime}$ with the exterior region $R_{\infty}$ at infinity is the same as
$(K,R_{1})$, so we have
$(K^{\prime},R_{\infty})=(rmK,R_{\infty})\cong_{{\mathbb{R}}^{2}}(K,R_{1})$.

Figure 5.

######  Proposition 6.1.

The figure eight knotoid $K$ as depicted in Figure 5 has the property that
$K\cong_{{\mathbb{S}}^{2}}rmK$, but a planar diagram $K^{\prime}$ spherically
equivalent to $rmK$ has the property that
$(K,R)\not\cong_{{\mathbb{R}}^{2}}(K^{\prime},R_{\infty})$.

###### Proof.

We have shown that $K\cong_{{\mathbb{S}}^{2}}rmK$, and below we show
$(K,R)\not\cong_{{\mathbb{R}}^{2}}(K,R_{1})$. It is well known that $K$ is
nontrivially colored by ${\bf R}_{5}$. A coloring of arcs are indicated in the
figure with variables. Specifically, an element $x$ of ${\bf R}_{5}$ is
assigned to the end points and the region $R_{0}$ that they belong to, and
another adjacent arc is colored by $y$. The colors of remaining arcs are
determined by crossings, and are as indicated in the figure. It is checked
that the indicated coloring is well defined modulo $5$, and uniquely
determined by a choice of $x,y\in{\bf R}_{5}$. The requirement that the color
of $R_{0}$ is $x$ determines all colors of the other regions uniquely, and is
computed as ${\rm Col}(R)=y$, ${\rm Col}(R_{1})=2y-x$, ${\rm Col}(R_{2})=y$
and ${\rm Col}(R_{3})=3x-2y$. Thus all colors, on arcs and regions, are
uniquely determines by $x,y\in{\bf R}_{5}$.

If we choose $(p_{\infty},p_{\iota},p_{\tau})=(y,x,x)=(1,0,0)$, for example,
we obtain a unique nontrivial coloring and that ${\rm Col}^{(1,0,0)}(K,R)=1$.
On the other hand, if we choose $R_{1}$ as the region at infinity, then for
the triple $(1,0,0)$, we have ${\rm Col}(R_{1})=2y-x=1$ and $x=0$, which
implies $y=3$, and we obtain ${\rm Col}^{(1,0,0)}(K,R)=1$ as well.

We apply the cocycle invariant in Definition 5.2, with the dihedral quandle
${\bf R}_{5}$ with the coefficient group of cohomology ${\mathbb{Z}}_{5}$. The
values of $\Psi_{\psi}^{(a,b,c)}$ are in the group ring
${\mathbb{Z}}[{\mathbb{Z}}_{5}]$, and written as polynomial in multiplicative
generator $u$ of ${\mathbb{Z}}_{5}$. There is a nontrivial 3-cocycle in
$Z^{3}_{\rm Q}({\bf R}_{5},{\mathbb{Z}}_{5})$. The general case of 3-cocycle
for ${\bf R}_{p}$ with prime $p$ is defined in [Mochi], simplified in
[AsamiSatoh, HashiTana] and given by the following formula:

$\psi(x,y,z)=(x-y)[(2z-y)^{p}+y^{2}-(2z)^{p})]/p=(x-y)\sum_{i=1}^{p-1}i^{(-1)}(-x)^{i}(2y)^{p-i}\in{\mathbb{Z}}_{p}.$

The cocycle weights for the top two positive crossings from left to right
computes, with given variables, $\psi(y,3y-2x,y)$ and $\psi(y,y,3y-2x)=0$.
Those bottom two negative crossing from left and right read $-\psi(x,y,x)$ and
$-\psi(y,3y-2x,2x-y)$. With these cocycle values we compute that
$\Psi_{\psi}^{(1,0,0)}(K,R)=u^{4}$, where $u$ is a multiplicative generator of
${\mathbb{Z}}_{5}$.

For $K^{\prime}$, the contributions of crossings are the same, when only the
outer arc is changed and labels are kept, and the region at infinity is
changed from $R$ to $R_{1}$. Similar computations with ${\rm
Col}(R_{1})=2y-x=1$ and $x=0$ shows that $\Psi_{\psi}^{(1,0,0)}(K,R_{1})=u$.
Hence we obtain $(K,R)\not\cong_{{\mathbb{R}}^{2}}(K,R_{1})$. ∎

######  Remark 6.2.

We conjecture that there are infinitely many spherically achiral knotoids that
are planarily chiral. A family of potential candidate is depicted in Figure 7,
that are generalizations of the figure eight knotoid, where the center arc is
cut into a pair of end points. (For some integers they have multiple
components.) The argument similar to that in proof of the preceding
proposition shows that they are spherically achiral.

We also conjecture that for any positive integer $n$, there is a planar
knotoid diagram $K_{n}$ with the exterior region $R_{0}$ at infinity that is
planarily achiral $(K,R_{0})\cong_{{\mathbb{R}}^{2}}(rmK,R_{0})$, such that
there are regions $R_{k}$, $k=1,\ldots,n$ with
$(K_{n},R_{i})\not\cong_{{\mathbb{R}}^{2}}(K_{n},R_{j})$ for $i\neq j$,
$i,j=0,1,\ldots,n$. Note that in this case all planar knotoids $(K_{n},R_{i})$
are spherically equivalent, so that they are all spherically achiral. A
potential example is depicted in Figure 7. A rotation shows that this diagram
is planarily achiral. This example is motivated by $8_{3}$ which is achiral.
By adding more twists on the top and bottom twists, we obtain similar
candidates.

Figure 6.

Figure 7.

##  7\. Triangular quandle $2$-cocycle invariant

As before, let $K$ be a spherical oriented knotoid diagram, $R_{\infty}$ be
the region specified as the region at infinity, $p_{\infty}$ be a point in
$R_{\infty}$, and $p_{\iota}$, $p_{\tau}$ be the initial and terminal end
points, respectively.

Let $\gamma_{\iota}$, $\gamma_{\tau}$ and $\gamma_{\iota\tau}$ be oriented
paths from $p_{\infty}$ to $p_{\iota}$, $p_{\infty}$ to $p_{\tau}$ and
$p_{\iota}$ to $p_{\tau}$, respectively, that goes under all arcs of $K$ and
away from crossings of $K$. We may assume that these arcs intersect with $K$
in finitely many double crossings. We may also assume that their self
intersections or intersections between them are away from those with $K$ and
arcs with $K$.

Let $X$ be a quandle and $\phi$ be its 2-cocycle with coefficient in an
abelian group $A$ in multiplicative notation. We define the triangular quandle
$2$-cocycle invariant for a knotoid $K$ as follows. Let $X$ be a quandle,
$(a,b,c)$ be a quandle triple, and ${\rm Col}^{(a,b,c)}_{X}(K)$ be the set of
end specified colorings with respect to $(a,b,c)$. Each arc of the paths
$\gamma_{h}$, $h=\iota,\tau,\iota\tau$ inherit the same color of the region
where it is included. Denote the same letter ${\cal C}$ for such a coloring.
Then for each crossing $\lambda\in\gamma_{h}\cap K$, the weight $B({\cal
C},\lambda)$ is defined as in [CJKLS] to be
$\phi(x(\lambda),y(\lambda))^{\epsilon(\lambda)}$, where $(x(c),y(c))$ are the
source colors at $\lambda$ for ${\cal C}$, and $\epsilon(c)$ denotes the sign
of $c$. Let $B_{\phi}({\cal C},\gamma_{h})=\prod_{\lambda\in\gamma_{h}\cap
K}\phi(x(\lambda),y(\lambda))^{\epsilon(\lambda)}\in A$.

Figure 8.

######  Definition 7.1.

For a planar knotoid $(K,R)$, define an element

$\Phi_{\phi}^{(a,b,c)}(K,R,\gamma_{h})=\sum_{{\cal C}\in{\rm
Col}^{(a,b,c)}_{X}(K)}B_{\phi}({\cal C},\gamma_{h})\in{\mathbb{Z}}[A]$

in the group ring ${\mathbb{Z}}[A]$. Also define

$\Phi_{\phi}(K,R,\gamma_{h})=\sum\Phi_{\phi}^{(a,b,c)}(K,R,\gamma_{h})(a\otimes
b\otimes c)\in{\mathbb{Z}}A[X].$

Next define the triangular 2-cocycle invariant to be the triple

$\Phi_{\phi}^{(a,b,c)}(K,R)=(\Phi_{\phi}^{(a,b,c)}(K,R,\gamma_{\iota}),\Phi_{\phi}^{(a,b,c)}(K,R,\gamma_{\iota\tau}),\Phi_{\phi}^{(a,b,c)}(K,R,\gamma_{\tau}))\in({\mathbb{Z}}[A])^{3}.$

Also define

$\Phi_{\phi}(K,R)=(\Phi_{\phi}(K,R,\gamma_{\iota}),\Phi_{\phi}(K,R,\gamma_{\iota\tau}),\Phi_{\phi}(K,R,\gamma_{\tau}))\in({\mathbb{Z}}A[X]^{\otimes
3})^{3}.$

Figure 9.

######  Lemma 7.2.

The triangular 2-cocycle invariant is well defined.

###### Proof.

It is sufficient to show that the invariant values do not change under
homotopy of the paths $\gamma_{h}$, $h=\iota,\tau,\iota\tau$. Generic homotopy
is realized by Reidemeister moves I, II, III and the move passing through the
end points of $K$. Reidemeister moves do not change the cocycle invariant
values for the same reason as the regular 2-cocycle invariant. When an arc
goes through an end point, it happens in the region that included the end
point, and the colors of the region, the end arc, and the portion of
$\gamma_{h}$ in the region, are all the same, before and after the move. When
$\gamma_{h}$ goes under the end arc of $K$, see Figure 9, the contribution of
the cocycle value at the crossing is trivial, since $\phi(x,x)=1$, $\phi$
being a quandle 2-cocycle. Hence the claim follows. ∎

Figure 10.

The triangular quandle cocycle invariant provides a stronger obstruction than
end specified quandle colorings, in the following sense.

######  Proposition 7.3.

There exists a quandle $X$ such that there is an infinite family of spherical
knotoids with distinct planar equivalence classes, that are distinguished by
triangular cocycle invariants of $X$, while they cannot be distinguished by
colorings only.

###### Proof.

Set $X={\mathbb{Z}}_{2}[t]/(t^{2}+t+1)$, a four-element Alexander quandle
denoted by $S_{4}$ in [CJKLS], also called the tetrahedral quandle, that
corresponds to rotations of a regular tetrahedron. We represent its elements
by $\\{0,1,t,1+t\\}$. We use the 2-cocycle of
$X={\mathbb{Z}}_{2}[t]/(t^{2}+t+1)$ with coefficient group
$A={\mathbb{Z}}_{2}$ given in [CJKS], $\phi=\prod\chi_{(a,b)}$, where the
product ranges over all $a,b\in\\{0,1,1+t\\}$ such that $a\neq b$, and $\chi$
denotes Kronecker’s delta.

In Figure 10, the knotoid product $T_{n,m}:=T_{n}\cdot T_{m}$ is depicted. The
arcs and regions are labeled as in Example 3.3 for the first factor $T_{n}$
and primed symbols for the second factor $T_{m}$ as depicted. We use the
computations of colorings by Alexander quandles as in Example 3.3. Consider
the colorings by $X$ specified by ${\cal C}(p_{\iota})=x=0$. Then one computes

${\cal C}(A_{1})=tx+(1-t)y=(1+t)y\in X,\ {\cal C}(A_{2})=0,\ {\cal
C}(A_{3})=y,$

and this pattern repeats in period $3$. We also compute

${\cal C}(R_{0})={\cal C}(R_{1})=tx+(1-t)y=(1+t)y,\ {\cal C}(R_{2})=ty,$

and starting ${\cal C}(R_{3})=x=0$, ${\cal C}(R_{4})=(1+t)y$ and ${\cal
C}(R_{5})=ty$, it repeats this pattern in period $3$. The colorings on the
part $T_{m}$ is similar.

There is a coloring of $T_{n}$ by $X$ if and only if

${\cal C}(A_{n-1})=y,\ {\rm and}\ {\cal C}(A_{n})={\cal C}(R_{0}).$

We choose $n=4+6k$ (since the pattern repeats by 3 and $n$ must be even) for
nonnegative integers $k$, then we have

${\cal C}(A_{n-1})={\cal C}(A_{3+6k})=y\ {\rm and}\ {\cal C}(A_{n})={\cal
C}(A_{4+6k})=(1+t)y={\cal C}(R_{0}),$

so that $T_{n}$ has a nontrivial coloring for all $y\neq 0$. If $y=0$, it
provides a trivial coloring by $0$ on all arcs.

For the colors $x=0$ and ${\cal C}(A_{n})={\cal C}(A_{4+6k})=(1+t)y={\cal
C}(R_{0})$, we have

${\cal C}(A_{1}^{\prime})={\cal C}(A_{n})=x^{\prime}=(1+t)y={\cal C}(R_{0}).$

Again from the computations in Example 3.3, in this case, we obtain

${\cal
C}(A_{1}^{\prime})=tx^{\prime}+(1+t)y^{\prime}=t(1+t)y+(1+t)y^{\prime}=y+(1+t)y^{\prime},\
{\cal C}(A_{2}^{\prime})=x^{\prime}=(1+t)y,\ {\cal
C}(A_{3}^{\prime})=y^{\prime},$

and this repeats in period 3. We also obtain

${\cal C}(R_{1}^{\prime})=tx^{\prime}+(1+t)y^{\prime}=y+(1+t)y^{\prime},\
{\cal C}(R_{2}^{\prime})=(1+t)x^{\prime}+ty^{\prime},\ {\cal
C}(R_{3}^{\prime})=x^{\prime},$

and this repeats in period 3. To obtain a coloring for $m=2$, we need

${\cal C}(A_{2}^{\prime})={\cal C}(R_{0}^{\prime}),\ {\rm and}\ {\cal
C}(A_{1}^{\prime})=y^{\prime}.$

From the above computation we have only one solution, $y^{\prime}=(1+t)y$,
giving a trivial coloring for the $T_{m}$ part. In this case

${\cal C}(A_{2}^{\prime})={\cal C}(p_{\tau})=(1+t)y.$

We choose a quandle triple $({\cal C}(p_{\infty}),{\cal C}(p_{\iota}),{\cal
C}(p_{\tau}))=(a,b,c)=(1+t,0,1+t)$. Then in the above computation we need
$x=0$, ${\cal C}(p_{\tau})={\cal C}(R_{0}^{\prime})=(1+t)y=1+t$ so that $y=1$
to have a coloring. Let $\phi$ be the 2-cocycle mentioned above, and let
$\zeta$ be a multiplicative generator of the coefficient group
${\mathbb{Z}}_{2}$.

Let $p_{\infty}$ be in $R_{4}$. Then there is a unique coloring, and the
cocycle value for $\Phi_{\phi}^{(a,b,c)}(K,R,\gamma_{\iota})$ is

$\phi({\cal C}(R_{4}),{\cal
C}(A_{3}))^{-1}=\phi(1+t,1)^{-1}=\zeta^{-1}=\zeta.$

The negative exponent of $\phi$ is due to the sign of the crossing from
$R_{4}$ to $R$. For other paths, we compute

$\displaystyle\Phi_{\phi}^{(a,b,c)}(K,R,\gamma_{\tau})$ $\displaystyle=$
$\displaystyle\phi({\cal C}(R_{4}),{\cal C}(A_{4}))=\phi(1+t,1+t)=1,$
$\displaystyle\Phi_{\phi}^{(a,b,c)}(K,R,\gamma_{\iota\tau})$ $\displaystyle=$
$\displaystyle\phi({\cal C}(A_{3}),{\cal C}(R_{4}))^{-1}\phi({\cal
C}(R_{4}),{\cal C}(A_{4}))=\phi(1+t,1)^{-1}\phi((1+t,1+t)=\zeta,$

so that we obtain $\Phi_{\phi}^{(a,b,c)}(K,R_{4})=(\zeta,1,\zeta)$.

Let $p_{\infty}$ be in $R_{1}^{\prime}$. Then there is a unique coloring,
which is the same as above. Since the coloring is trivial on the $T_{m}$ part,
the cocycle value is trivial for all paths. Hence we have
$\Phi_{\phi}^{(a,b,c)}(K,R_{4})=(1,1,1)$. Since there are infinitely many
choices for $n=4+6k$, the claim follows.

For both regions we verified that the number of colorings is one, therefore it
does not detect planar nonequivalence. It remains to show that the family
consists of distinct spherical equivalence classes. It is sufficient to show
this for $T_{n}$ for various $n=4+6k$. We use the colorings with colors of end
points specified, without region colors, to distinguish spherical equivalence.

By computations in Example 3.3, $T_{n}$ is colored nontrivially by the
dihedral quandle ${\bf R}_{\ell}$ with colors ${\cal C}(A_{0})=x$ and ${\cal
C}(A_{n-1})=y$ if and only if $(n-1)(x-y)=0$. Thus there is a nontrivial
coloring if and only if $n-1=0\in{\mathbb{Z}}_{\ell}$. This is satisfied if
$n=\ell+1$ and is not satisfied if $n<\ell+1$. We consider specified colors
${\cal C}(p_{\iota})=0$ and ${\cal C}(p_{\tau})=\ell+1$. For $n=4$, we choose
$\ell=3$, then $T_{n}$ has non-trivial coloring and $T_{n}$ does not for all
$n>3$. Inductively by choosing $\ell=5+6k$ for $k\in{\mathbb{N}}$, we obtain a
spherically nonequivalent family. ∎

##  8\. Concluding remarks

A natural question arises as to the interpretation of end specified quandle
invariants in terms of quandle homology theory. Since the boundary colors are
related to the boundary operator of second chain groups, such an
interpretation is expected. Applications to quandle homology from this point
of view would be an interesting topic of further research.

We also note that the invariants discussed in this paper extend naturally to
linkoids [gabrovvsek23]. Linkoids are the natural generalization of knotoids
in which one allows for several knotted components, either closed or with
endpoints. With the exception of the triangular quandle cocylcle invariants,
whose construction relies on there being exactly two endpoints, the end
specified quandle invariants can be defined analogously for linkoids. Moreover
the triangular quandle cocycle invariants extend to ‘multi-knotoids’, which
are linkoids with only a single open-ended component.

Consequently the scheme for deriving $\mathbb{R}$-valued entanglement measures
of open curves from the terms of ${\cal Col}_{X}(K,R_{\infty})$, as discussed
in Section 1, can be applied equally well to systems of several open curves.
The resulting entanglement measures can then be used to detect the planar
characteristics of the linkoids obtained from projections of these open
curves. This is a feature that is not shared by the other entanglement
measures studied so far, $\mathbb{R}$-valued or otherwise. Hence another
direction for future research is the implementation of these entanglement
measures for systems of open curves, to investigate the influence of this
planar characteristic on the topology of physical examples of tangled open
curve systems.

Acknowledgement We would like to thank Nataša Jonoska for organizing Workshop:
discussions of DNA knots, held at University of South Florida in January
22-24, 2023, where this work was initiated. Thanks also go to the participants
for valuable discussions. Mohamed Elhamdadi was partially supported by Simons
Foundation collaboration grant 712462.

## References

  * [1]
AsamiSoichiroSatohShinAn infinite family of non-invertible surfaces in
4-spaceThe Bulletin of the London Mathematical
Society3720052285–296@article{AsamiSatoh, author = {Asami, Soichiro}, author =
{Satoh, Shin}, title = {An infinite family of non-invertible surfaces in
4-space}, journal = {The Bulletin of the London Mathematical Society}, volume
= {37}, year = {2005}, number = {2}, pages = {285–296}}

  * [3]
Double branched covers of knotoidsBarbensiAgneseBuckDorothyHarringtonHeather
ALackenbyMarcarXiv preprint arXiv:1811.091212018@article{barbensi, title =
{Double branched covers of knotoids}, author = {Barbensi, Agnese}, author =
{Buck, Dorothy}, author = {Harrington, Heather A}, author = {Lackenby, Marc},
journal = {arXiv preprint arXiv:1811.09121}, year = {2018}}

  * [5]
CarterJ. ScottElhamdadiMohamedNikiforouMarina AppiouSaitoMasahicoExtensions of
quandles and cocycle knot invariantsJournal of Knot Theory and its
Ramifications1220036725–738@article{CENS, author = {Carter, J. Scott}, author
= {Elhamdadi, Mohamed}, author = {Nikiforou, Marina Appiou}, author = {Saito,
Masahico}, title = {Extensions of quandles and cocycle knot invariants},
journal = {Journal of Knot Theory and its Ramifications}, volume = {12}, year
= {2003}, number = {6}, pages = {725–738}}

  * [7]
Quandle cohomology and state-sum invariants of knotted curves and
surfacesCarterJ
ScottJelsovskyDanielKamadaSeiichiLangfordLaurelSaitoMasahicoTransactions of
the American Mathematical Society355103947–39892003@article{CJKLS, title =
{Quandle cohomology and state-sum invariants of knotted curves and surfaces},
author = {Carter, J Scott}, author = {Jelsovsky, Daniel}, author = {Kamada,
Seiichi}, author = {Langford, Laurel}, author = {Saito, Masahico}, journal =
{Transactions of the American Mathematical Society}, volume = {355}, number =
{10}, pages = {3947–3989}, year = {2003}}

  * [9]
CarterJ. ScottJelsovskyDanielKamadaSeiichiSaitoMasahicoComputations of quandle
cocycle invariants of knotted curves and surfacesAdv.
Math.1572001136–94@article{CJKS, author = {Carter, J. Scott}, author =
{Jelsovsky, Daniel}, author = {Kamada, Seiichi}, author = {Saito, Masahico},
title = {Computations of quandle cocycle invariants of knotted curves and
surfaces}, journal = {Adv. Math.}, volume = {157}, year = {2001}, number =
{1}, pages = {36–94}}

  * [11]
CarterJ. ScottKamadaSeiichiSaitoMasahicoGeometric interpretations of quandle
homologyJournal of Knot Theory and its
Ramifications1020013345–386@article{CKS-Geom, author = {Carter, J. Scott},
author = {Kamada, Seiichi}, author = {Saito, Masahico}, title = {Geometric
interpretations of quandle homology}, journal = {Journal of Knot Theory and
its Ramifications}, volume = {10}, year = {2001}, number = {3}, pages =
{345–386}}

  * [13]
CarterJ. ScottKamadaSeiichiSaitoMasahicoSurfaces in 4-spaceEncyclopaedia of
Mathematical Sciences142Low-Dimensional Topology, IIISpringer-Verlag,
Berlin2004xiv+213@book{CKS, author = {Carter, J. Scott}, author = {Kamada,
Seiichi}, author = {Saito, Masahico}, title = {Surfaces in 4-space}, series =
{Encyclopaedia of Mathematical Sciences}, volume = {142}, note = {Low-
Dimensional Topology, III}, publisher = {Springer-Verlag, Berlin}, year =
{2004}, pages = {xiv+213}}

  * [15]
CazetNicholasThe shadow quandle cocycle invariant of knotoidsarXiv:2207.02330
[math.GT]2022@article{Cazet, author = {Cazet, Nicholas}, title = {The shadow
quandle cocycle invariant of knotoids}, journal = {arXiv:2207.02330
[math.GT]}, year = {2022}}

  * [17]
ClarkW. EdwinDunningLarry A.SaitoMasahicoComputations of quandle 2-cocycle
knot invariants without explicit 2-cocyclesJournal of Knot Theory and its
Ramifications26201771750035, 22@article{CDS, author = {Clark, W. Edwin},
author = {Dunning, Larry A. }, author = {Saito, Masahico}, title =
{Computations of quandle 2-cocycle knot invariants without explicit
2-cocycles}, journal = {Journal of Knot Theory and its Ramifications}, volume
= {26}, year = {2017}, number = {7}, pages = {1750035, 22}}

  * [19]
FennRogerRourkeColinRacks and links in codimension twoJournal of Knot Theory
and Its Ramifications141992343–406@article{FR, author = {Fenn, Roger}, author
= {Rourke, Colin}, title = {Racks and Links in Codimension Two}, journal =
{Journal of Knot Theory and Its Ramifications}, volume = {1}, number = {4},
date = {1992}, pages = {343–406}}

  * [21]
Invariants of multi-linkoidsGabrovšekBoštjanGügümcüNeslihanMediterranean
Journal of Mathematics2031652023Springer@article{gabrovvsek23, title =
{Invariants of multi-linkoids}, author = {Gabrov{\v{s}}ek, Bo{\v{s}}tjan},
author = {G{\"u}g{\"u}mc{\"u}, Neslihan}, journal = {Mediterranean Journal of
Mathematics}, volume = {20}, number = {3}, pages = {165}, year = {2023},
publisher = {Springer}}

  * [23]
A systematic classification of knotoids on the plane and on the
sphereGoundaroulisDimosDorierJulienStasiakAndrzejarXiv preprint
arXiv:1902.072772019@article{systematic, title = {A systematic classification
of knotoids on the plane and on the sphere}, author = {Goundaroulis, Dimos},
author = {Dorier, Julien}, author = {Stasiak, Andrzej}, journal = {arXiv
preprint arXiv:1902.07277}, year = {2019}}

  * [25]
New invariants of knotoidsGügümcüNeslihanKauffmanLouis H.European Journal of
Combinatorics65186–2292017Elsevier@article{newinvariants, title = {New
invariants of knotoids}, author = {G\"{u}g\"{u}mc\"{u}, Neslihan}, author =
{Kauffman, Louis H.}, journal = {European Journal of Combinatorics}, volume =
{65}, pages = {186–229}, year = {2017}, publisher = {Elsevier}}

  * [27]
GügümcüNeslihanKauffmanLouis H.Parity, virtual closure and minimality of
knotoidsJournal of Knot Theory and its Ramifications30202111Paper No. 2150076,
28@article{GK, author = {G\"{u}g\"{u}mc\"{u}, Neslihan}, author = {Kauffman,
Louis H.}, title = {Parity, virtual closure and minimality of knotoids},
journal = {Journal of Knot Theory and its Ramifications}, volume = {30}, year
= {2021}, number = {11}, pages = {Paper No. 2150076, 28}}

  * [29]
GügümcüNeslihanNelsonSamBiquandle coloring invariants of knotoidsJournal of
Knot Theory and its Ramifications28201941950029, 18@article{GN, author =
{G\"{u}g\"{u}mc\"{u}, Neslihan}, author = {Nelson, Sam}, title = {Biquandle
coloring invariants of knotoids}, journal = {Journal of Knot Theory and its
Ramifications}, volume = {28}, year = {2019}, number = {4}, pages = {1950029,
18}}

  * [31]
GügümcüNeslihanPflumeRunaPointed quandles of
knotoidsarXiv:2404.140832024@article{GR, author = {G\"{u}g\"{u}mc\"{u},
Neslihan}, author = {Pflume, Runa}, title = {Pointed quandles of knotoids},
journal = {arXiv:2404.14083}, year = {2024}}

  * [33]
HashimotoYuTanakaKokoroShifting chain maps in quandle homology and cocycle
invariantsTransactions of the American Mathematical
Society3752022107261–7276@article{HashiTana, author = {Hashimoto, Yu}, author
= {Tanaka, Kokoro}, title = {Shifting chain maps in quandle homology and
cocycle invariants}, journal = {Transactions of the American Mathematical
Society}, volume = {375}, year = {2022}, number = {10}, pages = {7261–7276}}

  * [35]
JoyceDavidA classifying invariant of knots, the knot quandleJournal of Pure
and Applied Algebra231982137–65@article{Joyce, author = {Joyce, David}, title
= {A classifying invariant of knots, the knot quandle}, journal = {Journal of
Pure and Applied Algebra}, volume = {23}, year = {1982}, number = {1}, pages =
{37–65}}

  * [37]
Rail knotoidsKodokostasDimitriosLambropoulouSofiaJournal of Knot Theory and
Its Ramifications281319400192019World Scientific@article{rail, title = {Rail
knotoids}, author = {Kodokostas, Dimitrios}, author = {Lambropoulou, Sofia},
journal = {Journal of Knot Theory and Its Ramifications}, volume = {28},
number = {13}, pages = {1940019}, year = {2019}, publisher = {World
Scientific}}

  * [39]
Winding homology of knotoidsKutluayDenizarXiv preprint
arXiv:2002.078712020@article{winding, title = {Winding homology of knotoids},
author = {Kutluay, Deniz}, journal = {arXiv preprint arXiv:2002.07871}, year =
{2020}}

  * [41]
MatveevS. V.Distributive groupoids in knot theoryMatematicheskiĭ Sbornik.
Novaya Seriya119(161)1982178–88, 160@article{Matveev, author = {Matveev, S.
V.}, title = {Distributive groupoids in knot theory}, journal =
{Matematicheski\u{\i} Sbornik. Novaya Seriya}, volume = {119(161)}, year =
{1982}, number = {1}, pages = {78–88, 160}}

  * [43]
MochizukiTakuroSome calculations of cohomology groups of finite Alexander
quandlesJournal of Pure and Applied Algebra17920033287–330@article{Mochi,
author = {Mochizuki, Takuro}, title = {Some calculations of cohomology groups
of finite {A}lexander quandles}, journal = {Journal of Pure and Applied
Algebra}, volume = {179}, year = {2003}, number = {3}, pages = {287–330}}

  * [45]
New quantum invariants of planar knotoidsMoltmakerWoutvan der
VeenRolandCommunications in Mathematical
Physics4021695–7222023Springer@article{moltmaker, title = {New Quantum
Invariants of Planar Knotoids}, author = {Moltmaker, Wout}, author = { van der
Veen, Roland}, journal = {Communications in Mathematical Physics}, volume =
{402}, number = {1}, pages = {695–722}, year = {2023}, publisher = {Springer}}

  * [47]
PanagiotouEleniKauffmanLouis H.Knot polynomials of open and closed
curvesProceedings A4762020224020200124, 20@article{Eleni, author =
{Panagiotou, Eleni}, author = {Kauffman, Louis H.}, title = {Knot polynomials
of open and closed curves}, journal = {Proceedings A}, volume = {476}, year =
{2020}, number = {2240}, pages = {20200124, 20}}

  * [49]
Vassiliev measures of complexity of open and closed curves in
3-spacePanagiotouEleniKauffmanLouis H.Proceedings of the Royal Society
A4772254202104402021The Royal Society@article{Eleni2, title = {Vassiliev
measures of complexity of open and closed curves in 3-space}, author =
{Panagiotou, Eleni}, author = {Kauffman, Louis H.}, journal = {Proceedings of
the Royal Society A}, volume = {477}, number = {2254}, pages = {20210440},
year = {2021}, publisher = {The Royal Society}}

  * [51]
PrzytyckiJózef H.$3$-coloring and other elementary invariants of knotsKnot
theory ({W}arsaw, 1995)Banach Center Publ.42275–295@article{Jozef, author =
{Przytycki, J\'{o}zef H.}, title = {{$3$}-coloring and other elementary
invariants of knots}, booktitle = {Knot theory ({W}arsaw, 1995)}, series =
{Banach Center Publ.}, volume = {42}, pages = {275–295}}

  * [53]
@book{Rolf}

  * AUTHOR = Rolfsen, Dale, TITLE = Knots and links, NOTE = Mathematics Lecture Series, No. 7, PUBLISHER = Publish or Perish, Inc., Berkeley, Calif., YEAR = 1976, PAGES = ix+439,
  * [55]
TuraevVladimirKnotoidsOsaka Journal of
Mathematics4920121195–223@article{Turaev, author = {Turaev, Vladimir}, title =
{Knotoids}, journal = {Osaka Journal of Mathematics}, volume = {49}, year =
{2012}, number = {1}, pages = {195–223}}

  * [57]