# Crossing tribes of tangles in a thickened surface

Igor Nikonov

###### Abstract

Crossings of knot diagrams can be divided into classes (tribes) compatible
with Reidemeister moves. Tribes can be considered as localization of the
notion of weak chord index introduced by M. Xu. In the article we describe
tribes of crossings for tangles in a fixed surface and show that crossings are
differentiated by their component, order and homotopy types. As a consequence,
we conclude that there are no nontrivial indices on diagrams of classical
knots.

## Introduction

In 2004 V. Turaev [23] assigned an index $n(e)$ to each crossing $e$ of a flat
knot diagram. Using the index, he defined a polynomial invariant (index
polynomial), see also [24]. This polynomial was numerously rediscovered in
various forms afterwards [11, 2, 15, 8, 17]. A. Henrich [10] presented indices
valued in flat link diagrams. Other types of indices can be found in [7, 13,
3, 16, 5]. The notion of index was axiomatized by Z. Cheng [4], and finally,
M. Xu [25] found the general conditions for an index to define an invariant
polynomial.

In [21, 22] we localized the notion of index and defined tribes as classes of
crossings compatible in some sense with the Reidemeister moves (see Definition
2). It follows from the definition that crossings which belong to one tribe
have the same value for any (weak chord) index. The aim of this paper is to
describe tribe of crossings and the universal index for knots, links and
tangles in a fixed surface.

The paper is organized as follows. Section 1 contains the main definitions of
tangles, tribes and indices. Using technics of [19], in Section 2 we prove the
main theorem of the paper which shows that a tribe of crossings is determined
by the component, order and homotopy types. This result allows to give the
negative answer to the question [5, 6] whether nontrivial indices for
classical knots exist (we postpone this conclusion to Corollary 5 in Section
4.4). Section 3 is devoted to description of tribes for flat tangles in the
surface. There we also describe phratries of crossings which in some sense
substitute for the sign function of crossings in the situation when we don’t
have it. In Section 4 we find the universal index for tangles and flat tangles
in the surface. It appears that in order to pass from tribes to the index we
need to factorize the homotopy type by the action of the inner monodromy
group. We finish the paper with Section 5 where we describe the universal
index for diagrams of minimal genus of virtual and flat links.

##  1 Definitions

Let $F$ be an oriented compact connected surface with the boundary $\partial
F$ (perhaps empty).

######  Definition 1.

A _(oriented) tangle_ in the thickened surface $F\times(0,1)$ is an embedding
$T\colon M\to F\times(0,1)$ of an oriented compact $1$-dimensional manifold
$M$ such that $T(\partial M)\subset\partial F\times(0,1)$ and $T$ is
transversal to $\partial F$. The image of the map $T$ will be also called a
tangle.

A connected component of a tangle whose boundary is empty is called a _closed
component_ , otherwise, the components is called _long_.

If a tangle consists of one closed component then it is a _knot_ ; a tangle
consisting of one long component is a _long knot_ ; a tangle with several
component which are all closed, is a _link_.

We consider tangles up to isotopies of $F\times(0,1)$ with the boundary
$\partial F\times(0,1)$ fixed.

A _tangle diagram_ $D$ is the image of a tangle $T$ in general position under
the natural projection $F\times(0,1)\to F$. Combinatorially, a tangle diagram
$D$ is an embedded graph in $F$ with vertices of valency $4$ (called
_crossings_) and vertices of valency $1$ (forming the _boundary_ $\partial D$
of the diagram $D$) such that $D\cap\partial F=\partial D$ and each vertex of
valency $4$ carries the structure of under-overcrossing (Fig. 1).

Figure 1: A tangle diagram

Any tangle diagrams which correspond to isotopical tangles can be connected by
a sequence of isotopies of the surface $F$ identical on $\partial F$, and
_Reidemeister moves_ (Fig. 2).

1em

Figure 2: Reidemeister moves

Let $D$ be a tangle diagram. Denote the set of its crossings by
$\mathcal{V}(D)$. We assume that all tangle diagrams are oriented, hence, any
crossing $v\in\mathcal{V}(D)$ has a _sign_ (Fig. 3).

Figure 3: The sign of a crossing

Let $D=D_{1}\cup\cdots\cup D_{n}$ be a diagram of a tangle $T$ and $v$ be a
self-crossing of some component $D_{i}$. Then the oriented smoothing at $v$
splits the component $D_{i}$ into two the _left half_ $D^{l}_{v}$ and the
_right half_ $D^{r}_{v}$ (Fig. 4). We define also the _signed halves_
$D^{\pm}_{v}$ by the formula $D^{sgn(v)}_{v}=D^{l}_{v}$ and
$D^{-sgn(v)}_{v}=D^{r}_{v}$.

Figure 4: The left and the right halves of the diagram

Note that $D^{+}_{v}$ is the half of $D_{i}$ from the undercrossing of $v$ to
the overcrossing of $v$, and $D^{-}_{v}$ is the half from the overcrossing of
$v$ to the undercrossing of $v$.

###  1.1 Tribes and indices

Let $T$ be a tangle and $\mathfrak{T}$ be the set of its diagram. The set
$\mathfrak{T}$ can be considered as the objects of a diagram category whose
morphisms are compositions of isotopies and Reidemeister moves.

Given tangle diagrams $D$ and $D^{\prime}$ and a morphism $f\colon D\to
D^{\prime}$ between them, there is a correspondence
$f_{*}\colon\mathcal{V}(D)\to\mathcal{V}(D^{\prime})$ between the crossings of
the diagrams. The map $f_{*}$ is a partial bijection between the crossing sets
because some crossing can disappear or appear when $f$ is a first or second
Reidemeister move.

######  Remark 1.

Below we will often denote the crossing $f_{*}(v)$ in the diagram $D^{\prime}$
which corresponds to a crossing $v$ in $D$, by the same letter $v$ when the
morphism $f$ is clear from the context.

Recall the notion of a tribal system [21, 22].

######  Definition 2.

Let $T$ be a tangle and $\mathfrak{T}$ be the category of its diagrams. A
_tribal system_ on diagrams of the tangle $T$ is a family of partitions
$\mathfrak{C}(D)$ of the sets $\mathcal{V}(D)$, $D\in\mathfrak{T}$, into
subsets $C\in\mathfrak{C}(D)$ called _tribes_ such that

  * (T0)

Let $f\colon D\to D^{\prime}$ be a Reidemeister move,
$v_{1},v_{2}\in\mathcal{V}(D)$ and $v^{\prime}_{1}=f_{*}(v_{1})$ and
$v^{\prime}_{2}=f_{*}(v_{2})$ be the correspondent crossings in $D^{\prime}$.
Then if $v_{1}$ and $v_{2}$ belong to one tribe in $\mathfrak{C}(D)$ then
$v^{\prime}_{1}$ and $v^{\prime}_{2}$ also belong to one tribe in
$\mathfrak{C}(D^{\prime})$;

  * (T2)

If $f\colon D\to D^{\prime}$ is a decreasing second Reidemeister move and
$v_{1}$ and $v_{2}$ are the disappearing crossings then $v_{1}$ and $v_{2}$
belong to one tribe in $\mathfrak{C}(D)$.

Below in the paper, we will always consider the _finest tribal system_
$\mathfrak{C}^{u}$ on the diagram category $\mathfrak{T}$, i.e. the
equivalence relations on the crossing sets $\mathcal{V}(D)$,
$D\in\mathfrak{T}$, generated by the properties (T0) and (T2).

For crossings $v_{1},v_{2}\in\mathcal{V}(D)$, $D\in\mathfrak{T}$, we denote
$v_{1}\sim v_{2}$ if the crossings $v_{1},v_{2}$ belong to one tribe.

The sign function splits any tribe $C\in\mathfrak{C}(D)$, $D\in\mathfrak{T}$,
into two subsets $C^{+}$ and $C^{-}$ we call _phratries_. By definition

$C^{\pm}=\\{v\in C\mid sgn(v)=\pm 1\\}.$

The phratries $C^{+}$ and $C^{-}$ are called _dual_.

The notion of tribes is a localization of the notion a (weak chord) index
defined by M. Xu [25] (see also [4] for the original axiomatic definition of
chord index, and [5, 6] for examples of chord indices).

######  Definition 3.

Let $I$ be an arbitrary set. An _index on the diagrams of a tangle $T$ in the
surface $F$ with coefficients in the set $I$_ is a map $\iota$ which assigns
some value $\iota(v)$ to any crossing $v$ in any diagram $D$ of the tangle $T$
and possesses the following properties:

  * (I0)

for any Reidemeister move $f\colon D\to D^{\prime}$ and any crossings
$v\in\mathcal{V}(D)$ and $v^{\prime}\in\mathcal{V}(D^{\prime})$ such that
$v^{\prime}=f_{*}(v)$, one has $\iota(v)=\iota(v^{\prime})$;

  * (I2)

$\iota(v_{1})=\iota(v_{2})$ for any crossings $v_{1},\,v_{2}\in\mathcal{V}(D)$
to which a decreasing second Reidemeister move can be applied.

######  Example 1.

Let $K$ be a knot in the surface $F$. For any diagram $D$ of $K$ and any
crossing $v\in\mathcal{V}(D)$, define its index to be the intersection number
of two cycles: $Ind(v)=D\cdot D^{+}_{v}\in\mathbb{Z}$. Then the map $Ind$
obeys the conditions (I0) and (I2).

The following statement follows from the definitions.

######  Proposition 1.

Let $\iota$ be an index on the diagrams of a tangle $T$ in the surface $F$
with coefficients in a set $I$. Let $v$ and $w$ be two vertices of a diagram
$D$ of $T$. If $v$ and $w$ belong to one tribe then $\iota(v)=\iota(w)$.

######  Definition 4.

An index $\iota^{u}$ on the diagrams of a tangle $T$ in the surface $F$ with
coefficients in a set $I^{u}$ is called the _universal index_ on diagrams of
the tangle $T$ if for any index $\iota$ on $T$ with coefficients in a set $I$
there is a unique map $\psi\colon I^{u}\to I$ such that
$\iota=\psi\circ\iota^{u}$.

The main result of the paper is a description of the tribes (and phratries) of
the finest tribal system $\mathfrak{C}$ on diagrams of any tangle $T$ in the
surface $F$, and the universal index on $T$.

##  2 Crossing tribes of a tangle in the surface $F$

Let $T=K_{1}\cup K_{2}\cup\cdots\cup K_{n}$ be tangle with $n$ components in
the surface $F$ and $\mathfrak{T}$ be its diagram category.

######  Definition 5.

Let $D=D_{1}\cup\cdots\cup D_{n}$ be a diagram of the tangle $T$ and
$v\in\mathcal{V}(D)$ be a crossing of $D$. The _component type_ $\tau(v)$ of
the crossing $v$ is the pair $(i,j)$ where $D_{i}$ is the overcrossing
component at $v$ and $D_{j}$ is the undercrossing component at $v$ (Fig. 5).

Figure 5: A crossing of type $(i,j)$

Note that there are $n^{2}$ types of crossings.

The following statement is evident.

######  Proposition 2.

Let $v_{1},v_{2}\in\mathcal{V}(D)$, $D\in\mathfrak{T}$. If $v_{1}\sim v_{2}$
then $\tau(v_{1})=\tau(v_{2})$.

###  2.1 An auxiliary lemma

Let us formulate an auxiliary lemma.

######  Lemma 1.

Let $D$ be a tangle diagram in $F$. Let $a,b\in\mathcal{V}(D)$ be the ends of
two curves $\gamma_{1},\gamma_{2}\subset D$ such that (Fig. 6)

  1. 1.

$\gamma_{1}$ and $\gamma_{2}$ intersect transversely (i.e. they do not have
common edges);

  2. 2.

$\gamma_{2}$ overcrosses $\gamma_{1}$ at the crossings $a$ and $b$;

  3. 3.

$\gamma_{1}$ is homotopic to $\gamma_{2}$ in $F$ as a curve with fixed ends.

Figure 6: Paths $\gamma_{1}$ and $\gamma_{2}$

Then $a$ and $b$ belong to one tribe.

###### Proof.

The proof follows the reasoning of [19].

Without loss of generality, we assume that $\gamma_{1}$ and $\gamma_{2}$ are
oriented from $a$ to $b$.

Step 1. Pull the crossing $b$ to the crossing $a$ as shown in Fig. 7. The
pulling process consists second and third Reidemeister moves and defines a
morphism $f\colon D\to D^{\prime}$. The pulling contracts $\gamma_{1}$ and
transforms the path $\gamma_{2}$ to a path $\gamma^{\prime}_{2}$ in
$D^{\prime}$ which is homotopically equivalent to $\gamma_{2}\gamma_{1}^{-1}$.
Since $\gamma_{1}$ is homotopic to $\gamma_{2}$ then $\gamma^{\prime}_{2}$ can
be considered as a contractible loop.

Figure 7: Step 1: pulling the crossing $b$ to the crossing $a$

By the property (T0) $a\sim b$ in $D$ if and only if $a\sim b$ in
$D^{\prime}$. Thus, we reduce the situation to the case when $\gamma_{1}$ is a
small arc and $\gamma_{2}$ is a contractible loop in $F$ (up to contraction
$\gamma_{1}$ to a point).

Step 2. If $a$ and $b$ have the same sign we apply a first, a second and a
third Reidemeister move to create the crossing $c$ as shown in Fig. 8.

Figure 8: Step $2$: changing the crossing sign if necessary

By the property (T2) $b\sim c$ in $D^{\prime}$. Then it is suffice to prove
that $a\sim c$: in this case $a\sim c\sim b$ in $D^{\prime}$, hence $a\sim b$
in $D$ by the property (T0). Thus, below we will assume the $a$ and $b$ have
different signs.

Step 3. There is a homotopy which contracts $\gamma_{2}$ to a small arc with
ends $a$ and $b$ (Fig. 9).

Figure 9: Contracting homotopy

The contraction process can be considered as a homotopy of the curve in the
thickening $F\times(0,1)$. The homotopy can be represented as a sequence of
isotopies in $F\times(0,1)$ and self-intersections which look like a change of
under- and overcrossing in some crossing (see Fig. 10) after projection to
$F$.

Figure 10: Change of crossing

Then we replace each crossing change by an isotopy which adds a small loop
(see Fig. 11).

Figure 11: Adding a loop instead of crossing change

As a result we get an isotopy whose projection transforms the curve
$\gamma_{2}$ to a small arc with ”sprouts” (see Fig. 12 middle).

Figure 12: Contraction of $\gamma_{2}$

Then we apply second Reidemeister moves to each arc separating the bases of
sprouts (Fig. 12 right) and to the arc $ab$. Let us denote the appearing
crossings by $b_{1},a_{1},\dots,b_{n},a_{n}$. By the property (T2) we have
$b_{1}\sim a_{1}$, …, $b_{n}\sim a_{n}$.

Step 4. Let us prove that $a\sim b_{1}$, $a_{1}\sim b_{2}$,…, $a_{n}\sim b$.

Let $a$ and $b$ be the roots of a sprout which grows outside the small loop
which $\gamma_{2}$ was contracted to (Fig. 13 left). Then take the segment
$ab$ and pull it under the diagram along the sprout to its tip. Then in the
diagram obtained (Fig. 13 right) we have $a\sim b$ by the property (T2). And
by the property (T0), $a\sim b$ in the initial diagram.

Figure 13: A sprout growing outside

Let $a$ and $b$ be the roots of a sprout which grows outside the small loop
(Fig. 14 top left). Then pull the arc $ab$ up to the base of the sprout (Fig.
14 top left). For the appearing crossings $c,c^{\prime}$, $d,d^{\prime}$, we
have $c\sim c^{\prime}$ and $d\sim d^{\prime}$ by the property (T2) (Fig. 14
bottom left). We pull the arc $ab$ to the beginning of the sprout and see
(Fig. 14 bottom right) that $a\sim c^{\prime}$ and $b\sim d^{\prime}$ by the
property (T2). Then $a\sim c$ and $b\sim d$. After pulling the arc $cd$ to the
tip of the sprout we get the equivalence $c\sim d$ by the property (T2) (Fig.
14 top right). Hence, $a\sim b$.

Figure 14: A sprout growing inside

Applying the reasoning above to every sprout, we get $a\sim b_{1}$, $a_{1}\sim
b_{2}$,…$a_{n}\sim b$. On the other hand, $b_{1}\sim a_{1}$, …, $b_{n}\sim
a_{n}$. Then $a\sim b$. The lemma is proved. ∎

###  2.2 Self-crossings of a long component

Let $D_{i}$ be a component of the tangle diagram $D$ such that $\partial
D_{i}\neq\emptyset$. Let $s\in\partial D_{i}$ be the beginning point of the
oriented component $D_{i}$.

######  Definition 6.

A self-crossing $v$ of the component $D_{i}$ is called an _early
undercrossing_ if we pass the undercrossing of $v$ before its overcrossing
while moving along the component $D_{i}$ from $s$. Otherwise, the crossing $v$
is an _early overcrossing_.

Figure 15: An early undercrossing (left) and an early overcrossing (right)

Define the _order type_ $o$ of the crossing $v$ to be $o(v)=-1$ if $v$ is an
early undercrossing, and $o(v)=+1$ if $v$ is an early overcrossing.

The following proposition is checked directly.

######  Proposition 3.

Let $D_{i}$ be a long component of the tangle diagram $D$ and $v_{1}$ and
$v_{2}$ be self-crossing of $D_{i}$ (i.e. $\tau(v_{1})=\tau(v_{2})=(i,i)$). If
$v_{1}\sim v_{2}$ then $o(v_{1})=o(v_{2})$.

######  Definition 7.

Let $v$ be a self-crossing $v$ of the component $D_{i}$. Let $\gamma$ be the
arc of $D_{i}$ from the beginning point $s$ to the first appearance of $v$,
and $\alpha$ be the arc from the first appearance of $v$ to the second (Fig.
16). The _homotopy type_ of the crossing $v$ is the homotopy class

$h(v)=[\gamma\alpha\gamma^{-1}]\in\pi_{1}(F,s).$ (1)

Figure 16: The homotopy type of a crossing $v$

######  Proposition 4.

Let $v$ and $w$ be self-crossings of the long component $D_{i}$ of one order
type, i.e. $\tau(v)=\tau(w)=(i,i)$ and $o(v)=o(w)$. Then $v\sim w$ if and only
if $h(v)=h(w)$.

###### Proof.

By definition the homotopy type of a crossing $v$ does not change under
isotopies and Reidemeister moves which do not eliminate $v$.

Let $v$ and $w$ be crossings to which a second Reidemeister move can be
applied (Fig. 17). Denote the arc of $D_{i}$ from the beginning point $s$ to
$v$ by $\gamma$, and the arc between the appearances of $w$ by $\alpha$. We
ignore the small arcs between $v$ and $w$ by contracting them to a point. Then
$h(v)=h(w)=[\gamma\alpha\gamma^{-1}]$.

Figure 17: Invariance of homotopy type $h$ under a second Reidemeister move

The invariance of the homotopy type established above means that all crossings
of one tribe have the same homotopy type. Thus, $v\sim w$ implies $h(v)=h(w)$.

Now, assume that $v$ and $w$ are early overcrossings and $h(v)=h(w)$. Pull the
undercrossings of $v$ and $w$ to the starting point $s$ and get a diagram
$D^{\prime}$ (Fig. 18). The pulling process does not change the homotopy types
$h(v)$ and $h(w)$. Assume that the undercrossing of $v$ preceeds the
undercrossing of $w$ in the component $D_{i}$. Denote the arc of $D^{\prime}$
between the overcrossing of $w$ and the undercrossing of $v$ by $\alpha$ and
the arc between the undercrossing of $v$ and the undercrossing of $w$ by
$\beta$.

Figure 18: Diagram $D^{\prime}$

Then $h(v)=[\alpha]$ and $h(w)=[\alpha\beta]$. Since $h(v)=h(w)$, we have
$[\beta]=1$, that is the arc $\beta$ is contractible. By Lemma 1, $v\sim w$ in
the diagram $D^{\prime}$. Then by the property (T0) $v\sim w$ in $D$.

The case of early undercrossings is proved analogously. ∎

###  2.3 Self-crossings of a closed component

Let $D_{i}$ be a closed component of the tangle diagram $D$, i.e. $\partial
D_{i}=\emptyset$.

Fix an arbitrary non-crossing point $z$ in $D_{i}$.

Let $v$ be a self-crossing on the component $D_{i}$. Denote the arc of $D_{i}$
which connects $z$ with the overcrossing of $v$ and does not contain the
undercrossing of $v$, by $\gamma^{o}_{v,z}$; the arc which connects $z$ with
the undercrossing of $v$ and does not contain the overcrossing of $v$, by
$\gamma^{u}_{v,z}$. Let $\alpha_{z,v}$ be the arc between the under- and
overcrossings of $v$ which does not include $z$ (Fig. 19). We orient the
curves $\gamma^{o}_{v,z}$ and $\gamma^{u}_{v,z}$ from $z$ to $v$, and the arc
$\alpha_{z,v}$ keeps the orientation of the component $D_{i}$.

Figure 19: Arcs $\gamma^{o}_{v,z}$, $\gamma^{u}_{v,z}$ and $\alpha_{z,v}$

Recall that $D^{+}_{v}$ is the half of $D_{i}$ from the undercrossing of $v$
to the overcrossing of $v$, and $D^{-}_{v}$ starts in the overcrossing of $v$
and ends in the undercrossing of $v$. The crossing $v$ is called a _$z$
-overcrossing_ if $z\in D^{+}_{v}$ (i.e. one passes first the overcrossing of
$v$ when moves from $z$ along $D_{i}$), and a _$z$ -undercrossing_ if $z\in
D^{-}_{v}$.

Denote the homotopy class of the component $D_{i}$ by
$\kappa_{z}=[D_{i}]\in\pi_{1}(F,z)$. Let $\bar{\pi}_{D}(F,z)$ be the quotient
of $\pi_{1}(F,z)$ by the adjoint action of the element $\kappa_{z}$. Then
$\bar{\pi}_{D}(F,z)$ consists of orbits
$\bar{x}=\\{\kappa_{z}^{n}x\kappa_{z}^{-n}\mid n\in\mathbb{Z}\\}$,
$x\in\pi_{1}(F,z)$.

######  Definition 8.

Let $v$ be a self-crossing of the component $D_{i}$. The loops
$\hat{D}^{+}_{v,z}=\gamma^{u}_{v,z}D^{+}_{v}(\gamma^{u}_{v,z})^{-1}$ and
$\hat{D}^{-}_{v,z}=\gamma^{o}_{v,z}D^{-}_{v}(\gamma^{o}_{v,z})^{-1}$ are
called the _based signed halves_ of the crossing $v$ with the base point $z$
(Fig. 20). Let us denote the homotopy classes of the based signed halves
$D^{\pm}_{v,z}$ in $\pi_{1}(F,z)$ by $\delta^{\pm}_{v,z}$.

Figure 20: Based signed halves

The orbit

$h_{D,z}(v)=\bar{\delta}^{+}_{v,z}\in\bar{\pi}_{D}(F,z)$ (2)

of the homotopy class of the based half $\hat{D}^{+}_{v,z}$ is called the
_homotopy type_ of the crossings $v$.

######  Lemma 2.

1\. Let $v$ be a self-crossing of a closed component $D_{i}$. If $v$ is a
$z$-overcrossing then $\delta^{-}_{v,z}\delta^{+}_{v,z}=\kappa_{z}$, if $v$ is
a $z$-undercrossing then $\delta^{+}_{v,z}\delta^{-}_{v,z}=\kappa_{z}$.

2\. Let $v$ and $w$ be a self-crossings of a closed component $D_{i}$ and
$z,z^{\prime}$ be points in $D_{i}$ which are not crossings. Then
$h_{D,z}(v)=h_{D,z}(w)$ if and only if
$h_{D,z^{\prime}}(v)=h_{D,z^{\prime}}(w)$.

###### Proof.

1\. Let $v$ be a $z$-overcrossing. Then
$D^{+}_{v}=(\gamma^{u}_{v,z})^{-1}\gamma^{o}_{v,z}$ and
$D^{-}_{v}=\alpha_{v,z}$. Hence,
$\delta^{+}_{v,z}=[\gamma^{u}_{v,z}(\gamma^{u}_{v,z})^{-1}\gamma^{o}_{v,z}(\gamma^{u}_{v,z})^{-1}]=[\gamma^{o}_{v,z}(\gamma^{u}_{v,z})^{-1}]$
and $\delta^{-}_{v,z}=[\gamma^{o}_{v,z}\alpha_{v,z}(\gamma^{o}_{v,z})^{-1}]$.
Thus,

$\delta^{-}_{v,z}\delta^{+}_{v,z}=[\gamma^{o}_{v,z}\alpha_{v,z}(\gamma^{o}_{v,z})^{-1}\gamma^{o}_{v,z}(\gamma^{u}_{v,z})^{-1}]=[\gamma^{o}_{v,z}\alpha_{v,z}(\gamma^{u}_{v,z})^{-1}]=\kappa_{z}.$

The proof for a $z$-undercrossing is analogous.

2\. Let $\gamma_{z,z^{\prime}}$ is an arc of $D_{i}$ from $z$ to $z^{\prime}$.
Then the formula $x\mapsto(\gamma_{z,z^{\prime}})^{-1}x\gamma_{z,z^{\prime}}$
defines an isomorphism $\phi\colon\pi_{1}(F,z)\to\pi_{1}(F,z^{\prime})$. Since
$\phi(\kappa_{z})=\kappa_{z^{\prime}}$, the isomorphism $\phi$ induces a
bijection $\bar{\phi}\colon\bar{\pi}_{D}(F,z)\to\bar{\pi}_{D}(F,z^{\prime})$.

Since the paths $\gamma^{u}_{v,z^{\prime}}$ and
$(\gamma_{z,z^{\prime}})^{-1}\gamma^{u}_{v,z}$ connects $z^{\prime}$ with $v$,
we have
$(\gamma_{z,z^{\prime}})^{-1}\gamma^{u}_{v,z}=\kappa_{z^{\prime}}^{p}\gamma^{u}_{v,z^{\prime}}$
for some $p\in\mathbb{Z}$. Then

$\bar{\phi}(h_{D,z}(v))=\overline{[(\gamma_{z,z^{\prime}})^{-1}\gamma^{u}_{v,z}D^{+}_{v}(\gamma^{u}_{v,z})^{-1}\gamma_{z,z^{\prime}}]}=\overline{[\kappa_{z^{\prime}}^{p}\gamma^{u}_{v,z^{\prime}}D^{+}_{v}(\gamma^{u}_{v,z^{\prime}})^{-1}\kappa_{z^{\prime}}^{-p}]}=\\\
\overline{[\gamma^{u}_{v,z^{\prime}}D^{+}_{v}(\gamma^{u}_{v,z^{\prime}})^{-1}]}=h_{D,z^{\prime}}(v).$

Analogously, $\bar{\phi}(h_{D,z}(w))=h_{D,z^{\prime}}(w)$. Since $\bar{\phi}$
is a bijection, $h_{D,z}(v)=h_{D,z}(w)$ if and only if
$h_{D,z^{\prime}}(v)=h_{D,z^{\prime}}(w)$. ∎

######  Proposition 5.

Let $v$ and $w$ be self-crossings of the closed component $D_{i}$. Then $v\sim
w$ if and only if $h_{D,z}(v)=h_{D,z}(w)$.

###### Proof.

1\. Let us show first that $v\sim w$ implies $h_{D,z}(v)=h_{D,z}(w)$. Define
an equivalence relation on the crossings of the component $K_{i}$ of the
tangle $T$ for all diagrams of $T$ by the formula: $v\sim_{h}w$ if and only if
$h_{D,z}(v)=h_{D,z}(w)$, $v,w\in\mathcal{V}(D_{i})\subset\mathcal{V}(D)$,
$D\in\mathfrak{T}$. By Lemma 2, the equivalence relation $\sim_{h}$ does not
depend on the choice of the point $z$.

Let us check that the relation $\sim_{h}$ obeys the properties (T0) and (T2).

Let $f\colon D\to D^{\prime}$ be an isotopy or a Reidemeister move. We can
assume that $f$ does not move the point $z$ (by Lemma 2 we can choose any
appropriate point). Then $\kappa_{z}=\kappa^{\prime}_{z}\in\pi_{1}(F,z)$.
Hence, the sets of orbits $\bar{\pi}_{D}(F,z)$ and
$\bar{\pi}_{D^{\prime}}(F,z)$ are identical. For any crossing $v$ in the
component $D_{i}$ which survives under the move $f$, we have
$\delta^{+}_{v,z}=(\delta^{\prime})^{+}_{f_{*}(v),z}\in\pi_{1}(F,z)$, hence,
$h_{D,z}(v)=h_{D^{\prime},z}(f_{*}(v))$ where $f_{*}(v)$ is the correspondent
vertex in $D^{\prime}_{i}$, and $(\delta^{\prime})^{+}_{f_{*}(v),z}$ is the
homotopy class of $(\widehat{D^{\prime}})^{+}_{f_{*}(v),z}$. Thus, the
condition $v\sim_{h}w$ is equivalent to the condition
$f_{*}(v)\sim_{h}f_{*}(w)$. So, the property (T0) is fulfilled.

Let $v,w$ be two self-crossings of $D_{i}$ to which a second Reidemeister move
can be applied (Fig. 21). Then
$[\hat{D}^{+}_{v,z}]=[\hat{D}^{+}_{w,z}]\in\pi_{1}(F,z)$. Hence,
$h_{D,z}(v)=h_{D,z}(w)$. Thus, the property (T2) holds.

Figure 21: Based signed halves $\hat{D}^{+}_{v,z}$ (blue) and
$\hat{D}^{+}_{w,z}$ (red)

Since, the equivalence relation $\sim$ is the finest with the properties (T0)
and (T2), the condition $v\sim w$ implies $v\sim_{h}w$, i.e.
$h_{D,z}(v)=h_{D,z}(w)$.

2\. Now, assume that $h_{D,z}(v)=h_{D,z}(w)$.

If $\delta^{+}_{v,z}=\delta^{+}_{w,z}$ for some base point $z$ then pull the
overcrossings of $v$ and $w$ to the point $z$ along the arcs
$\gamma^{u}_{v,z}$ and $\gamma^{u}_{w,z}$ and get a new diagram $D^{\prime}$
(Fig. 22).

Figure 22: Diagram $D^{\prime}$

Then $\delta^{+}_{v,z}=(\delta^{\prime})^{+}_{v,z}=[\alpha]$ and
$\delta^{+}_{w,z}=(\delta^{\prime})^{+}_{w,z}=[\alpha\beta]$. Since
$\delta^{+}_{v,z}=\delta^{+}_{w,z}$, the arc $\beta$ is contractible. Then
$v\sim w$ in $D^{\prime}$ by Lemma 1, and $v\sim w$ in $D$ by the property
(T0).

Assume then that
$\delta^{+}_{w,z}=\kappa_{z}^{p}\delta^{+}_{v,z}\kappa_{z}^{-p}$. We can
suppose that the undercrossing of $v$ is close to $z$ so
$\hat{D}^{+}_{v,z}=\alpha_{v,z}$ up to arcs in a contractible neighbourhood of
$z$ (Fig. 23 left). For brevity denote the arc $\alpha_{v,z}$ by $\alpha$.

Figure 23: Construction of the crossing $v_{1}$

Choose a point $z_{1}$ on $D_{i}$ near $z$ and pull a part of the segment
$zz_{1}$ along the loop $\alpha$ and create a crossing $v_{1}$ as shown in
Fig. 23 right. Then $(\delta^{\prime})^{+}_{v_{1},z^{\prime}}=[\alpha]$ and
$(\delta^{\prime})^{+}_{v,z^{\prime}}=[\alpha\alpha^{-1}\alpha\alpha^{-1}\alpha]=[\alpha]$.
Then as we proved above, $v\sim v_{1}$ in $D^{\prime}$. On the other hand,
$(\delta^{\prime})^{+}_{v,z}=\delta^{+}_{v,z}$ and
$(\delta^{\prime})^{+}_{v_{1},z}=[\kappa^{\prime}_{z}\alpha(\kappa^{\prime}_{z})^{-1}]=\kappa_{z}\delta^{+}_{v,z}\kappa_{z}^{-1}$.

Repeating the process, we will get a diagram $\tilde{D}$ with new crossings
$v_{1},v_{2},\dots,v_{p}$ such that $v_{i}\sim v_{i+1}$ in $\tilde{D}$ and
$\tilde{\delta}^{+}_{v_{i},z}=\kappa^{i}\delta^{+}_{v,z}\kappa^{-i}$. Hence,
$v_{p}\sim v$ in $\tilde{D}$. On the other hand,
$\tilde{\delta}^{+}_{v_{p},z}=\kappa^{i}\delta^{+}_{v,z}\kappa^{-i}=\delta^{+}_{w,z}=\tilde{\delta}^{+}_{w,z}$,
therefore, $v_{n}\sim w$ in $\tilde{D}$. Thus, $v\sim w$ in $\tilde{D}$ and
$v\sim w$ in $D$ by the property (T0).

The case $p<0$ can be proved analogously. ∎

###  2.4 Mixed crossings

Let $D_{i}$ and $D_{j}$ be two components of the tangle diagram $D$.

Choose a point $z_{i}$ in the component $D_{i}$ to be equal to the beginning
point $s_{i}$ if the component $D_{i}$ is not closed, and an arbitrary non
crossing point if $D_{i}$ is closed. By the same rule, choose a point $z_{j}$
in $D_{j}$.

Let $v$ be a crossing of the components $D_{i}$ and $D_{j}$. Denote the arc in
$D_{i}$ from $z_{i}$ to $v$ by $\gamma_{z_{i},v}$, and the arc in $D_{i}$ from
$z_{j}$ to $v$ by $\gamma_{z_{j},v}$ (Fig. 24)

Figure 24: Paths $\gamma_{z_{i},v}$ and $\gamma_{z_{j},v}$

Let $\kappa_{i}\in\pi_{1}(F,z_{i})$ (respectively,
$\kappa_{j}\in\pi_{1}(F,z_{j})$) be the homotopy class of $D_{i}$
(respectively, the homotopy class of $D_{j}$) if it is closed and $1$
otherwise.

Let $\pi_{1}(F,z_{i},z_{j})$ be the set of homotopy classes of continuous
paths in $F$ from $z_{i}$ to $z_{j}$. Then the fundamental group
$\pi_{1}(F,z_{i})$ acts on this set from the left, and the group
$\pi_{1}(F,z_{j})$ acts on it from the right. Consider the set
$\bar{\pi}_{D}(F,z_{i},z_{j})=\kappa_{i}\backslash\pi_{1}(F,z_{i},z_{j})/\kappa_{j}$
consisting of cosets

$\bar{x}=\\{\kappa_{i}^{p}x\kappa_{j}^{q}\mid p,q\in\mathbb{Z}\\},\
x\in\pi_{1}(F,z_{i},z_{j}).$

######  Definition 9.

Let $v$ be a crossing of the components $D_{i}$ and $D_{j}$. The _homotopy
type_ of the crossing $v$ is the element

$h_{D,z_{i},z_{j}}(v)=\overline{[\gamma_{z_{i},v}\gamma_{z_{j},v}^{-1}]}\in\bar{\pi}_{D}(F,z_{i},z_{j}).$
(3)

######  Proposition 6.

Let $v$ and $w$ be crossings of the components $D_{i}$ and $D_{j}$ with
$D_{i}$ overcrossing $D_{j}$. Then $v\sim w$ if and only if
$h_{D,z_{i},z_{j}}(v)=h_{D,z_{i},z_{j}}(w)$.

###### Proof.

Assume that $D_{i}$ is closed and $z_{i}$, $z_{i}^{\prime}$ be two non
crossing points in $D_{i}$. Denote the path from $z_{i}$ to $z_{i}^{\prime}$
by $\gamma_{z_{i},z_{i}^{\prime}}$. The map
$x\mapsto(\gamma_{z_{i},z_{i}^{\prime}})^{-1}x$ defines a bijection
$\phi\colon\pi_{1}(F,z_{i},z_{j})\to\pi_{1}(F,z_{i}^{\prime},z_{j})$ and a
bijection between the sets
$\bar{\phi}\colon\bar{\pi}_{D}(F,z_{i},z_{j})\to\bar{\pi}_{D}(F,z_{i}^{\prime},z_{j})$.

For a crossing $v$ of the components $D_{i}$ and $D_{j}$, we have
$\gamma_{z_{i}^{\prime},v}=\kappa_{i}^{p}(\gamma_{z_{i},z_{i}^{\prime}})^{-1}\gamma_{z_{i},v}$,
hence $\bar{\phi}(h_{D,z_{i},z_{j}}(v))=h_{D,z_{i}^{\prime},z_{j}}(v)$. Thus,
the condition $h_{D,z_{i},z_{j}}(v)=h_{D,z_{i},z_{j}}(w)$ is equivalent to the
condition $h_{D,z_{i}^{\prime},z_{j}}(v)=h_{D,z_{i}^{\prime},z_{j}}(w)$.
Analogously, we can change the point $z_{j}$ if the component $D_{j}$ is
closed.

For any crossings $v$ and $w$ of component type $(i,j)$ in a diagram $D$ of
the tangle $T$, denote $v\sim_{h}w$ if and only if
$h_{D,z_{i},z_{j}}(v)=h_{D,z_{i},z_{j}}(w)$. We prove that the equivalence
relation $\sim_{h}$ obeys (T0) and (T2).

Let $f\colon D\to D^{\prime}$ be an isotopy or a Reidemeister move. We can
suppose that $f$ does not move points $z_{i}$ and $z_{j}$ (otherwise choose
other base points). Then the sets $\bar{\pi}_{D}(F,z_{i},z_{j})$ and
$\bar{\pi}_{D^{\prime}}(F,z_{i},z_{j})$ coincide. The paths
$\gamma_{z_{i},v}\gamma_{z_{j},v}^{-1}$ in $D$ and
$\gamma^{\prime}_{z_{i},v}(\gamma^{\prime}_{z_{j},v})^{-1}$ in $D^{\prime}$
are homotopic. Hence, for any crossing $v$ of component type $(i,j)$ we have
$h_{D,z_{i},z_{j}}(v)=h_{D^{\prime},z_{i},z_{j}}(f_{*}(v))$ if the
correspondent crossings $f_{*}(v)$ in $D^{\prime}$ exists. Thus, the condition
$v\sim_{h}w$ in $D$ implies $f_{*}(v)\sim_{h}f_{*}(w)$ in $D$. The property
(T0) is fulfilled.

Let $v$ and $w$ be two crossings of component type $(i,j)$ to which one can
apply a second Reidemeister move (Fig. 25).

Figure 25: Mixed crossings in a second Reidemeister move

The paths $\gamma_{z_{i},v}\gamma_{z_{j},v}^{-1}$ and
$\gamma_{z_{i},w}\gamma_{z_{j},w}^{-1}$ are homotopic, hence $v\sim_{h}w$.
Thus, the property (T2) holds.

Since $\sim$ is the finest equivalence with the properties (T0) and (T2),
$v\sim w$ implies $v\sim_{h}w$, i.e.
$h_{D,z_{i},z_{j}}(v)=h_{D,z_{i},z_{j}}(w)$.

Now, let $h_{D,z_{i},z_{j}}(v)=h_{D,z_{i},z_{j}}(w)$. Assume first that
$[\gamma_{z_{i},v}\gamma_{z_{j},v}^{-1}]=[\gamma_{z_{i},w}\gamma_{z_{j},w}^{-1}]$.

Pull the crossings $v$ and $w$ to the point $z_{i}$ along the paths
$\gamma_{z_{i},v}$ and $\gamma_{z_{i},w}$ and get a diagram $D^{\prime}$ (Fig.
26). Let $\gamma$ be the arc in $D_{j}$ from $z_{j}$ to $w$ and $\alpha$ be
the arc from $w$ to $v$.

Figure 26: Diagram $D^{\prime}$

Then $[\gamma_{z_{i},v}\gamma_{z_{j},v}^{-1}]=[\gamma\alpha]$ and
$\gamma_{z_{i},w}\gamma_{z_{j},w}^{-1}]=[\gamma]$. By assumption,
$[\gamma\alpha]=[\gamma]$, hence, $\alpha$ is contractible. Then $v\sim w$ by
Lemma 1.

Let
$[\kappa_{i}^{p}\gamma_{z_{i},v}\gamma_{z_{j},v}^{-1}]=[\gamma_{z_{i},w}\gamma_{z_{j},w}^{-1}]$,
$p\in\mathbb{Z}$. Then pull the crossings $v$ and $w$ to the point $z_{i}$
along the paths $\kappa_{i}^{p}\gamma_{z_{i},v}$ and $\gamma_{z_{i},w}$, and
apply the arguments of the previous case. Then we get $v\sim w$. Analogously,
the equality
$[\gamma_{z_{i},v}\gamma_{z_{j},v}^{-1}\kappa_{j}^{q}]=[\gamma_{z_{i},w}\gamma_{z_{j},w}^{-1}]$,
$q\in\mathbb{Z}$ implies $v\sim w$.

In general case,
$[\kappa_{i}^{p}\gamma_{z_{i},v}\gamma_{z_{j},v}^{-1}\kappa_{j}^{q}]=[\gamma_{z_{i},w}\gamma_{z_{j},w}^{-1}]$,
$p,q\in\mathbb{Z}$. With a second Reidemeister move, create a crossing $v_{1}$
near $v$ such that $[\gamma_{z_{i},v_{1}}]=[\kappa_{i}^{p}\gamma_{z_{i},v}]$
and $[\gamma_{z_{j},v_{1}}]=[\gamma_{z_{j},v}]$ (we identify homotopy classes
of paths from $z_{i}$ and $z_{j}$ to $v$ and $v_{1}$ by merging $v$ with
$v_{1}$), see Fig. 27. Then $v\sim v_{1}$ and $v_{1}\sim w$ by the previous
cases. Thus, $v\sim w$.

Figure 27: Crossing $v_{1}$

∎

######  Remark 2.

Instead of the element $h_{D,z_{i},z_{j}}(v)$ we can take the element

$h_{D,z_{j},z_{i}}(v)=\overline{[\gamma_{z_{j},v}\gamma_{z_{i},v}^{-1}]}\in\bar{\pi}_{D}(F,z_{j},z_{i})$

where
$\bar{\pi}_{D}(F,z_{j},z_{i})=\kappa_{j}\backslash\pi_{1}(F,z_{j},z_{i})/\kappa_{i}$,
as the homotopy type of a mixed crossing $v$. Since the classes
$h_{D,z_{i},z_{j}}(v)$ and $h_{D,z_{j},z_{i}}(v)$ are inverse to each other,
the condition $h_{D,z_{i},z_{j}}(v)=h_{D,z_{i},z_{j}}(w)$ is equivalent to the
condition $h_{D,z_{j},z_{i}}(v)=h_{D,z_{j},z_{i}}(w)$.

###  2.5 Main theorem

We can summarize Propositions 4, 5 and 6 in the following theorem.

######  Theorem 1.

Let $T$ be an oriented tangle in the surface $F$ and $D$ be its diagram. Then
two crossings $v$ and $w$ of the diagram $D$ belong to one tribe if and only
if they have identical component type, order type (if $v$ and $w$ are self-
crossings of a long component) and homotopy type.

For the phratries of the crossings of a tangle we get immediately the
following description.

######  Corollary 1.

Let $T$ be an oriented tangle in the surface $F$ and $D$ be its diagram. Then
two crossings $v$ and $w$ of the diagram $D$ belong to one phratry if and only
if they have identical signs, component type, order type (if $v$ and $w$ are
self-crossings of a long component) and homotopy type.

######  Corollary 2.

Let $K$ be a classical knot. Then all the crossings of a diagram of $K$ belong
to one tribe.

###### Proof.

For a classical knots we have $F=\mathbb{R}^{2}$ or $S^{2}$. In both cases
$\pi_{1}(F,z)=1$, hence, the homotopy type $h$ is trivial. For a knot, the
component type $\tau$ is trivial too. And we have no order type. Thus, all
crossing belong to one tribe. ∎

We postpone the further corollaries of Theorem 1 to Section 4.4.

##  3 Tribes and phratries of flat tangles in the surface $F$

_Flat tangles_ in the surface $F$ can be defined as immersions of an oriented
compact 1-manifold into the thickened surface $F\times(0,1)$ up to homotopies
fixed on the boundary $\partial F\times(0,1)$. Combinatorially, one defines
flat tangles as equivalence classes of tangle diagrams modulo isotopies,
Reidemeister moves (Fig. 2) and crossing changes (Fig. 10).

Flat tangle diagrams are obtained from tangle diagrams by forgetting under-
overcrossing structure (Fig. 28). A flat diagram can be considered as the
equivalence class of a tangle diagram modulo crossing change operations
applied to the crossings of the diagram. We identify the crossing sets of all
the diagrams in this equivalence class.

Figure 28: A flat tangle diagram

######  Proposition 7.

Let $D=D_{1}\cup\cdots\cup D_{n}$ be a diagram of a tangle $T$ in the surface
$F$ and $v\in\mathcal{V}(D)$ be a crossing. After the crossing change applied
to $v$ we get a tangle diagram $\bar{D}$ with the same set of crossing as $D$.

  1. 1.

Let $w\in\mathcal{V}(D)=\mathcal{V}(\bar{D})$ be a crossing different from
$v$. Then the component, order and homotopy types $\tau_{\bar{D}}(w)$,
$o_{\bar{D}}(w)$, $h_{\bar{D}}(w)$ of the crossing $w$ in the diagram
$\bar{D}$ coincide with the correspondent types $\tau_{D}(w)$, $o_{D}(w)$,
$h_{D}(w)$ of $w$ in the diagram $D$:

$\tau_{\bar{D}}(w)=\tau_{D}(w),\quad o_{\bar{D}}(w)=o_{D}(w),\quad
h_{\bar{D}}(w)=h_{D}(w).$

  2. 2.

Let $\tau_{D}(v)=(i,j)$, $o_{D}(v)$, $h_{D}(v)$ be the types of the crossing
$v$ in $D$. Then

     * •

$\tau_{\bar{D}}(v)=(j,i)$;

     * •

$o_{\bar{D}}(v)=-o_{D}(v)$ if $i=j$ and the component $D_{i}$ is long;

     * •

the homotopy type $h_{\bar{D}}(v)$ is equal to:

       * –

$h_{D}(v)\in\pi(F,s_{i})$ if $i=j$, $\partial D_{i}\neq\emptyset$ and $s_{i}$
is the beginning point of the component $D_{i}$;

       * –

$h_{D}(v)\in\bar{\pi}(F,z_{i},z_{j})$ if $i\neq j$ (by Remark 2 we can choose
the set $\bar{\pi}(F,z_{i},z_{j})$ for the homotopy index)

       * –

$\kappa_{i}h_{D,z_{i}}^{-1}(v)\in\bar{\pi}_{1}(F,z_{i})$ if $i=j$, $\partial
D_{i}=\emptyset$ and $z_{i}\in D_{i}\setminus\mathcal{V}(D)$

###### Proof.

The proposition follows from the definitions. For example, if $v$ is a self-
crossing of a closed component $D_{i}$ then its homotopy type in $\bar{D}$ is

$h_{\bar{D}}(v)=h_{\bar{D},z_{i}}(v)=\bar{\delta}^{\prime+}_{v,z}=\bar{\delta}^{-}_{v,z}=\kappa_{i}(\bar{\delta}^{+}_{v,z})^{-1}=\kappa_{i}h_{D,z_{i}}^{-1}(v),$

since $D^{\prime+}_{v}=D^{-}_{v}$ and $\delta^{+}_{v,z}\delta^{-}_{v,z}$ or
$\delta^{-}_{v,z}\delta^{+}_{v,z}$ is equal to $\kappa_{i}$ by Lemma 2. ∎

Let $T=K_{1}\cup\cdots\cup K_{n}$ be a flat tangle in the surface $F$ and
$\mathfrak{T}$ be the category of flat diagrams of $T$. Let us describe the
finest tribal system on $T$ (the definition of tribes for flat diagrams
repeats verbally Definition 2).

######  Definition 10.

Let $D=D_{1}\cup\cdots\cup D_{n}$ be a diagram of the flat tangle $T$. Let $v$
be a crossing of the diagram $D$. Then $v$ is an intersection point of
components $D_{i}$ and $D_{j}$.

The _flat component type_ of the crossing $v$ is the unordered pair
$\bar{\tau}(v)=\\{i,j\\}$.

Let $i=j$, the component $D_{i}$ be long, and $s_{i}$ be the beginning point
of $D_{i}$. The _flat homotopy type_ $\bar{h}(v)$ of the crossing $v$
coincides with the homotopy type $h(v)\in\pi_{1}(F,s_{i})$ defined by the
formula (1).

Let $i=j$ and the component $D_{i}$ be closed. The _flat homotopy type_
$\bar{h}(v)$ of the crossing $v$ is the element $h_{D,z}(v)$ defined by the
formula (2) and considered as an element in the quotient set

$\bar{\bar{\pi}}_{D}(F,z)=\bar{\pi}_{D}(F,z)/\sigma.$

where $\sigma$ is the involution $\sigma(x)=\kappa_{i}x^{-1}$,
$x\in\bar{\pi}_{D}(F,z)$.

Let $i\neq j$. The _flat homotopy type_
$\bar{h}(v)=\bar{h}_{D,z_{i},z_{j}}(v)$ of the crossing $v$ coincides with the
homotopy type $h_{D,z_{i},z_{j}}(v)\in\bar{\pi}_{D}(F,z_{i},z_{j})$ defined by
the formula (3).

Now we can formulate an analogue of Theorem 1 for flat tangles.

######  Theorem 2.

Let $T$ be an oriented flat tangle in the surface $F$ and $D$ be its diagram.
Then two crossings $v$ and $w$ of the diagram $D$ belong to one tribe if and
only if they have identical flat component types and flat homotopy types.

###### Proof.

By Proposition 7, if $v$ and $w$ belong to one tribe of flat crossings then
their flat component type and flat homotopy type coincide.

Let $\bar{\tau}(v)=\bar{\tau}(w)$ and $\bar{h}(v)=\bar{h}(w)$. Then lift the
diagram $D$ to a diagram $\tilde{D}$ of a tangle by choosing arbitrarily the
under-overcrossing structure in all the crossings of $D$ except $v$ and $w$.
For the crossings $v$ and $w$ we choose the under-overcrossings by the
following rule:

  * •

Let $v$ and $w$ are mixed crossings of components $D_{i}$ and $D_{j}$, $i<j$.
Then choose the under-over crossings so that $\tau(v)=\tau(w)=(i,j)$ in the
diagram $\tilde{D}$.

  * •

Let $v$ and $w$ are self-crossings of a long component $D_{i}$. Then choose
the under-over crossings so that $o(v)=o(w)=+1$ in the diagram $\tilde{D}$.

  * •

Let $v$ and $w$ are self-crossings of a closed component $D_{i}$. Let
$\alpha\in\bar{\pi}_{D}(F,z)$ be one of the two elements which correspond to
the element $h(v)=h(w)\in\bar{\bar{\pi}}_{D}(F,z)$. Then choose the under-over
crossings so that $\bar{\delta}^{+}_{v,z}=\bar{\delta}^{+}_{w,z}=\alpha$ in
the diagram $\tilde{D}$.

Thus, we obtain a tangle diagram $\tilde{D}$ with crossings $v$ and $w$ which
have identical component, order and homotopy types. Then by Theorem 1 $v$ and
$w$ belong to one tribe of crossings of tangle diagrams. This means that there
exists a sequence of intermediate diagrams
$\tilde{D},\tilde{D}_{1},\dots,\tilde{D}_{n},\tilde{D}$ and crossings in them
which establishes the equivalence $v\sim w$ by sequental application of
properties (T0) and (T2). Then the flattening $D,D_{1},\dots,D_{n},D$ of this
sequence prove the equivalence of $v$ and $w$ as crossings of flat tangle
diagrams. ∎

Crossings of flat diagrams have no signs, so we can not split tribes into
phratries like we did in Definition 2. But the difference between two parts of
a tribe reveals when one considers an oriented parity of crossings [20]:
parity values of two crossings in a second Reidemeister move are opposite
although the crossings belong to one tribe. Thus, we need another definition
of tribe splitting.

######  Definition 11.

Let $T$ be a (flat) tangle and $\mathfrak{T}$ be the category of its diagrams.
A _phratry system_ on diagrams of the tangle $T$ is a family of partitions
$\mathfrak{P}(D)$ of the sets $\mathcal{V}(D)$, $D\in\mathfrak{T}$, into
subsets $P\in\mathfrak{P}(D)$ called _phratries_ such that

  * $(\Phi^{\ast})$

There is an involution $\ast\colon\mathfrak{P}(D)\to\mathfrak{P}(D)$. For a
phratry $P\in\mathfrak{P}(D)$, the phratry $P^{*}$ is called the _dual
phratry_ to $P$; if $P^{*}=P$ then the phratry $P$ is called _self-dual_. The
union $P\cup P^{*}$ is called a _tribe_ ;

  * $(\Phi 0)$

Let $f\colon D\to D^{\prime}$ be a Reidemeister move,
$v_{1},v_{2}\in\mathcal{V}(D)$ and $v^{\prime}_{1}=f_{*}(v_{1})$ and
$v^{\prime}_{2}=f_{*}(v_{2})$ be the correspondent crossings in $D^{\prime}$.
Then if $v_{1}$ and $v_{2}$ belong to one phratry in $\mathfrak{P}(D)$ then
$v^{\prime}_{1}$ and $v^{\prime}_{2}$ also belong to one phratry in
$\mathfrak{P}(D^{\prime})$; if $v_{1}$ and $v_{2}$ belong to dual phratries
then $v^{\prime}_{1}$ and $v^{\prime}_{2}$ also belong to dual phratries in
$\mathfrak{P}(D^{\prime})$;

  * $(\Phi 2)$

If $f\colon D\to D^{\prime}$ is a decreasing second Reidemeister move and
$v_{1}$ and $v_{2}$ are the disappearing crossings then $v_{1}$ and $v_{2}$
belong to dual phratries in $\mathfrak{P}(D)$.

######  Remark 3.

1\. A tribal system $\mathfrak{C}$ in Definition 2 defines a phratry system
with phratries $C^{\pm}$, $C\in\mathfrak{C}(D)$, $D\in\mathfrak{T}$. The
phratries $C^{+}$ and $C^{-}$ are dual.

2\. Any phratry system $\mathfrak{P}$ induces a tribal system with tribes
$P\cup P^{*}$, $P\in\mathfrak{P}(D)$, $D\in\mathfrak{T}$.

Below we will always consider the _finest phratry system_ $\mathfrak{P}^{u}$,
i.e. the partitions of the crossing sets $\mathcal{V}(D)$, $D\in\mathfrak{T}$,
generated by the properties $(\Phi 0)$ and $(\Phi 2)$.

######  Remark 4.

Let us give a graph interpretation of the finest phratry and tribal systems.
Let $T$ be a (flat) tangle in the surface $F$. We construct a family of simple
unoriented graphs $G(D)$ whose vertex sets $V(G(D))$ are the crossing sets
$\mathcal{V}(D)$, $D\in\mathfrak{T}$. The sets of edges $E(G(D))$ of the
graphs $G(D)$ are defined inductively by the following rules:

  * •

($R_{2}$-duality) If $v_{1}$ and $v_{2}$ are crossings in a diagram $D$ to
which a decreasing second Reidemeister move can be applied then $v_{1}v_{2}\in
E(G(D))$;

  * •

(Reidemeister translation) Let $f\colon D\to D^{\prime}$ be a Reidemeister
move, $v_{1},v_{2}\in\mathcal{V}(D)$ and $v^{\prime}_{1}=f_{*}(v_{1})$ and
$v^{\prime}_{2}=f_{*}(v_{2})$ be the correspondent crossings in $D^{\prime}$.
If $v_{1}v_{2}\in E(G(D))$ then $v^{\prime}_{1}v^{\prime}_{2}\in
E(G(D^{\prime}))$;

  * •

(transitive closure) Let $v_{1}$, $v_{2}$ and $v_{3}$ be crossings of a
diagram $D\in\mathfrak{T}$ such that $v_{1}v_{2},v_{2}v_{3}\in E(G(D))$. Then
$v_{1}v_{3}\in E(G(D))$.

Then for each diagram $D\in\mathfrak{T}$ the graph $G(D)$ defines a partition
of the crossing set $\mathcal{V}(D)$ into subsets which are the vertex sets of
the connected components of $G(D)$. This partition is the partition
$\mathfrak{C}^{u}(D)$ of the finest tribal system $\mathfrak{C}^{u}$.

In order to describe the finest phratries, let us assign weights
$\epsilon(v_{1}v_{2})\in\mathbb{Z}_{2}$ to the edges $v_{1}v_{2}\in E(G(D))$,
$D\in\mathfrak{T}$, of the graphs $G(D)$. We define weights inductively by the
following rules:

  * •

($R_{2}$-duality) If $v_{1}$ and $v_{2}$ are crossings in a diagram $D$ to
which a decreasing second Reidemeister move can be applied then
$\epsilon(v_{1}v_{2})=1$;

  * •

(Reidemeister translation) Let $f\colon D\to D^{\prime}$ be a Reidemeister
move, $v_{1},v_{2}\in\mathcal{V}(D)$ and $v^{\prime}_{1}=f_{*}(v_{1})$ and
$v^{\prime}_{2}=f_{*}(v_{2})$ be the correspondent crossings in $D^{\prime}$.
If $v_{1}v_{2}\in E(G(D))$ then
$\epsilon(v^{\prime}_{1}v^{\prime}_{2})=\epsilon(v_{1}v_{2})$;

  * •

(transitive closure) Let $v_{1}$, $v_{2}$ and $v_{3}$ be crossings of a
diagram $D\in\mathfrak{T}$ such that $v_{1}v_{2},v_{2}v_{3}\in E(G(D))$. Then
$\epsilon(v_{1}v_{3})=\epsilon(v_{1}v_{2})+\epsilon(v_{2}v_{3})$.

For some components of the graphs $G(D)$, $D\in\mathfrak{T}$, it may happen
that the weights are not consistent. For example, consider the flat diagram
$D$ in Fig. 29. It has three crossings $u,v,w$. One can eliminate one of the
crossings with a first Reidemeister move and apply a second Reidemeister move
to the other two. Then $G(D)$ is the triangle $uvw$ and the weights of all the
edges are equal to $1$. This contradicts the transitive closure rule of weight
assignment.

Figure 29: A diagram with a self-dual tribe

If we encounter a contradiction of weights for some component
$G^{\prime}\subset G(D)$ then we call the component $G^{\prime}$ _self-dual_
and its vertex set $C^{\prime}=V(G^{\prime})$ is called a _self-dual tribe_.
Then the tribe $C^{\prime}$ consists of one phratry of the finest phratry
system $\mathfrak{P}^{u}$ such that $(C^{\prime})^{*}=C^{\prime}$.

Let $G^{\prime}\subset G(D)$ is a component which is not self-dual. Then due
to the transitive closure rule, the vertex set $C^{\prime}=V(G^{\prime})$
splits into two subsets $C^{\prime}_{1}$ and $C^{\prime}_{2}$ such that any
two crossings from one subset are connected by an even edge, and two crossings
from different subsets are connected by an odd edge. The subsets
$C^{\prime}_{1}$ and $C^{\prime}_{2}$ are dual phratries of the finest phratry
system $\mathfrak{P}^{u}$.

Let us describe the finest phratry system for diagrams of a flat tangle.

Let $T=K_{1}\cup\cdots\cup K_{n}$ be an oriented flat tangle in the surface
$F$ and $\mathfrak{T}$ be the category of flat diagrams of $T$. In order to
distinguish phratries we need to refine the flat component and homotopy types.

######  Definition 12.

1\. Let $D=D_{1}\cup\cdots\cup D_{n}$ be a diagram of the tangle $T$, and $v$
be a crossing of $D$. Then $v$ is an intersection point of components $D_{i}$
and $D_{j}$. We order the component according to the orientation as shown in
Fig. 30. The _refined flat component type_ of the crossing $v$ is the ordered
pair $\tau^{f}(v)=(i,j)$.

Figure 30: A crossing of refined flat component type $(i,j)$

Note that there are $n^{2}$ possible refined flat component types.

2\. Let $i=j$ and $\partial D_{i}\neq\emptyset$. Let $s_{i}$ be the starting
point of $D_{i}$. The crossing $v$ split the component $D_{i}$ into two halves
$D^{l}_{v}$ and $D^{r}_{v}$ (Fig. 4) one of which is closed (it does not
contain $s_{i}$) and the other is not (it contains $s_{i}$). We denote the
closed half by $D^{c}_{v}$.

We define the _refined order type_ $o^{f}(v)$ of the crossing $v$ to be equal
to $+1$ if $D^{c}_{v}=D^{l}_{v}$ and to be equal to $-1$ if
$D^{c}_{v}=D^{r}_{v}$ (Fig. 31)

Figure 31: Refined order type

3\. Let $i=j$ and $\partial D_{i}=\emptyset$. Choose a non crossing point
$z_{i}$ in the component $D_{i}$. Let $\kappa_{z_{i}}$ be the homotopy class
of $D_{i}$ in the group $\pi_{1}(F,z_{i})$. Let $\bar{\pi}_{D}(F,z_{i})$ be
the quotient of $\pi_{1}(F,z_{i})$ by the adjoint action of the element
$\kappa_{z_{i}}$.

For the crossing $v$, choose a path $\gamma_{v,z_{i}}$ which connects $z_{i}$
with $v$ in $D_{i}$. Define the based left half as the loop
$\hat{D}^{l}_{v,z_{i}}=\gamma_{v,z_{i}}D^{l}_{v}(\gamma_{v,z_{i}})^{-1}$ (Fig.
32). Analogously,
$\hat{D}^{r}_{v,z_{i}}=\gamma_{v,z_{i}}D^{r}_{v}(\gamma_{v,z_{i}})^{-1}$.

Figure 32: Based left half $\hat{D}^{l}_{v,z_{i}}$

The homotopy classes $\delta^{l}_{v,z_{i}}=[\hat{D}^{l}_{v,z_{i}}]$ and
$\delta^{r}_{v,z_{i}}=[\hat{D}^{r}_{v,z_{i}}]$ in $\pi_{1}(F,z_{i})$ depend on
the choice of $\gamma_{v,z}$ but their images $\bar{\delta}^{l}_{v,z_{i}}$ and
$\bar{\delta}^{r}_{v,z_{i}}$ in $\bar{\pi}_{D}(F,z_{i})$ do not. Note that
$\delta^{l}_{v,z_{i}}\delta^{r}_{v,z_{i}}=\kappa_{z_{i}}$ if $z_{i}\in
D^{r}_{v}$ and $\delta^{r}_{v,z_{i}}\delta^{l}_{v,z_{i}}=\kappa_{z_{i}}$ if
$z_{i}\in D^{l}_{v}$.

The element

$h^{f}_{D}(v)=\bar{\delta}^{l}_{v,z_{i}}\in\bar{\pi}_{D}(F,z_{i})$ (4)

is called the _refined flat homotopy type_ of the self-crossing $v$ of the
closed component $D_{i}$.

If $v$ is a self-crossing of a long component $D_{i}$ with the beginning point
$s$ then the _refined flat homotopy type_ $h^{f}(v)$ of the crossing $v$
coincides with the homotopy type $h(v)$ defined by the formula (1):
$h^{f}_{D}(v)=h(v)\in\pi_{1}(F,s)$.

If $v$ is a mixed crossing of components $D_{i}$ and $D_{j}$, $i<j$ then the
_refined flat homotopy type_ $h^{f}_{D}(v)$ of the crossing $v$ coincides with
the homotopy type $h_{D,z_{i},z_{j}}(v)$ defined by the formula (3):
$h^{f}_{D}(v)=h_{D,z_{i},z_{j}}(v)\in\bar{\pi}(F,z_{i},z_{j})$.

We consider the following involutions on the sets which the types take values
in:

$(i,j)^{*}=(j,i),\quad 1\leq i,j\leq n,$

for the refined flat component type,

$o^{*}=-o,\quad o\in\\{-1,+1\\},$

for the refined flat order type, and

$\bar{x}^{*}=\overline{\kappa_{z_{i}}x^{-1}},\quad\bar{x}\in\bar{\pi}_{D}(F,z_{i})$

for the homotopy type of self-crossings of a closed component.

We define the involution to be the identity on the sets $\pi_{1}(F,s)$ and
$\bar{\pi}(F,z_{i},z_{j})$ where the homotopy type of self-crossings of long
components and mixed crossings takes value.

######  Lemma 3.

Let $v$ and $w$ be crossings of a flat tangle diagram $D$ to which a second
Reidemeister move can be applied. Then

  * •

$\tau^{f}(w)=\tau^{f}(v)^{*}$;

  * •

$o^{f}(w)=o^{f}(v)^{*}$ if $v$ and $w$ are self-crossings of a long component;

  * •

$h_{D}(w)=h_{D}(v)^{*}$.

###### Proof.

The first statement of the lemma follows from the definition of the refined
flat component type (Fig. 33).

Figure 33: Mixed crossing in second Reidemeister moves

If $v$ and $w$ a self-crossing of a long component then their refined order
types are opposite by the definition of the type (see Fig. 34).

Figure 34: Refined order type in a second Reidemeister move

Let $v$ and $w$ a self-crossing of a closed component. Then the left based
half $\hat{D}^{l}_{w,z_{i}}$ is homotopic to the based right half
$\hat{D}^{r}_{v,z_{i}}$ (Fig. 35).

Figure 35: Based halves $\hat{D}^{r}_{v,z}$ (blue) and $\hat{D}^{l}_{w,z}$
(red)

Hence,

$h^{f}_{D,z_{i}}(w)=\overline{[\hat{D}^{l}_{w,z_{i}}]}=\overline{[\hat{D}^{r}_{v,z_{i}}]}=\overline{\kappa_{z_{i}}[\hat{D}^{l}_{w,z_{i}}]^{-1}}=\sigma\left(\overline{[\hat{D}^{l}_{v,z_{i}}]}\right)=\sigma\left(h^{f}_{D,z_{i}}(v)\right).$

By Theorem 2, the homotopy types of $v$ and $w$ coincide if the crossings are
mixed or they are self-crossings of a long component. ∎

Now, we can give a description of crossing phratries of flat tangle diagrams.

######  Theorem 3.

Let $T$ be an oriented flat tangle in the surface $F$ and $D$ be its diagram.
Then two crossings $v$ and $w$ of the diagram $D$ belong to one phratry if and
only if they have identical refined flat component types, refined order types
(if $v$ and $w$ are self-crossing of a long component) and refined flat
homotopy types.

###### Proof.

Let $G(D)$ be the weighted graph from Remark 4. We claim that for any
crossings $v$ and $w$ which belong to one tribe (i.e. a connected component of
$G(D)$) their refined flat component, order and homotopy types coincide if the
edge weight $\epsilon(vw)$ is zero, and the types are dual (i.e. differ by the
correspondent involutions) if the edge $vw$ is odd ($\epsilon(vw)=1$). This
statement can be proved by the induction which defines the weights of the
graph $G(D)$:

  * •

$R_{2}$-duality is proved in Lemma 3;

  * •

Reidemeister translation follows from the fact that for any Reidemeister move
$f\colon D\to D^{\prime}$ and any vertices $v\in\mathcal{V}(D)$ and
$v^{\prime}\in\mathcal{V}(D)$ such that $v^{\prime}=f_{*}(v)$ we have
$\tau^{f}(v^{\prime})=\tau^{f}(v)$, $o^{f}(v^{\prime})=o^{f}(v)$ and
$h^{f}_{D^{\prime}}(v^{\prime})=h^{f}_{D}(v)$;

  * •

transitive closure holds because the square of each of the involutions of the
crossing types is identity.

Now, if $v$ and $w$ belong to one phratry the they are connected by an even
edge of the graph $G(D)$. Thus, their refined flat types coincide.

Next, assume that the refined flat types of crossings $v$ and $w$ coincide.
Then by definition $\bar{\tau}(v)=\bar{\tau}(w)$ and $\bar{h}(v)=\bar{h}(w)$.
Hence, $v$ and $w$ belong to one tribe by Theorem 2. This means $v$ and $w$
belong either to one phratry or to dual phratries.

If $v$ and $w$ are mixed crossings then the dual phratries are distinguished
by the refined component type. Since $\tau^{f}(v)=\tau^{f}(w)$, they belong to
one phratry.

If $v$ and $w$ are self-crossings of a long component then the dual phratries
are distinguished by the refined order type. Since $o^{f}(v)=o^{f}(w)$, they
belong to one phratry.

Let $v$ and $w$ are self-crossings of a closed component $D_{i}$. If
$h_{D}(v)^{*}\neq h_{D}(v)$ then the dual phratries can be distinguished by
the homotopy type. Since $h^{f}_{D}(v)=h^{f}_{D}(w)$, they belong to one
phratry.

Assume that $h_{D}(v)^{*}=h_{D}(v)$. Then
$\hat{D}^{l}_{v,z_{i}}=\hat{D}^{r}_{v,z_{i}}=\hat{D}^{l}_{w,z_{i}}=\hat{D}^{r}_{w,z_{i}}$.
Lift the flat tangle diagram $D$ to a tangle diagram $\tilde{D}$ so that $v$
and $w$ be positive crossings in $\tilde{D}$. Then

$h_{\tilde{D}}(v)=\overline{[\hat{\tilde{D}}^{+}_{v,z_{i}}]}=\overline{[\hat{D}^{l/r}_{v,z_{i}}]}=\overline{[\hat{D}^{l/r}_{w,z_{i}}]}=\overline{[\hat{\tilde{D}}^{+}_{w,z_{i}}]}=h_{\tilde{D}}(w).$

By Corollary 1 the crossings $v$ and $w$ belong to one phratry in $\tilde{D}$.
Thus, they belong to one phratry in $D$. ∎

Let us describe the self-dual phratries of crossings. By Theorem 3 this can
happen only for self-crossings of a closed component.

######  Proposition 8.

Let $D$ be a flat tangle diagram, $D_{i}$ be a closed component of $D$ and $v$
be a self-crossing of $D_{i}$. Choose a non-crossing point $z_{i}$ in $D_{i}$.
Let $\kappa_{z_{i}}\in\pi_{1}(F,z_{i})$ be the homotopy class of $D_{i}$ and
$\delta^{l}_{v,z_{i}}=[\hat{D}^{l}_{v,z_{i}}]\in\pi_{1}(F,z_{i})$ be the
homotopy class of a based left half at the crossing $v$. Then $v$ belongs to a
self-dual phratry if and only if

$(\delta^{l}_{v,z_{i}})^{2}=\kappa_{z_{i}}.$ (5)

###### Proof.

For brevity denote $a=\delta^{l}_{v,z_{i}}$ and $b=\delta^{r}_{v,z_{i}}$.
Without loss of generality assume that $z_{i}\in D^{r}_{v}$. Then
$ab=\kappa_{z_{i}}$.

By Theorem 3 the phratry of the crossing $v$ is self-dual if and only if
$\bar{a}=\bar{a}^{*}$. Since $\bar{a}^{*}=\bar{b}$, the condition is
$b=\kappa^{l}_{z_{i}}a\kappa^{-l}_{z_{i}}$ for some $l\in\mathbb{Z}$. If $l=0$
then $a=b$ and $\kappa_{z_{i}}=a^{2}$.

Let $l>0$. Then $b(ab)^{l}=(ab)^{l}a$. Consider the covering $\tilde{F}\to F$
which correspond to the subgroup $H=\left<a,b\right>\subset\pi_{1}(F,z_{i})$
generated by $a$ and $b$. Then $\pi_{1}(\tilde{F})=H$ and
$H_{1}(\tilde{F},\mathbb{Z})=H/[H,H]$. In the group $H/[H,H]$ we have the
relation $b+l(a+b)=l(a+b)+a$ which implies $a=b\in H/[H,H]$. Then the homology
group $H_{1}(\tilde{F},\mathbb{Z})$ is cyclic, and the surface $\tilde{F}$ is
a cylinder. Hence, $H=\pi_{1}(\tilde{F})=H_{1}(\tilde{F},\mathbb{Z})$, and
$a=b$ in $H$. Thus, $a=b$ in $\pi_{1}(F,z_{i})$ and $\kappa_{z_{i}}=ab=a^{2}$.

The case $l<0$ is proved analogously. Thus, the self-duality condition is
equivalent to $\kappa_{z_{i}}=a^{2}$. ∎

######  Corollary 3.

1\. A closed component $D_{i}$ can have a self-dual phratry of self-crossings
if and only if its homotopy type $\kappa_{z_{i}}$ is a square of an element of
$\pi_{1}(F,z_{i})$.

2\. A closed component $D_{i}$ has at most one self-dual phratry.

The second statement of the corollary follows from the fact that any element
in the fundamental group of a surface has at most one square root.

##  4 Universal index on tangles in the surface $F$

Below we describe the universal index $\iota^{u}$ on tangles in the surface
$F$.

Let $T=K_{1}\cup\cdots\cup K_{n}$ be a tangle in the surface $F$ and
$\mathfrak{T}$ be its diagram category. Let $D=D_{1}\cup\cdots\cup D_{n}$ be a
diagram of $T$.

By Theorem 1 the component type $\tau$ is an index with coefficients in
$\\{1,\dots,n\\}^{2}$. Then the universal index must distinguish the component
type of crossings. Thus, we can consider crossings of different types
separately.

###  4.1 Self-crossings of a long component

Let $D_{i}$ be a long component and $s_{i}\in\partial D_{i}$ be its beginning
point.

The order type and the homotopy type define an index $o\times h$, $(o\times
h)(v)=(o(v),h(v))$, with coefficients in
$\mathbb{Z}_{2}\times\pi_{1}(F,s_{i})$. Let us prove it is universal for the
self-crossings of a long component.

######  Proposition 9.

Let $\iota$ be an index with coefficients in a set $I$. Let $v$ and $w$ be
self-crossings of the long component $D_{i}$. If $o(v)=o(w)$ and $h(v)=h(w)$
then $\iota(v)=\iota(w)$.

The proposition follows from Theorem 1 and Proposition 1.

######  Corollary 4.

The index $o\times h$ is universal for self-crossings of the long component
$D_{i}$.

###### Proof.

Let $\iota$ be an index with coefficients in a set $I$. We need to construct a
map $\psi\colon\mathbb{Z}_{2}\times\pi_{1}(F,s_{i})\to I$ such that for any
self-crossing $v$ of $D_{i}$ one has $\iota(v)=\psi\circ(o\times h)(v)$.

Let $\epsilon\in\\{-1,+1\\}$ and $\alpha\in\pi_{1}(F,s_{i})$. By applying
second Reidemeister moves, create a crossing $v$ such that $o(v)=\epsilon$ and
$h(v)=\alpha$ as shown in Fig. 36 if the diagram $D$ has no such a crossing.
Then set $\psi(\epsilon,\alpha)=\iota(v)$.

Figure 36: A crossing with a given homotopy type $\alpha$

By Proposition 9 the map $\psi$ is well-defined, and by definition
$\iota(v)=\psi\circ(o\times h)$. ∎

###  4.2 Self-crossings of a closed component

Let $D_{i}$ be a closed component. Unlike the long case, we can not use the
group $\bar{\pi}_{D}(F,z_{i})$ where the homotopy type takes values, as the
coefficient set for a homotopy index because the point $z_{i}$ is not fixed.

Choose an arbitrary point $z^{0}_{i}\in F\setminus\partial F$. The following
lemma shows that we can reduce (is some sense) the diagrams of the tangle $T$
to diagrams which contains the point $z^{0}_{i}$.

######  Lemma 4.

1\. There exists a tangle diagram $\tilde{D}$ such that
$z^{0}_{i}\in\tilde{D}_{i}$ and $\tilde{D}$ is obtained from $D$ by applying
increasing second Reidemeister moves.

2\. For any morphism $f\colon D\to D^{\prime}$, i.e. a sequence of
Reidemeister moves

$D=D^{0}\stackrel{{\scriptstyle
f_{1}}}{{\rightarrow}}D^{1}\rightarrow\cdots\rightarrow
D^{n-1}\stackrel{{\scriptstyle f_{n}}}{{\rightarrow}}D^{n}=D^{\prime},$

there exists a sequence of morphisms

$\tilde{D}^{0}\stackrel{{\scriptstyle\tilde{f}_{1}}}{{\rightarrow}}\tilde{D}^{1}\rightarrow\cdots\rightarrow\tilde{D}^{n-1}\stackrel{{\scriptstyle\tilde{f}_{n}}}{{\rightarrow}}\tilde{D}^{n}$

such that $z^{0}_{i}\in\tilde{D}^{k}_{i}$ and $\tilde{D}^{k}$ is obtained from
$D^{k}$ by applying increasing second Reidemeister moves, $k=0,\dots,n$.

###### Proof.

1\. The diagram $D$ is a projection of a tangle $T\subset F\times(0,1)$.
Choose points $(z_{i},s)\in T$ and $(z^{0}_{i},s_{0})\not\in T$ and connect
them by a path $\gamma$ such that $\gamma\cap T=(z_{i},s)$. Then pull a sprout
from the point $(z_{i},s)$ to the point $(z^{0}_{i},s_{0})$ along $\gamma$. We
obtain a tangle $\tilde{T}$ (Fig. 37). Its projection $\tilde{D}$ to $F$ obeys
the required conditions.

Figure 37: The tangle $\tilde{T}$

2\. Denote the natural projection from $F\times(0,1)$ to $F$ by $p$. Let
$T\subset F\times(0,1)$ be a tangle such that $D=p(T)$. Let $H_{t}\colon
F\times[0,1]\to F\times[0,1]$, $t\in[0,1]$, be an isotopy which represents the
morphism $f\colon D\to D^{\prime}$. This means that $D^{i}=p(H_{t_{i}}(T))$
for some $t_{0}=0<t_{1}<\cdots<t_{n}=1$ such that for any $i=1,\dots,n$ the
interval $(t_{i-1},t_{i})$ contains exactly one singular value of the map
$p\circ H_{t}\circ T$ which corresponds to the Reidemeister move $f_{i}$.

Push the tangle $T$ to the top by the map
$(z,s)\mapsto(z,\frac{2}{3}+\frac{s}{3})$ and replace the isotopy $H_{t}$ with
the isotopy

$\hat{H}_{t}(z,s)=\left\\{\begin{array}[]{cl}H_{t}(z,3s-2)&s\in[\frac{2}{3},1],\\\
H_{(3s-1)t}(z,0)&s\in[\frac{1}{3},\frac{2}{3}],\\\
(z,s)&s\in[0,\frac{1}{3}],\end{array}\right.$

which is still in the “abyssal” layer $F\times[0,1/3]$. Then $\hat{H}_{t}$
also represents the morphism $f$.

Consider the tangle $\tilde{T}$ from the first part of the proof and “drop the
anchor” from the point $(z^{0}_{i},s_{0})$ to the point
$(z^{0}_{i},\frac{1}{3})$ and get a tangle $\hat{T}$ (Fig. 38). Then define
$\tilde{D}^{i}=p(\hat{H}_{t_{i}}(\hat{T}))$. By definitions of $\hat{H}_{t}$
and $\hat{T}$ the diagrams $D^{i}$ and $\tilde{D}^{i}$ differs by a sprout
which ends in the point $p(z^{0}_{i},\frac{1}{3})=z^{0}_{i}$. One can pull
this sprout with second Reidemeister moves. Thus, the diagrams $\tilde{D}^{i}$
satisfy the required conditions.

Figure 38: The tangle $\hat{T}$

∎

######  Remark 5.

The sprout which differs $\tilde{T}$ from $T$ goes along the curve
$\gamma\gamma_{0}$ where $\gamma_{0}$ is the vertical segment connecting the
points $(z^{0}_{i},s_{0})$ and $(z^{0}_{i},\frac{1}{3})$. Then the sprout
which differs $\tilde{T}^{\prime}$ from $T^{\prime}$ goes along
$\hat{H}_{1}(\gamma\gamma_{0})$. Let $\gamma_{1}$ be defined by the formula
$\gamma_{1}(t)=\hat{H}_{1-t}(z_{i},s)$, $t\in[0,1]$. Then the path
$\hat{H}_{1}(\gamma\gamma_{0})$ is homotopic to the path
$\gamma_{1}\gamma\gamma_{0}$, and the projection
$p(\hat{H}_{1}(\gamma\gamma_{0}))$ is homotopic to
$p(\gamma_{1}\gamma\gamma_{0})=p(\gamma_{1})p(\gamma)$.

Thus, the diagram $\tilde{D}$ is obtained from $D$ by pulling a sprout along a
path in the homotopy class $p(\gamma)$, and $\tilde{D}^{\prime}$ is obtained
from $D^{\prime}$ by pulling a sprout along $p(\gamma_{1})p(\gamma)$. In
particular case, when $D$ contains $z^{0}_{i}$ one can choose
$z_{i}=z^{0}_{i}$. Then $p(\gamma)$ is trivial, and the homotopy type of the
sprout in $\tilde{D}^{\prime}$ is $p(\gamma_{1})$, i.e. the projection of the
path that the point $(z_{i},s)$ walks during the homotopy $\hat{H}_{t}$
(oriented in the opposite direction).

Let us define the based diagram category $\mathfrak{T}_{z^{0}_{i}}$ which
consists of the diagrams $D$ of the tangle $T$ for which the $i$-th component
$D_{i}$ includes $z^{0}_{i}$ as a non-crossing point. The morphisms of
$\mathfrak{T}_{z^{0}_{i}}$ are compositions of isotopies and Reidemeister
moves between diagrams in $\mathfrak{T}_{z^{0}_{i}}$.

Fix a diagram $D^{0}=D^{0}_{1}\cup\cdots\cup
D^{0}_{n}\in\mathfrak{T}_{z^{0}_{i}}$. Denote by
$\mathfrak{T}^{0}_{z^{0}_{i}}$ the subcategory consisting of the diagrams in
$\mathfrak{T}_{z^{0}_{i}}$ which are connected by a morphism with the diagram
$D^{0}$.

Let us denote by $\bar{\pi}^{0}(F,z^{0}_{i})$ the quotient of the group
$\pi_{1}(F,z^{0}_{i})$ by the adjoint action of the element
$\kappa_{z^{0}_{i}}=[D^{0}_{i}]$. For any self-crossing $v$ of the closed
component $D_{i}$ of a tangle diagram $D=D_{1}\cup\cdots\cup
D_{n}\in\mathfrak{T}^{0}_{z^{0}_{i}}$, its homotopy type $h_{D,z^{0}_{i}}(v)$
is an element of $\bar{\pi}^{0}(F,z^{0}_{i})$ (note that
$\bar{\pi}^{0}(F,z^{0}_{i})=\bar{\pi}_{D}(F,z^{0}_{i})$ because $D_{i}$ and
$D^{0}_{i}$ are homotopic as loops with the end $z^{0}_{i}$). Let us denote
this element by $h^{0}_{i}(v)$.

######  Proposition 10.

The map $h^{0}_{i}$ is the universal index for self-crossings of the closed
component $D_{i}$ on the diagram category $\mathfrak{T}^{0}_{z^{0}_{i}}$.

###### Proof.

The map $h^{0}_{i}$ is an index with coefficients in
$\bar{\pi}^{0}(F,z^{0}_{i})$ by Theorem 1. Then we need to show that for any
index $\iota$ with coefficients in a set $I$ there exists a unique map
$\psi^{0}\colon\bar{\pi}^{0}(F,z^{0}_{i})\to I$ such that
$\iota(v)=\psi^{0}(h^{0}_{i}(v))$ for any self-crossing $v\in\mathcal{V}(D)$,
$D\in\mathfrak{T}^{0}_{z^{0}_{i}}$, of the component $D_{i}$.

Like in corollary 4, for an element $\alpha\in\bar{\pi}^{0}(F,z^{0}_{i})$ we
define the map $\psi^{0}$ by the formula $\psi^{0}(\alpha)=\iota(v)$ where $v$
is a crossing with the homotopy type $h^{0}_{i}(v)=h_{D,z^{0}_{i}}(v)=\alpha$.
Let us prove that the map $\psi^{0}$ is well-defined.

Let $\alpha\in\bar{\pi}^{0}(F,z^{0}_{i})$. Let $v\in\mathcal{V}(D)$ and
$v^{\prime}\in\mathcal{V}(D^{\prime})$,
$D,D^{\prime}\in\mathfrak{T}^{0}_{z^{0}_{i}}$ be two crossings such that
$h^{0}_{i}(v)=h^{0}_{i}(v^{\prime})=\alpha$. Since
$D,D^{\prime}\in\mathfrak{T}^{0}_{z^{0}_{i}}$ there exist a morphism $f\colon
D\to D^{\prime}$ in the category $\mathfrak{T}^{0}_{z^{0}_{i}}$. Then there is
a tangle $T\in F\times(0,1)$ and an isotopy $H_{t}\colon F\times[0,1]\to
F\times[0,1]$, $t\in[0,1]$, which realizes $f$. That is
$p(T)=D,p(H_{1}(T))=D^{\prime}$ and for some $s_{0}\in(0,1)$ one has
$(z^{0}_{i},s_{0})\in T_{i}$ where $T_{i}$ is the $i$-th component of $T$ and
$H_{t}(z^{0}_{i},s_{0})=(z^{0}_{i},s_{0})$ for any $t$. Like in Lemma 4, we
can assume that the isotopy $H_{t}$ does not move the abyssal layer
$F\times[0,\frac{1}{3}]$. Then we sink the point $(z^{0}_{i},s_{0})$ and tie a
crossing with the homotopy type $\alpha$ in the abyssal layer (Fig. 39). We
get a tangle $\tilde{T}$ and a crossing $w$ in the diagram
$\tilde{D}=p(\tilde{T})$ such that $h^{0}_{i}(w)=\alpha$.

Figure 39: The tangle $\tilde{T}$

The isotopy $H_{1}$ does not disturb the crossing $w$. Then it corresponds to
a crossing $w^{\prime}$ in the diagram
$\tilde{D}^{\prime}=p(\tilde{T}^{\prime})$ where
$\tilde{T}^{\prime}=H_{1}(\tilde{T})$. By construction
$h^{0}_{i}(w^{\prime})=h^{0}_{i}(w)=\alpha$. By property (I0) we have
$\iota(w)=\iota(w^{\prime})$. On the other hand, by Theorem 1
$h^{0}_{i}(v)=h^{0}_{i}(w)$ implies $v\sim w$ in $\tilde{D}$. Then
$\iota(v)=\iota(w)$. Analogously, $\iota(v^{\prime})=\iota(w^{\prime})$.
Hence, $\iota(v)=\iota(v^{\prime})$.

Thus, we has proved that $\psi^{0}$ is well-defined. The equality
$\iota(v)=\psi^{0}(h^{0}_{i}(v))$ follows from the definition. ∎

Now let us extend the homotopy index to all diagrams. Let $\iota$ be an index
with coefficients in a set $I$ on the diagram category $\mathfrak{T}$.

Let $D\in\mathfrak{T}$ be a diagram of the tangle $T$. Choose a non-crossing
point $z_{i}\in D_{i}$. There is a morphism in $\mathfrak{T}$ from $D^{0}$ to
$D$. The morphism is realized by an isotopy $H_{t}$, $t\in[0,1]$ of
$F\times[0,1]$. We can suppose that $H_{1}$ maps $z^{0}_{i}$ to the point
$z_{i}$. More precisely, let $T^{0}\subset F\times(0,1)$ be a tangle such that
$p(T^{0})=D^{0}$ and $y^{0}_{i}=p^{-1}(z^{0}_{i})\cap T^{0}$. Then
$D=p(H_{1}(T^{0}))$ and we suppose that $p(H_{1}(y^{0}_{i}))=z_{i}$. Denote
the point $H_{1}(y^{0}_{i})$ by $y_{i}$.

By Lemma 4 there is a morphism $f\colon D^{0}\to\tilde{D}$ in
$\mathfrak{T}^{0}_{z^{0}_{i}}$ where the diagram $\tilde{D}$ is obtained from
$D$ by pulling a sprout from the point $z_{i}$ to the point $z^{0}_{i}$. By
Remark 5 the sprout is pulled along a path which is homotopic to the path
$\bar{\gamma}_{D}=p(H_{1-t}(y_{i}))$, $t\in[0,1]$.

Since the pulling process consists of increasing second Reidemeister moves,
the set of crossings $\mathcal{V}(D)$ embeds in the set of crossings
$\mathcal{V}(\tilde{D})$. By property (I0) the value $\iota(v)$ of a crossing
$v$ in $D$ coincides with the value $\iota(v)$ of this crossing in the diagram
$\tilde{D}$. Since $\tilde{D}\in\mathfrak{T}^{0}_{z^{0}_{i}}$, by Proposition
10 for any $v\in\tilde{D}$ $\iota(v)=\psi^{0}(h^{0}_{i}(v))$. By definition of
$h^{0}_{i}$, for any $v\in\mathcal{V}(D)$

$h^{0}_{i}(v)=h_{\tilde{D},z^{0}_{i}}(v)=[\hat{D}^{+}_{v,z^{0}_{i}}]=\bar{\gamma}_{D}^{-1}[\hat{D}^{+}_{v,z_{i}}]\bar{\gamma}_{D}=\bar{\gamma}_{D}^{-1}h_{D,z_{i}}(v)\bar{\gamma}_{D}.$

The elements $h^{0}_{i}(v)$ depends on $\gamma_{D}$ and hence on the isotopy
$H_{t}$.

Let $H^{\prime}_{t}$, $t\in[0,1]$, be another isotopy from $D^{0}$ to $D$ such
that $H^{\prime}_{1}(y^{0}_{i})=y_{i}$. Denote the path
$p(H^{\prime}_{1-t}(y_{i}))$ by $\bar{\gamma}^{\prime}_{D}$. Then
$H^{-1}_{t}\circ H_{t}$ is an isotopy from the diagram $D^{0}$ to itself, and
$\bar{\gamma}=\bar{\gamma}_{D}^{-1}\bar{\gamma}^{\prime}_{D}$ is the homotopy
class of the path of the base point $z^{0}_{i}$ under this isotopy.

On the other hand, if $H^{0}_{t}$ be an isotopy of $D^{0}$ to itself such that
$H^{0}_{1}(y^{0}_{i})=y^{0}_{i}$ and $\bar{\gamma}=p(H^{0}_{t}(y^{0}_{i}))$,
$t\in[0,1]$, then the composition $H^{\prime}_{t}=H_{t}\circ H^{0}_{t}$ is an
isotopy from $D^{0}$ to $D$ with
$\bar{\gamma}^{\prime}_{D}=\bar{\gamma}_{D}\bar{\gamma}^{-1}$.

These reasonings lead us to the following definition.

######  Definition 13.

Let $D=D_{1}\cup\cdots\cup D_{n}$ be a tangle diagram, $D_{i}$ be a closed
component of $D$ and $z\in D_{i}$ be a non-crossing point. Let
$T=T_{1}\cup\cdots\cup T_{n}\subset F\times(0,1)$ be a tangle such that
$p(T)=D$ and $p(T_{i})=D_{i}$. Let $y=p^{-1}(z)\cap T_{i}$. Define the _inner
monodromy group_ $IM_{i}(D,z)$ as the subgroup of $\pi_{(}F,z)$ consisting of
the homotopy classes of the loop $\gamma(t)=p(H_{t}(y))$, $t\in[0,1]$, where
$H_{t}\colon F\times[0,1]\to F\times[0,1]$ is an isotopy such that
$H_{1}(T)=T$ and $H_{1}(y)=y$.

######  Definition 14.

Let $\hat{\pi}_{D^{0}}(F,z^{0}_{i})$ the quotient set of the group
$\pi_{1}(F,z^{0}_{i})$ by the adjoint action of the inner monodromy group
$IM_{i}(D^{0},z^{0}_{i})$. For any crossing $v\in\mathcal{V}(D)$ of a diagram
$D\in\mathfrak{T}$, we define its _homotopy index_ $h_{i}(v)$ as the element

$h_{i}(v)=\widehat{\bar{\gamma}_{D}^{-1}[\hat{D}^{+}_{v,z_{i}}]\bar{\gamma}_{D}}=\widehat{\bar{\gamma}_{D}^{-1}h_{D,z_{i}}(v)\bar{\gamma}_{D}}\in\hat{\pi}_{D^{0}}(F,z^{0}_{i}).$
(6)

######  Proposition 11.

The map $h_{i}$ is the universal index for self-crossings of the closed
component $D_{i}$ on the diagram category $\mathfrak{T}$.

###### Proof.

The factorization by the adjoint action in $\hat{\pi}_{D^{0}}(F,z^{0}_{i})$
ensures that $h_{i}(v)$ is well defined. The index property (I0) holds by
definition and the property (I2) follows from the correspondent property of
the homotopy type $h_{D,z_{i}}$.

Let $\iota$ be an index with coefficients in a set $I$ on the diagram category
$\mathfrak{T}$. We define a map $\psi\colon\hat{\pi}_{D^{0}}(F,z^{0}_{i})\to
I$ by the formula $\psi(\hat{\alpha})=\iota(v)$,
$\alpha\in\pi_{1}(F,z^{0}_{i})$ where $v$ is a crossing of a diagram
$D\in\mathfrak{T}^{0}_{z^{0}_{i}}$ such that
$[\hat{D}^{+}_{v,z^{0}_{i}}]=\alpha$.

Let us prove that the map $\psi$ is well-defined. Let $v\in\mathcal{V}(D)$,
$D\in\mathfrak{T}$, be a crossing. The reasonings above show that the value
$\iota(v)$ is determined by the homotopy class
$\bar{\gamma}_{D}^{-1}[\hat{D}^{+}_{v,z_{i}}]\bar{\gamma}_{D}$. Since we can
choose any class $\bar{\gamma}_{D}$ induced from an isotopy from $D^{0}$ to
$D$ and such classes differ by elements of the inner monodromy group, the
value $\iota(v)$ can be restored from any representative of the orbit of the
homotopy class by the adjoint action of $IM_{i}(D^{0},z^{0}_{i})$, that is
from the element $h_{i}(v)$. Thus, $\psi$ if well-defined.

The equality $\iota=\psi\circ h_{i}$ follows from the definition. ∎

Let us enumerate some properties of the inner monodromy group.

######  Proposition 12.

Let $D$ be a tangle diagram of the tangle $T$, $D_{i}$ be a closed component
of $D$ and $z\in D_{i}$ be a non-crossing point.

  1. 1.

Let $D^{\prime}\in\mathfrak{T}$ be another diagram of $T$ and $z^{\prime}\in
D^{\prime}_{i}$ be a non-crossing point in the $i$-th component. Then
$IM_{i}(D^{\prime},z^{\prime})\simeq IM_{i}(D,z)$.

  2. 2.

Let $\kappa_{z}=[D_{i}]\in\pi_{1}(F,z)$ be the homotopy class of the component
$D_{i}$. Then $\kappa_{z}\in IM_{i}(D,z)$.

  3. 3.

$IM_{i}(D,z)\subset C(\kappa_{z})$ where
$C(\kappa_{z})=\\{\alpha\in\pi_{1}(F,z)\mid\alpha\kappa_{z}\alpha^{-1}=\kappa_{z}\\}$
is the centralizer of the element $\kappa_{z}$.

###### Proof.

1\. Since $D$ and $D^{\prime}$ are diagrams of one tangle, there is a morphism
$f\colon D\to D^{\prime}$. Let $H_{t}$, $t\in[0,1]$, be an isotopy of
$F\times[0,1]$ which realizes the morphism $f$. That means $D=p(T)$ for a
tangle $T$ and $D^{\prime}=p(H_{1}(T))$. We can also assume that
$z^{\prime}=p(H_{1}(y))$ where $y=p^{-1}(z)\cap T$. Let
$\gamma(t)=p(H_{t}(y))$, $t\in[0,1]$ be the path which the point $z$ pass
during the isotopy. Then the isomorphism

$\gamma_{*}\colon\pi_{1}(F,z)\to\pi_{1}(F,z^{\prime}),\quad\alpha\mapsto\gamma^{-1}\alpha\gamma,$

maps the subgroup $IM_{i}(D,z)$ onto the subgroup
$IM_{i}(D^{\prime},z^{\prime})$.

2\. Let $T\subset F\times(0,1)$ be a tangle such that $p(T)=D$. Consider an
isotopy $H_{t}$ which pushes the $i$-th component $T_{i}$ along itself so that
the points of $T_{i}$ make a full turn, and does not move the other
components. Then the corresponding loop $\gamma\in IM_{i}(D,z)$ coincides with
$\kappa_{z}$.

3\. Let $T\subset F\times(0,1)$ be a tangle such that $p(T)=D$ and let $y\in
T$ be the point such that $p(y)=z$. Consider an isotopy $H_{t}$, $t\in[0,1]$,
such that $H_{1}(T)=T$ and $H_{1}(y)=y$. Let $\gamma(t)=p(H_{t}(y))$ and
$\gamma^{\leq t}$ be the part of $\gamma$ connecting $z$ with
$z(t)=p(H_{t}(y))$. Let $\kappa(t)\in\pi(F,z(t))$ be the homotopy type of
$p(H_{t}(T_{i}))$. Then $\tilde{\kappa}(t)=\gamma^{\leq
t}\kappa(t)(\gamma^{\leq t})^{-1}$, $t\in[0,1]$ is a homotopy between
$\kappa_{z}$ and $\gamma\kappa_{z}\gamma^{-1}$. Hence,
$\gamma\kappa_{z}\gamma^{-1}=\kappa_{z}$ and $\gamma\in C(\kappa_{z})$. Thus,
$IM_{i}(D,z)\subset C(\kappa_{z})$. ∎

######  Remark 6.

If $\kappa_{z}$ is not contractible then $C(\kappa_{z})=\left<\alpha\right>$
is a cyclic group generated by a prime homotopy class $\alpha$ such that
$\kappa_{z}=\alpha^{k}$ for some $k\in\mathbb{Z}$.

If $\kappa_{z}$ is contractible then $C(\kappa_{z})=\pi_{1}(F,z)$. But the
inner monodromy group can differ from the centralizer as the following example
shows (Fig. 40). Consider a knot $K$ in a surface of genus $2$. The homotopy
type of $K$ is trivial. Let us show that $IM(K,z)=\left<\alpha\right>$ where
$\alpha$ is the homotopy of the simple curve close to $K$ which separates the
surface into two equal parts. The curve $\alpha$ belongs to $IM(K,z)$ because
one can drag the knot along $\alpha$.

Figure 40: A knot in a surface of genus $2$

Consider the linking invariant [22] (index polynomial)
$lk_{K}(h^{0})\in\mathbb{Z}[\pi_{1}(F,z)]$ corresponding to the homotopy index
$h^{0}$. The knot has two negative crossings with the homotopy index equal to
$\alpha$. Then $lk_{K}(h^{0})=-\alpha-\alpha^{-1}$.

Assume $\gamma\in IM(K,z)$. Then then there is a diagram $K^{\prime}$ obtained
from $K$ by pulling a sprout along $\gamma$ which is connected to $K$ by an
isotopy which does not move the base point $z$. Then
$lk_{K^{\prime}}(h^{0})=lk_{K}(h^{0})$.

On the other hand, the crossings have the homotopy index
$\gamma^{-1}\alpha^{\pm}\gamma$ in $K^{\prime}$, so
$lk_{K^{\prime}}(h^{0})=-\gamma^{-1}\alpha\gamma-\gamma^{-1}\alpha^{-1}\gamma$.
Then $\gamma^{-1}\alpha\gamma=\alpha^{\pm}$. The equality
$\gamma^{-1}\alpha\gamma=\alpha^{-1}$ is impossible because
$\alpha\neq-\alpha$ in $H_{1}(F,\mathbb{Z})$. Hence,
$\gamma^{-1}\alpha\gamma=\alpha$, i.e. $\gamma$ and $\alpha$ commute. Then
$\gamma=\alpha^{k}$ for some $k\in\mathbb{Z}$.

###  4.3 Mixed crossings

Let us define a universal index for mixed crossing. Let
$i,j\in\\{1,\dots,n\\}$ and $i\neq j$.

Fix a diagram $D^{0}=D^{0}_{1}\cup\cdots\cup D^{0}_{n}$ of the tangle $T$.
Choose con-crossing points $z^{0}_{i}$ and $z^{0}_{j}$ in the components
$D^{0}_{i}$ and $D^{0}_{j}$. If the component $D^{0}_{i}$ ($D^{0}_{j}$) is
long then we set $z^{0}_{i}=s^{0}_{i}$ ($z^{0}_{j}=s^{0}_{j}$) where
$s^{0}_{i}$ ($s^{0}_{j}$) is the beginning point of the component $D^{0}_{i}$
($D^{0}_{j}$).

Let us define the inner monodromy group for the case of mixed crossing.

######  Definition 15.

Let $D=D_{1}\cup\cdots\cup D_{n}$ be a tangle diagram, $D_{i}$ and $D_{j}$ be
a closed component of $D$ and $z_{i}\in D_{i}$ and $z_{j}\in D_{j}$ be non-
crossing points. If a component is long then we choose the beginning point of
the component for $z_{i}$ or $z_{j}$. Let $T=T_{1}\cup\cdots\cup T_{n}\subset
F\times(0,1)$ be a tangle such that $p(T)=D$, $p(T_{i})=D_{i}$ and
$p(T_{j})=D_{j}$. Let $y_{i}=p^{-1}(z_{i})\cap T_{i}$ and
$y_{j}=p^{-1}(z_{j})\cap T_{j}$. Define the _inner monodromy group_
$IM_{ij}(D,z_{i},z_{j})$ as the subgroup of
$\pi_{1}(F,z_{i})\times\pi_{1}(F,z_{j})$ consisting of the pairs
$(\gamma_{i},\gamma_{j})$ where $\gamma_{i}(t)=p(H_{t}(y_{i}))$ and
$\gamma_{j}(t)=p(H_{t}(y_{j}))$, $t\in[0,1]$. Here $H_{t}\colon
F\times[0,1]\to F\times[0,1]$ is an isotopy such that $H_{1}(T)=T$,
$H_{1}(y_{i})=y_{i}$ and $H_{1}(y_{j})=y_{j}$.

######  Proposition 13.

Let $D=D_{1}\cup\cdots\cup D_{n}$ be a tangle diagram, $D_{i}$ and $D_{j}$ be
components of $D$ and $z_{i}\in D_{i}$ and $z_{j}\in D_{j}$ be non-crossing
points chosen as above. Let $\kappa_{z_{i}}\in\pi_{1}(F,z_{i})$ be the
homotopy class of $D_{i}$ if the component is closed, and $1$ otherwise. The
element $\kappa_{z_{j}}\in\pi_{1}(F,z_{j})$ is defined analogously.

  1. 1.

Let $D^{\prime}\in\mathfrak{T}$ be another diagram of $T$ and
$z^{\prime}_{i}\in D^{\prime}_{i}$, $z^{\prime}_{j}\in D^{\prime}_{j}$ be a
non-crossing point in the $i$-th component. Then
$IM_{ij}(D^{\prime},z^{\prime}_{i},z^{\prime}_{j})\simeq
IM_{ij}(D,z_{i},z_{j})$.

  2. 2.

$(\kappa_{z_{i}},1),(1,\kappa_{z_{j}})\in IM_{ij}(D,z_{i},z_{j})$.

  3. 3.

$p_{1}(IM_{ij}(D,z_{i},z_{j}))=IM_{i}(D,z_{i})$ and
$p_{2}(IM_{ij}(D,z_{i},z_{j}))=IM_{j}(D,z_{j})$ where $p_{1},p_{2}$ are the
natural projections from $\pi_{1}(F,z_{i})\times\pi_{1}(F,z_{j})$ to the
groups $\pi_{1}(F,z_{i})$ and $\pi_{1}(F,z_{j})$

The proposition is proved analogously to Proposition 12.

######  Remark 7.

Let $D=D_{1}\cup\cdots\cup D_{n}$ be a diagram of a tangle $T\subset
F\times(0,1)$ and $z_{1},\dots,z_{n}$ be non-crossing points in the components
$D_{1},\dots,D_{n}$. As usual, for long components suppose that the chosen
points coincide with the beginning points. Then one can define the _general
inner monodromy group_ $IM(D,z_{1},\dots,z_{n})$ as the subgroup in
$\prod_{i=1}^{n}\pi_{1}(F,z_{i})$ which consists of tuples
$(\gamma_{i})_{i=1}^{n}$ of the homotopy classes of the paths which the chosen
points $z_{i}$ (more accurately, their liftings $y_{i}$ to $T$) pass during
some isotopy $H_{t}$, $t\in[0,1]$, such that $H_{1}(T)=T$ and
$H_{1}(y_{i})=y_{i}$, $i=1,\dots,n$. The general inner monodromy group has the
following properties:

  1. 1.

For another diagram $D^{\prime}$ of $T$ and another choice of points
$z^{\prime}_{1},\dots,z^{\prime}_{n}$ the groups
$IM(D^{\prime},z^{\prime}_{1},\dots,z^{\prime}_{n})$ and
$IM(D,z_{1},\dots,z_{n})$ are isomorphic.

  2. 2.

$(1,\dots,1,\kappa_{z_{i}},1,\dots,1)\in IM(D,z_{1},\dots,z_{n})$ where
$\kappa_{z_{i}}=[D_{i}]\in\pi_{1}(F,z_{i})$.

  3. 3.

$IM(D,z_{1},\dots,z_{n})\subset\prod_{i=1}^{n}C(\kappa_{z_{i}})$ (for long
components we set $C(\kappa_{z_{i}})=\\{1\\}$).

  4. 4.

$p_{i}(IM(D,z_{1},\dots,z_{n}))=IM_{i}(D,z_{i})$ and
$p_{ij}(IM(D,z_{1},\dots,z_{n}))=IM_{ij}(D,z_{i},z_{j})$ where $p_{i},p_{ij}$
are the natural projections from $\prod_{i=1}^{n}\pi_{1}(F,z_{i})$ to
$\pi_{1}(F,z_{i})$ and $\pi_{1}(F,z_{i})\times\pi_{1}(F,z_{j})$.

Now we can define the homotopy index $h$ for the mixed crossings of the
component type $(i,j)$.

######  Definition 16.

Let $\pi_{1}(F,z^{0}_{i},z^{0}_{j})$ be the set of homotopy classes of
continuous paths in $F$ from $z^{0}_{i}$ to $z^{0}_{j}$. Let
$\hat{\pi}_{D^{0}}(F,z^{0}_{i},z^{0}_{j})$ be the quotient of
$\pi_{1}(F,z^{0}_{i},z^{0}_{j})$ by the action
$(\gamma_{1},\gamma_{2})\cdot\beta=(\gamma_{1}\beta\gamma_{2}^{-1})$ of the
subgroup $IM_{ij}(D,z_{i},z_{j})$.

Let $D=D_{1}\cup\cdots\cup D_{n}$ be a diagram of the tangle $T$ and $z_{i}\in
D_{i}$ and $z_{j}\in D_{j}$ be non-crossing points which coincide with the
beginning point of a component if it is long. Let $H_{t}$ be an isotopy from
$D$ to $D^{0}$ which moves $z_{i}$ to $z^{0}_{i}$ and $z_{j}$ to $z^{0}_{j}$,
and $\bar{\gamma}_{D,i}$ and $\bar{\gamma}_{D,j}$ be the corresponding paths.

For any crossing $v$ in $D$ of the component type $(i,j)$, we define its
_homotopy index_ $h_{ij}(v)$ as the element

$h_{ij}(v)=\widehat{\bar{\gamma}_{D,i}^{-1}h_{D,z_{i},z_{j}}(v)\bar{\gamma}_{D,j}}\in\hat{\pi}_{D^{0}}(F,z^{0}_{i},z^{0}_{j})$
(7)

where $h_{D,z_{i},z_{j}}(v)$ is defined by the formula (3).

######  Proposition 14.

The map $h_{ij}$ is the universal index for mixed crossings of the component
type $(i,j)$ on the diagram category $\mathfrak{T}$.

The proposition is proved analogously to Proposition 11.

###  4.4 Universal index

Collecting the results of Corollary 1 and Propositions 11 and 14, we come to
the following theorem.

######  Theorem 4.

Let $T=T_{1}\cup\cdots\cup T_{n}$ be a tangle in the surface $F$ and
$D^{0}=D^{0}_{1}\cup\cdots\cup D^{0}_{n}$ be its diagram. Let
$I^{u}=\bigsqcup_{i,j=1}^{n}I^{u}_{ij}$ where

$I^{u}_{ij}=\left\\{\begin{array}[]{cl}\mathbb{Z}_{2}\times\pi_{1}(F,s_{i}),&i=j\mbox{
and }\partial D_{i}\neq\emptyset,\\\ \hat{\pi}_{D^{0}}(F,z^{0}_{i}),&i=j\mbox{
and }\partial D_{i}=\emptyset,\\\
\hat{\pi}_{D^{0}}(F,z^{0}_{i},z^{0}_{j}),&i\neq j.\end{array}\right.$

For a diagram $D$ of the tangle $T$, define a map
$\iota^{u}\colon\mathcal{V}(D)\to I^{u}$ by the formula

$\iota^{u}(v)=\left\\{\begin{array}[]{cl}(o(v),h(v)),&\tau(v)=(i,i)\mbox{ and
}\partial D_{i}\neq\emptyset,\\\ h_{i}(v),&\tau(v)=(i,i)\mbox{ and }\partial
D_{i}=\emptyset,\\\ h_{ij}(v),&\tau(v)=(i,j),\ i\neq j.\end{array}\right.$

Then $\iota^{u}$ is the universal index on the diagram category of the tangle
$T$.

######  Corollary 5.

Let $K$ be a classical knot. Then there are no nontrivial indices on crossings
of diagrams of $K$.

###### Proof.

For a classical knots we have $F=\mathbb{R}^{2}$ or $S^{2}$. In both cases
$\pi_{1}(F,z)=1$, hence, the homotopy index $h$ is trivial. For a knot, the
component type $\tau$ is trivial too, and there is no order type. Thus, the
universal index is constant, and all other indices are constant too. ∎

For classical links and long knots we have analogous statements.

######  Corollary 6.

The only nontrivial index for long classical knots is the order type $o$.

######  Corollary 7.

The component type $\tau$ is the universal index of classical links.

######  Example 2.

Let $F=S^{1}\times[0,1]$ be an annulus. Its fundamental group is
$\pi_{1}(F)=H_{1}(F)=\mathbb{Z}$. Since the group is commutative, the adjoint
action on it is trivial.

Let $K$ be a knot in $F$ and $D$ be a diagram of $K$. For a crossing
$v\in\mathcal{V}(D)$, its homotopy index coincides with the homology class of
the signed half $D^{+}_{v}$: $h(v)=[D^{+}_{v}]\in H_{1}(F)=\mathbb{Z}$.

######  Corollary 8.

The map $h$ with coefficients in $\mathbb{Z}$ is the universal index for knots
in the annulus.

Note that $sgn\cdot h$ is an ordered parity in the sense of [19].

Let $K$ be a long knot in the annulus $F$. Then $\partial K\subset\partial
F=S^{1}\times\\{0,1\\}$. Let $D$ be a diagram of $K$ and $v$ be a crossing of
$D$. The vertex $v$ splits the diagram $D$ into two halves, one of which is
closed. We denote the close half by $D^{c}_{v}$.

The homotopy index of the crossing $v$ is the homology class of its closed
half: $h(v)=[D^{c}_{v}]\in H_{1}(F)=\mathbb{Z}$.

######  Corollary 9.

The map $h\times o\colon\mathcal{V}(D)\to\mathbb{Z}\times\mathbb{Z}_{2}$ where
$o$ is the order type, is the universal index for long knots in the annulus.

######  Example 3.

Let $F=T^{2}$ be the torus. Then $\pi_{1}(F)=H_{1}(F)=\mathbb{Z}^{2}$. Since
the group is commutative then the adjoint action on it is trivial.

Let $K$ be a knot in the (thickened) torus. Then the homotopy index of
crossings diagrams is the homology type of their positive half:
$h(v)=[D^{+}_{v}]\in H_{1}(T^{2})=\mathbb{Z}^{2}$. Thus, we have a universal
$\mathbb{Z}^{2}$-valued index for knots in the torus.

Note that $sgn\cdot h$ is an oriented parity in the sense of [19].

###  4.5 Universal index of flat tangles in the surface $F$

Let us formulate a result analogous to that of Section 4.4. The flat case is
simpler because closed components of a flat tangle are essentially free loops
in the surface $F$, and long components are curves in $F$ defined up to
homotopy with fixed ends.

Let $T=K_{1}\cup\cdots\cup K_{n}$ be a flat tangle in the surface $F$ and
$\mathfrak{T}$ be the category of flat diagrams of $T$.

Fix a diagram $D^{0}=D^{0}_{1}\cup\cdots\cup D^{0}_{n}\in\mathfrak{T}$ and
choose points $z^{0}_{i}\in D^{0}_{i}$, $i=1,\dots,n$. If some component
$D^{0}_{i}$ is long then we set $z^{0}_{i}=s^{0}_{i}$ where $s^{0}_{i}$ is the
beginning point of the component $D^{0}_{i}$. Let $\kappa^{0}_{i}=[D^{0}_{i}]$
be the homotopy class of the $i$-th component in the group
$\pi_{1}(F,z^{0}_{i})$ if $D^{0}_{i}$ is closed, and
$\kappa^{0}_{i}=1\in\pi_{1}(F,z^{0}_{i})$ is $D^{0}_{i}$ is long,
$i=1,\dots,n$. Let $C^{0}_{i}$ be the subgroup in $\pi_{1}(F,z^{0}_{i})$ which
is equal to the centralizer $C(\kappa^{0}_{i})$ is the component $D^{0}_{i}$
is closed, and equal to $\\{1\\}$ if the component is long.

Let us define sets $\bar{I}^{u}_{ij}$, $i,j\in\\{1,\dots,n\\}$ as follows

$I^{u}_{ij}=\left\\{\begin{array}[]{cl}\hat{\bar{\pi}}_{D^{0}}(F,z^{0}_{i}),&i=j,\\\
\hat{\bar{\pi}}_{D^{0}}(F,z^{0}_{i},z^{0}_{j}),&i\neq j.\end{array}\right.$

where $\hat{\bar{\pi}}_{D^{0}}(F,z^{0}_{i})=\pi_{1}(F,z^{0}_{i})$ for a long
component $D^{0}_{i}$; $\hat{\bar{\pi}}_{D^{0}}(F,z^{0}_{i})$ is the quotient
set of the group $\pi_{1}(F,z^{0}_{i})$ by the adjoint action of the subgroup
$C^{0}_{i}$ and the involution $\sigma(x)=\kappa^{0}_{i}x^{-1}$,
$x\in\pi_{1}(F,z^{0}_{i})$, for a closed component $D^{0}_{i}$; and
$\hat{\bar{\pi}}_{D^{0}}(F,z^{0}_{i},z^{0}_{j})$ is the quotient set
$C^{0}_{i}\backslash\pi_{1}(F,z_{i},z_{j})/C^{0}_{j}$. Let
$\bar{I}^{u}=\bigsqcup_{i,j=1}^{n}\bar{I}^{u}_{ij}$.

Given a flat diagram $D=D_{1}\cup\cdots\cup D_{n}$ of the flat tangle $T$,
choose non-crossing points $z_{i}\in D_{i}$, $i=1,\dots,n$ which coincide with
the beginning points for the long components of $D$. For each closed component
$D_{i}$, choose a path $\bar{\gamma}_{D,i}$ such that it connects $z_{i}$ with
$z^{0}_{i}$ and
$\bar{\gamma}_{D,i}^{-1}[D_{i}]\bar{\gamma}_{D,i}=\kappa^{0}_{i}$. Such paths
exist because $D$ is homotopic to $D^{0}$ in $F$.

We define a map $\bar{\iota}^{u}\colon\mathcal{V}(D)\to\bar{I}^{u}$ as
follows: for a crossing $v$ of the diagram $D$ the value $\bar{\iota}^{u}(v)$
is the image of the flat homotopy type $\bar{h}(v)$ (see Definition 10) under
the natural projection of $\pi_{1}(F,s^{0}_{i})$ or
$\bar{\bar{\pi}}_{D^{0}}(F,z^{0}_{i})$ to
$\hat{\bar{\pi}}_{D^{0}}(F,z^{0}_{i})$, or the natural projection of
${\bar{\pi}}_{D^{0}}(F,z^{0}_{i},z^{0}_{j})$ to
$\hat{\bar{\pi}}_{D^{0}}(F,z^{0}_{i},z^{0}_{j})$.

######  Theorem 5.

The map $\bar{\iota}$ with the coefficients in $\bar{I}^{u}$ is the universal
index on the diagrams of the flat tangle $T$.

The proof follows the proof of Corollary 4 and Propositions 11 and 14. First
we prove that the flat homotopy type $\bar{h}$ is the universal index on the
diagram category which consists of diagrams $D$ that can be connected with
$D^{0}$ by a homotopy which does not move the base points $z^{0}_{i}$. Then we
pass to the general category $\mathfrak{T}$.

Note that the flat analogues of the inner monodromy groups
$IM_{i}(D^{0},z^{0}_{i})$ and $IM_{ij}(D^{0},z^{0}_{i},z^{0}_{j})$ coincide
with the centralizers $C(\kappa^{0}_{i})$ and $C(\kappa^{0}_{i})\times
C(\kappa^{0}_{j})$ correspondingly.

##  5 Index for virtual and flat knots

###  5.1 Virtual knots

Recall that _oriented virtual links_ $\mathcal{L}$ are equivalence classes of
pairs $(F,L)$ where $F$ is an oriented closed surface and $L\subset
F\times(0,1)$ is an oriented link in thickening of $F$, considered modulo
isotopies, diffeomorphisms and stabilization/destabilization operations [1],
see Fig. 41.

Figure 41: Stabilization operation

Given a virtual link $\mathcal{L}$, we consider its diagram category
$\mathfrak{L}$ whose objects are pairs $(F,D)$ where $F$ is a surface and
$D\subset F$ is a link diagram (a general projection of $L$ for some
representative $(F,L)$ of $\mathcal{L}$). The morphisms in $\mathfrak{L}$ are
compositions of surface isotopies, Reidemeister moves, diffeomorphisms of
pairs $(F,D)$ and (de)stabilizations.

G. Kuperberg proved the following theorem [18].

######  Theorem 6.

Let $(F,L)$ and $(F^{\prime},L^{\prime})$ be two representatives of a virtual
link $\mathcal{L}$ and the genus of $F$ and $F^{\prime}$ be minimal among all
representatives of $\mathcal{L}$. Then there exists an orientation preserving
diffeomorphism $\Phi\colon F\times[0,1]\to F^{\prime}\times[0,1]$ such that
$\Phi(L)=L^{\prime}$.

Kuperberg’s theorem allows us to introduce the diagram subcategory
$\mathfrak{L}_{min}$ consisting of diagrams $D$ of $\mathcal{L}$ in surfaces
$F$ of minimal genus. The morphisms in $\mathfrak{L}_{min}$ are isotopies,
Reidemeister moves and diffeomorphisms.

Note that we don’t have stabilization morphisms in $\mathfrak{L}_{min}$, and
the surfaces $F$ of all objects $(F,D)$ in $\mathfrak{L}_{min}$ are
diffeomorphic. Unlike diagram categories of links in a fixed surface, the
category $\mathfrak{L}_{min}$ has new morphisms — diffeomorphisms of thickened
surfaces.

Let us describe the universal index in the category $\mathfrak{L}_{min}$.

######  Definition 17.

Let $D=D_{1}\cup\cdots\cup D_{n}$ be a link diagram in a surface $F$ of
minimal genus. Let $L=L_{1}\cup\cdots\cup L_{n}$ be the corresponding link in
$F\times(0,1)$. Choose non-crossing points $z_{i}\in D_{i}$, $i=1,\dots,n$.
Let $y_{i}=p^{-1}(z_{i})\cap L_{i}$ be the corresponding points in $L$. Let
$\widetilde{Sym}(L,y_{1},\dots,y_{n})$ be the subgroup in the diffeomorphism
group $Diff^{+}(F\times[0,1])$ which consists of diffeomorphisms $\Phi\colon
F\times[0,1]\to F\times[0,1]$ such that $\Phi(L_{i})=L_{i}$ and
$\Phi(y_{i})=y_{i}$, $i=1,\dots,n$.

Every diffeomorphism $\Phi\in\widetilde{Sym}(L,y_{1},\dots,y_{n})$ induces the
automorphisms $\Phi_{*}\in Aut(\pi_{1}(F,z_{i}))$ and $\Phi_{*}\in
Aut(\pi_{1}(F,z_{i},z_{j}))$. The group

$Sym(D,z_{i})=\\{\Phi_{*}\mid\Phi\in\widetilde{Sym}(L,y_{1},\dots,y_{n})\\}\subset
Aut(\pi_{1}(F,z_{i}))$

is the _symmetry group_ for the component $D_{i}$ and the group

$Sym(D,z_{i},z_{j})=\\{\Phi_{*}\mid\Phi\in\widetilde{Sym}(L,y_{1},\dots,y_{n})\\}\subset
Aut(\pi_{1}(F,z_{i},z_{j}))$

is the _symmetry group_ for the components $D_{i}$ and $D_{j}$.

######  Proposition 15.

Let $D=D_{1}\cup\cdots\cup D_{n}\in\mathfrak{L}_{min}$ be a link diagram and
$z_{i}\in D_{i}$, $i=1,\dots,n$, be non-crossing points. Then

  1. 1.

The groups $Sym(D,z_{i})$ and $Sym(D,z_{i},z_{j})$ are well defined, and for
any other diagram $D^{\prime}$ and a set of non-crossing points
$z^{\prime}_{i}\in D^{\prime}_{i}$, $i=1,\dots,n$, we have $Sym(D,z_{i})\simeq
Sym(D^{\prime},z^{\prime}_{i})$ and $Sym(D,z_{i},z_{j})\simeq
Sym(D^{\prime},z^{\prime}_{i},z^{\prime}_{j})$ for all $1\leq i,j\leq n$.

  2. 2.

Let $\gamma\in IM_{i}(D,z_{i})$. Then $Ad_{\gamma}\in Sym(D,z_{i})$ where
$Ad_{\gamma}(\alpha)=\gamma\alpha\gamma^{-1}$, $\alpha\in\pi_{1}(F,z_{i})$, is
the adjoint action.

  3. 3.

Let $(\gamma_{i},\gamma_{j})\in IM_{ij}(D,z_{i},z_{j})$. Then
$Ad_{\gamma_{i},\gamma_{j}}\in Sym(D,z_{i},z_{j})$ where
$Ad_{\gamma_{i},\gamma_{j}}(\alpha)=\gamma_{i}\alpha\gamma_{j}^{-1}$,
$\alpha\in\pi_{1}(F,z_{i},z_{j})$.

  4. 4.

Let $\kappa_{z_{i}}=[D_{i}]\in\pi_{1}(F,z_{i})$. Then for any $\phi\in
Sym(D,z_{i})$ we have $\phi(\kappa_{z_{i}})=\kappa_{z_{i}}$.

###### Proof.

1\. Let $D^{\prime}=D^{\prime}_{1}\cup\cdots\cup D^{\prime}_{n}$ be a diagram
in a surface $F^{\prime}$, and $L^{\prime}=L^{\prime}_{1}\cup\cdots\cup
L^{\prime}_{n}\in F^{\prime}\times(0,1)$ be a corresponding link, i.e.
$p(L^{\prime})=D^{\prime}$. Let $y^{\prime}_{i}=p^{-1}(z^{\prime}_{i})\cap
L^{\prime}$, $i=1,\dots,n$. By Kuperberg’s theorem there exists a
diffeomorphism $\Psi\colon F\times[0,1]\to F^{\prime}\times[0,1]$ such that
$\Psi(L_{i})=L^{\prime}_{i}$. We can suppose that $\Psi(y_{i})=y^{\prime}_{i}$
for all $i$. Then for any $\Phi\in\widetilde{Sym}(L,y_{1},\dots,y_{n})$ the
composition $\Phi^{\prime}=\Psi\Phi\Psi^{-1}$ belongs to
$\widetilde{Sym}(L^{\prime},y^{\prime}_{1},\dots,y^{\prime}_{n})$ and vice
versa. Then we have a sequence of isomorphisms of groups

$\pi_{1}(F,z_{i})\stackrel{{\scriptstyle
p_{*}}}{{\leftarrow}}\pi_{1}(F\times[0,1],y_{i})\stackrel{{\scriptstyle\Psi_{*}}}{{\rightarrow}}\pi_{1}(F^{\prime}\times[0,1],y^{\prime}_{i})\stackrel{{\scriptstyle
p_{*}}}{{\rightarrow}}\pi_{1}(F^{\prime},z^{\prime}_{i})$

which induces an isomorphism $\Psi_{*}\colon Aut(\pi_{1}(F,z_{i}))\to
Aut(\pi_{1}(F^{\prime},z^{\prime}_{i}))$. The isomorphism $\Psi_{*}$
identifies $Sym(D,z_{i})$ and $Sym(D^{\prime},z^{\prime}_{i})$.

Analogously, the groups $Sym(D,z_{i},z_{j})$ and
$Sym(D^{\prime},z^{\prime}_{i},z^{\prime}_{j})$ are also isomorphic.

2\. Let $\gamma\in IM_{i}(D,z_{i})$. By Definition 13 there exist an isotopy
$H_{t}$, $t\in[0,1]$, such that $H_{1}(L)=L$ and $H_{1}(y_{i})=y_{i}$. Without
loss of generality, we can assume that $H_{1}(y_{j})=y_{j}$ for all $j$. Then
$H_{1}\in\widetilde{Sym}(L,y_{1},\dots,y_{n})$ and
$(H_{1})_{*}=Ad_{\gamma^{-1}}\in Sym(D,z_{i})$. Hence,
$Ad^{-1}_{\gamma^{-1}}=Ad_{\gamma}\in Sym(D,z_{i})$.

3\. The third statement is proved analogously to the previous one.

4\. Let $\Phi\in\widetilde{Sym}(L,y_{1},\dots,y_{n})$. Then

$\Phi_{*}(\kappa_{z_{i}})=\Phi_{*}p_{*}[L_{i}]=p_{*}\Phi_{*}[L_{i}]=p_{*}[\Phi(L_{i})]=p_{*}[L_{i}]=\kappa_{z_{i}}.$

∎

Let $\mathcal{L}$ be a virtual link. Fix a diagram
$D^{0}=D^{0}_{1}\cup\cdots\cup D^{0}_{n}$ in a surface $F^{0}$ of minimal
genus. Choose non-crossing points $z^{0}_{i}\in D^{0}_{i}$, $i=1,\dots,n$. Let
$L^{0}=L^{0}_{1},\cup\cdots\cup L^{0}_{n}\subset F^{0}\times(0,1)$ be a link
such that $p(L^{0})=D^{0}$. Let $y^{0}_{i}=p^{-1}(z^{0}_{i})\cap L^{0}_{i}$,
$i=1,\dots,n$.

Denote the sets of orbits by action of the symmetry groups by

$\pi^{v}_{D^{0}}(F,z^{0}_{i})=\pi_{1}(F^{0},z^{0}_{i})/Sym(D^{0},z^{0}_{i}),\quad
i=1,\dots,n,$

and

$\pi^{v}_{D^{0}}(F,z^{0}_{i},z^{0}_{j})=\pi_{1}(F^{0},z^{0}_{i},z^{0}_{j})/Sym(D^{0},z^{0}_{i},z^{0}_{j}),\quad
i,j=1,\dots,n.$

######  Definition 18.

Let $(D,F)\in\mathfrak{L}_{min}$ be a virtual link diagram. Let $L\subset
F\times(0,1)$ be a link such that $p(L)=D$. Choose non-crossing points
$z_{i}\in D_{i}$ and let $y_{i}\in L_{i}$ be such that $p(y_{i})=z_{i}$,
$i=1,\dots,n$.

Let $\Psi_{D}\colon F^{0}\times[0,1]\to F\times[0,1]$ be a diffeomorphism such
that $\Psi_{D}(L^{0}_{i})=L_{i}$ and $\Psi_{D}(y^{0}_{i})=y_{i}$.

Let $v$ be a self-crossing of a component $D_{i}$. The _homotopy index_
$h^{v}_{i}(v)$ of the crossing $v$ is defined by the formula

$h^{v}_{i}(v)=(\Psi_{D})^{-1}_{*}h_{D,z_{i}}(v)=(\Psi_{D})^{-1}_{*}[\hat{D}^{+}_{v,z_{i}}]\in\pi^{v}_{D^{0}}(F^{0},z^{0}_{i}).$
(8)

Let $v\in\mathcal{V}(D)$ be a mixed crossing of component type
$\tau(v)=(i,j)$. The _homotopy index_ $h^{v}_{ij}(v)$ of the crossing $v$ is
defined by the formula

$h^{v}_{ij}(v)=(\Psi_{D})^{-1}_{*}h_{D,z_{i},z_{j}}(v)\in\pi^{v}_{D^{0}}(F^{0},z^{0}_{i},z^{0}_{j}).$
(9)

where the homotopy class $h_{D,z_{i},z_{j}}(v)\in\pi_{1}(F,z_{i},z_{j})$ is
defined by the formula (3).

Note that by definition of the groups $\pi^{v}_{D^{0}}(F^{0},z^{0}_{i})$ and
$\pi^{v}_{D^{0}}(F^{0},z^{0}_{i},z^{0}_{j})$, the homotopy index is well
defined.

######  Theorem 7.

Let $\mathcal{L}$ be a virtual link and $(F^{0},D^{0})\in\mathfrak{L}_{min}$.
Let

$I^{v}=\bigsqcup_{i=1}^{n}\pi^{v}_{D^{0}}(F^{0},z^{0}_{i})\sqcup\bigsqcup_{i\neq
j}\pi^{v}_{D^{0}}(F^{0},z^{0}_{i},z^{0}_{j}).$

For a diagram $(F,D)\in\mathfrak{L}_{min}$ define a map
$\iota^{v}\colon\mathcal{V}(D)\to I^{v}$ by the formula

$\iota^{v}(v)=\left\\{\begin{array}[]{cl}h^{v}_{i}(v),&\tau(v)=(i,i),\\\
h^{v}_{ij}(v),&\tau(v)=(i,j),\ i\neq j.\end{array}\right.$

Then $\iota^{v}$ is the universal index on the diagram category
$\mathfrak{L}_{min}$.

###### Proof.

The index properties for $\iota^{v}$ for isotopies and Reidemeister moves
follow from the corresponding properties of the homotopy type. The property
(I0) for diffeomorphisms follows from the definition of the homotopy index
$h^{v}$.

Let us prove that $\iota^{v}$ is universal. Let $\iota$ be an index with
coefficients in a set $I$ on the diagram category $\mathfrak{L}_{min}$. We
need to construct a map $\phi\colon I^{v}\to I$ such that
$\iota(v)=\psi(\iota^{v}(v))$ for any crossing $v\in\mathcal{V}(D)$,
$(F,D)\in\mathfrak{L}_{min}$.

Firstly, consider the case of self-crossings of an $i$-th component. Let
$\alpha\in\pi_{1}(F^{0},z^{0}_{i})$. Pull a sprout from an arc near
$z^{0}_{i}$ along a representative path for $\alpha$ and get a diagram
${D^{0}}^{\prime}$ with a self-crossing $v\in\mathcal{V}(D^{\prime})$ of the
$i$-th component such that
$[\widehat{({D^{0}}^{\prime})}^{+}_{v,z^{0}_{i}}]=\alpha$ (Fig. 42). Note that
the diagram ${D^{0}}^{\prime}$ is obtained from $D^{0}$ by increasing second
Reidemeister moves. Set $\psi(\alpha)=\iota(v)$.

Figure 42: Construction of the crossing $v$

Let us show that $\psi$ is well defined. If $w$ another crossing such that
$[\hat{D}^{+}_{w,z^{0}_{i}}]=\alpha$ then
$h_{i}(v)=h_{i}(w)\in\hat{\pi}_{D^{0}}(F^{0},z^{0}_{i})$, hence,
$\iota(v)=\iota(w)$ by Proposition 11.

Let ${L^{0}}^{\prime}$ be a link which corresponds to the diagram
${D^{0}}^{\prime}$. We can suppose that the link ${L^{0}}^{\prime}$ is
obtained from $L^{0}$ by pulling a sprout from $y^{0}_{i}$ to a close
neighbourhood of $y^{0}_{i}$.

Let $\Phi\in\widetilde{Sym}(L^{0},y^{0}_{1},\dots,y^{0}_{n})$. Since
$\Phi(y^{0}_{i})=y^{0}_{i}$, the link $\Phi({L^{0}}^{\prime})$ differ from
$\Phi(L^{0})=L^{0}$ by a sprout, and the diagram
${D^{0}}^{\prime\prime}=\Phi({D^{0}}^{\prime})$ differs by a sprout from
$D^{0}$. The diffeomorphism $\Phi$ maps the crossing $v$ to a crossing
$w=\Phi_{*}(v)\in\mathcal{V}({D^{0}}^{\prime\prime})$. The homotopy class of
the based half of the crossing $w$ is equal to
$[\widehat{({D^{0}}^{\prime\prime})}^{+}_{w,z^{0}_{i}}]=\Phi_{*}([\widehat{({D^{0}}^{\prime})}^{+}_{v,z^{0}_{i}}])=\Phi_{*}(\alpha)$.
On the other hand, by the property (I0), $\iota(w)=\iota(v)$. Thus,
$\psi(\Phi_{*}(\alpha))=\psi(\alpha)$, and $\psi$ is well defined.

Let $(F,D)\in\mathfrak{L}_{min}$ and $v\in\mathcal{V}(D)$. Let $L\subset
F\times(0,1)$ be a link such that $p(L)=D$ and $\Psi_{D}$ be a diffeomorphism
which maps $L^{0}$ to $L$. Let
$\alpha=\Psi^{-1}_{*}([\hat{D}^{+}_{v,z_{i}}])\in\pi_{1}(F^{0},z^{0}_{i})$. By
definition the orbit of $\alpha$ in $\pi^{v}_{D^{0}}(F^{0},z^{0}_{i})$ is the
homotopy index $h^{v}_{i}(v)$.

Create a crossing $w\in\mathcal{V}({D^{0}}^{\prime})$ such that
$[\widehat{({D^{0}}^{\prime})}^{+}_{w,z^{0}_{i}}]=\alpha$ as before. Let
$D^{\prime}=p(\Psi({L^{0}}^{\prime}))$. The diagram $D^{\prime}$ differs from
$D$ by a sprout. Let $w^{\prime}=\Psi_{*}(w)\in\mathcal{V}(D^{\prime})$. By
construction
$h_{D^{\prime},z_{i}}(w^{\prime})=\Psi_{*}(\alpha)=h_{D,z_{i}}(v)=h_{D^{\prime},z_{i}}(v)$.
Then $w^{\prime}$ and $v$ belong to one tribe in $D^{\prime}$ and by
Proposition 1 and the property (I0)
$\iota(v)=\iota(w^{\prime})=\iota(w)=\psi(\iota^{v}(v))$. ∎

######  Example 4.

Consider the link $D=D_{1}\cup D_{2}$ in the torus in Fig. 43. The link is not
classical, so the surface genus is minimal.

Figure 43: A link in the torus

The homotopy group of the torus does not depend on the base point and is equal
to $\pi_{1}(T^{2},z)=H_{1}(T^{2},\mathbb{Z})=\mathbb{Z}^{2}$. The symmetry
groups are $Sym_{1}(D)=Sym_{2}(D)=Sym_{12}(D)=Sym_{21}(D)=\mathbb{Z}_{2}$ (the
nontrivial element of the groups acts like $x\mapsto-x$ on
$\pi_{1}(T^{2})=\mathbb{Z}^{2}$). Thus, $\pi^{v}_{D}=\mathbb{Z}_{2}^{2}$. That
means there are 16 types of the crossings in diagrams in the torus of the
virtual link. The crossings in Fig. 43 present 4 different types (the
crossings with the same component type have the same index value).

######  Remark 8.

Although we have described the index only for the subcategory
$\mathfrak{L}_{min}$ in the category $\mathfrak{L}$ of all diagrams of the
virtual link, this is enough to calculate link invariants which use an index
in their construction. Indeed, if $Inv$ is an invariant (e.g. an index
polynomial) which relies on an index $\iota$ with coefficients in a set $I$ on
the diagrams of a link $\mathcal{L}$, then $Inv$ can be calculated on a
diagram $(F,D)$ of the minimal genus. For such a diagram,
$\iota=\psi\circ\iota^{v}$ for some $\psi\colon I^{v}\to I$, hence, the
invariant $Inv$ is expressed by the index $\iota^{v}$, i.e. by the component
index $\tau$ and homotopy index $h^{v}$.

###  5.2 Flat knots

A _flat link diagram_ is a pair $(F,D)$ where $F$ is an oriented closed
connected surface and $D\subset F$ is an embedded $4$-valent graph. A _flat
link_ is an equivalence class of flat link diagrams modulo isotopies of the
surface, Reidemeister moves, orientation preserving diffeomorphisms of pairs
$(F,D)$ and (de)stabilizations. We assume that the components of a flat link
are numbered and oriented.

An analogue of Kuperberg’s theorem is the following statement [9] (see also
[12] for the knot case).

######  Theorem 8.

Let $(F,D)$, $D=D_{1}\cup\cdots\cup D_{n}$, and $(F^{\prime},D^{\prime})$,
$D^{\prime}=D^{\prime}_{1}\cup\cdots\cup D^{\prime}_{n}$, be representatives
of a flat link $\mathcal{L}$ with $n$ components such that the genus of $F$
and $F^{\prime}$ is minimal among all diagrams of $\mathcal{L}$. Then there is
an orientation preserving diffeomorphism $\Phi\colon F\to F^{\prime}$ such
that for all $i=1,\dots,n$ the free loops $\Phi(D_{i})$ and $D^{\prime}_{i}$
are homotopic in $F^{\prime}$.

Let $\mathcal{L}$ be a flat link. Consider the category $\mathfrak{L}_{min}$
whose objects are the diagrams $(F,D)$ of $\mathcal{L}$ of minimal genus, and
the morphisms are compositions of isotopies, Reidemeister moves and
diffeomorphisms. By Theorem 8 any two diagrams $(D,F)$ and
$(D^{\prime},F^{\prime})$ in $\mathfrak{L}_{min}$ are connected with a
morphism of $\mathfrak{L}_{min}$.

We will define the universal index on the diagram category
$\mathfrak{L}_{min}$.

######  Definition 19.

Let $(F,D)$, $D=D_{1}\cup\cdots\cup D_{n}$, be a flat link diagram. Consider
the following subgroup in $Diff^{+}(F)$:

$\widetilde{Sym}^{f}(D)=\\{\Phi\in Diff^{+}(F)\mid\Phi(D_{i})\sim D_{i},\
i=1,\dots,n\\}.$

Choose non-crossing points $z_{i}\in D_{i}$, $i=1,\dots,n$. Let
$\kappa_{z_{i}}=[D_{i}]\in\pi_{1}(F,z_{i})$.

Let $\Phi\in\widetilde{Sym}^{f}(D)$. By definition, there exist paths
$\gamma_{i}$ from $z_{i}$ to $\Phi(z_{j})$ such that
$\gamma_{i}\Phi(\kappa_{z_{i}})\gamma_{i}^{-1}=\kappa_{z_{i}}$. Note that the
path $\gamma_{i}$ is defined up to multiplication by elements from
$IM_{i}(D,z_{i})$. Denote the map

$\alpha\mapsto\gamma_{i}\Phi(\alpha)\gamma_{i}^{-1},\quad\alpha\in\pi_{1}(F,z_{i}),$

by $\Phi_{\gamma_{i}}\in Aut(\pi_{1}(F,z_{i}))$. The set

$Sym^{f}(D,z_{i})=\\{\Phi_{\gamma_{i}}\mid\Phi\in\widetilde{Sym}^{f}(D),\
\gamma_{i}\Phi(\kappa_{z_{i}})\gamma_{i}^{-1}=\kappa_{z_{i}}\\}\subset
Aut(\pi_{1}(F,z_{i}))$

is called the _flat symmetry group_ of the component $D_{i}$.

Denote the map

$\alpha\mapsto\gamma_{i}\Phi(\alpha)\gamma_{j}^{-1},\quad\alpha\in\pi_{1}(F,z_{i},z_{j}),$

by $\Phi_{\gamma_{i},\gamma_{j}}\in Aut(\pi_{1}(F,z_{i},z_{j}))$. The set

$Sym^{f}(D,z_{i},z_{j})=\\{\Phi_{\gamma_{i},\gamma_{j}}\mid\Phi\in\widetilde{Sym}^{f}(D)\\}\subset
Aut(\pi_{1}(F,z_{i}))$

is called the _flat symmetry group_ of the components $D_{i}$, $D_{j}$.

######  Proposition 16.

Let $D=D_{1}\cup\cdots\cup D_{n}\in\mathfrak{L}_{min}$ be a flat link diagram
and $z_{i}\in D_{i}$, $i=1,\dots,n$, be non-crossing points. Then

  1. 1.

The sets $Sym^{f}(D,z_{i})$ and $Sym^{f}(D,z_{i},z_{j})$ are groups.

  2. 2.

For any other diagram $D^{\prime}$ and a set of non-crossing points
$z^{\prime}_{i}\in D^{\prime}_{i}$, $i=1,\dots,n$, we have
$Sym^{f}(D,z_{i})\simeq Sym^{f}(D^{\prime},z^{\prime}_{i})$ and
$Sym^{f}(D,z_{i},z_{j})\simeq
Sym^{f}(D^{\prime},z^{\prime}_{i},z^{\prime}_{j})$ for all $1\leq i,j\leq n$.

  3. 3.

Let $\gamma\in IM_{i}(D,z_{i})$. Then $Ad_{\gamma}\in Sym^{f}(D,z_{i})$ where
$Ad_{\gamma}(\alpha)=\gamma\alpha\gamma^{-1}$, $\alpha\in\pi_{1}(F,z_{i})$, is
the adjoint action.

  4. 4.

Let $(\gamma_{i},\gamma_{j})\in IM_{ij}(D,z_{i},z_{j})$. Then
$Ad_{\gamma_{i},\gamma_{j}}\in Sym^{f}(D,z_{i},z_{j})$ where
$Ad_{\gamma_{i},\gamma_{j}}(\alpha)=\gamma_{i}\alpha\gamma_{j}^{-1}$,
$\alpha\in\pi_{1}(F,z_{i},z_{j})$.

  5. 5.

For any $\phi\in Sym^{f}(D,z_{i})$ we have
$\phi(\kappa_{z_{i}})=\kappa_{z_{i}}$.

###### Proof.

1\. Let $\Phi_{\gamma_{i}},\Phi^{\prime}_{\gamma^{\prime}_{i}}\in
Sym^{f}(D,z_{i})$. Then

$\Phi_{\gamma_{i}}\circ\Phi^{\prime}_{\gamma^{\prime}_{i}}=(\Phi\circ\Phi^{\prime})_{\gamma_{i}\Phi(\gamma_{i}^{\prime})}\in
Sym^{f}(D,z_{i})$

and $(\Phi_{\gamma_{i}})^{-1}=(\Phi^{-1})_{\Phi^{-1}(\gamma_{i}^{-1})}$.

2\. Assume first that $D^{\prime}\subset F^{\prime}$ and there exists a
diffeomorphism $\Psi\colon F\to F^{\prime}$ such that
$D^{\prime}_{i}=\Psi(D_{i})$ and $\Psi(z_{i})=z^{\prime}_{i}$, $i=1,\dots,n$.
Then the map

$\Phi\mapsto\Psi\circ\Phi\circ\Psi^{-1}$

identifies the groups $\widetilde{Sym}^{f}(D)$ and
$\widetilde{Sym}^{f}(D^{\prime})$, and the map

$\Phi_{\gamma_{i}}\mapsto(\Psi\circ\Phi\circ\Psi^{-1})_{\Psi(\gamma_{i})}$

identifies the groups $Sym^{f}(D,z_{i})$ and
$Sym^{f}(D^{\prime},z^{\prime}_{i})$.

Now, let $D^{\prime}\subset F$ and $D^{\prime}_{i}\sim D_{i}$, $i=1,\dots,n$.
Choose paths $\beta_{i}$ from $z_{i}$ to $z^{\prime}_{i}$ such that
$\beta_{i}\kappa^{\prime}_{z^{\prime}_{i}}\beta_{i}^{-1}=\kappa_{z_{i}}$. Let
$H_{t}\colon F\to F$, $t\in[0,1]$ be an isotopy which moves the points $z_{i}$
to $z^{\prime}_{i}$ along the paths $\beta_{i}$. Then
$H_{1}(\kappa_{z_{i}})=\kappa^{\prime}_{z^{\prime}_{i}}\in\pi_{1}(F,z^{\prime}_{i})$.
The isomorphism between $Sym^{f}(D,z_{i})$ and
$Sym^{f}(D^{\prime},z^{\prime}_{i})$ is given by the formula
$\Phi_{\gamma_{i}}\mapsto(H_{1}\circ\Phi\circ
H_{1}^{-1})_{H_{1}(\gamma_{i})}$.

The isomorphisms $Sym^{f}(D,z_{i},z_{j})\simeq
Sym^{f}(D^{\prime},z^{\prime}_{i},z^{\prime}_{j})$ are proved analogously.

3\. Let $\gamma\in IM_{i}(D,z_{i})$. Consider an isotopy $H_{t}$, $t\in[0,1]$,
which pulls the point $z_{i}$ along the path $\gamma^{-1}$. Then
$Ad_{\gamma}=(H_{1})_{1}$ where $1\in\pi_{1}(F,z_{i})$ is the trivial loop.

4\. The fourth statement is proved analogously to the previous one.

5\. The last statement holds by definition. ∎

Let $\mathcal{L}$ be a flat link. Fix a diagram
$(F^{0},D^{0})\in\mathfrak{L}_{min}$, $D^{0}=D^{0}_{1}\cup\cdots\cup
D^{0}_{n}$. Choose non-crossing points $z^{0}_{i}\in D^{0}_{i}$,
$i=1,\dots,n$. Let
$\kappa^{0}_{z^{0}_{i}}=[D^{0}_{i}]\in\pi_{1}(F^{0},z^{0}_{i})$.

Denote the sets of orbits by action of the symmetry groups by

$\pi^{f}_{D^{0}}(F,z^{0}_{i})=\pi_{1}(F^{0},z^{0}_{i})/Sym^{f}(D^{0},z^{0}_{i})/\sigma,\quad
i=1,\dots,n,$

and

$\pi^{f}_{D^{0}}(F,z^{0}_{i},z^{0}_{j})=\pi_{1}(F^{0},z^{0}_{i},z^{0}_{j})/Sym^{f}(D^{0},z^{0}_{i},z^{0}_{j}),\quad
i,j=1,\dots,n.$

The involution $\sigma$ on $\pi_{1}(F^{0},z^{0}_{i})$ is defined by the
formula $\sigma(\alpha)=\kappa^{0}_{z^{0}_{i}}\alpha^{-1}$.

######  Definition 20.

Let $(D,F)\in\mathfrak{L}_{min}$ be a flat link diagram. Choose non-crossing
points $z_{i}\in D_{i}$, $i=1,\dots,n$. Let
$\kappa_{z_{i}}=[D_{i}]\in\pi_{1}(F,z_{i})$.

By Theorem 8 there exist a diffeomorphism $\Psi_{D}\colon F^{0}\to F$ and
paths $\gamma_{i}$ from $z_{i}$ to $\Psi_{D}(z^{0}_{i})$ such that
$\gamma_{i}^{-1}\kappa_{z_{i}}\gamma_{i}=(\Psi_{D})_{*}(\kappa^{0}_{z^{0}_{i}})\in\pi_{1}(F,\Psi_{D}(z^{0}_{i}))$.

Let $v$ be a self-crossing of a component $D_{i}$. The _flat homotopy index_
$h^{f}_{i}(v)$ of the crossing $v$ is defined by the formula

$h^{f}_{i}(v)=(\Psi_{D})_{\gamma_{i}}^{-1}h_{D,z_{i}}(v)=(\Psi_{D})^{-1}_{\gamma_{i}}[\hat{D}^{+}_{v,z_{i}}]\in\pi^{f}_{D^{0}}(F^{0},z^{0}_{i}).$
(10)

where
$(\Psi_{D})_{\gamma_{i}}\colon\pi_{1}(F^{0},z^{0}_{i})\to\pi_{1}(F,z_{i})$ is
given by the formula

$\alpha\mapsto\gamma_{i}\cdot(\Psi_{D})_{*}(\alpha)\cdot\gamma_{i}^{-1}.$

Let $v\in\mathcal{V}(D)$ be a mixed crossing of component type
$\tau(v)=(i,j)$. The _homotopy index_ $h^{v}_{ij}(v)$ of the crossing $v$ is
defined by the formula

$h^{f}_{ij}(v)=(\Psi_{D})^{-1}_{\gamma_{i},\gamma_{j}}h_{D,z_{i},z_{j}}(v)\in\pi^{v}_{D^{0}}(F^{0},z^{0}_{i},z^{0}_{j}).$
(11)

where the homotopy class $h_{D,z_{i},z_{j}}(v)\in\pi_{1}(F,z_{i},z_{j})$ is
defined by the formula (3) and the map $(\Psi_{D})_{\gamma_{i},\gamma_{j}}$ is
defined by the formula

$\alpha\mapsto\gamma_{i}\cdot(\Psi_{D})_{*}(\alpha)\cdot\gamma_{j}^{-1}.$

######  Theorem 9.

Let $\mathcal{L}$ be a flat link and $(F^{0},D^{0})\in\mathfrak{L}_{min}$. Let

$I^{f}=\bigsqcup_{i=1}^{n}\pi^{f}_{D^{0}}(F^{0},z^{0}_{i})\sqcup\bigsqcup_{i\neq
j}\pi^{f}_{D^{0}}(F^{0},z^{0}_{i},z^{0}_{j}).$

For a diagram $(F,D)\in\mathfrak{L}_{min}$ define a map
$\iota^{f}\colon\mathcal{V}(D)\to I^{f}$ by the formula

$\iota^{f}(v)=\left\\{\begin{array}[]{cl}h^{f}_{i}(v),&\tau(v)=(i,i),\\\
h^{f}_{ij}(v),&\tau(v)=(i,j),\ i\neq j.\end{array}\right.$

Then $\iota^{f}$ is the universal index on the diagram category
$\mathfrak{L}_{min}$.

The theorem is proved analogously to Theorem 7.

######  Example 5.

Consider the flat knot $K$ if a surface $F$ of genus $2$ presented in Fig. 44.
The diagram has a $\mathbb{Z}_{5}$-symmetry, and any diffeomorphism
$\Phi\in\widetilde{Sym}^{f}(K)$ is isotopic to a rotation of the diagram. The
internal monodromy group is
$IM(K)=C(\kappa)=\langle\kappa\rangle\subset\pi_{1}(F,z)$ where $\kappa=[K]$.

Figure 44: A flat knot in a surface of genus two

The fundamental group has a presentation with generators $a,b,c,d,e$ (a
generator is a loop which intersects the corresponding edge once and does not
intersect other edges) and relations $abcde=1$ and
$a^{-1}b^{-1}c^{-1}d^{-1}e^{-1}=1$. The class $\kappa$ is equal to $cebda$.
The homotopy indices of the crossings are $h^{f}(v_{1})=c$, $h^{f}(v_{2})=a$,
$h^{f}(v_{3})=a^{-1}da$, $h^{f}(v_{4})=a^{-1}d^{-1}bda$,
$h^{f}(v_{5})=cec^{-1}$.

Let $\Phi\in\widetilde{Sym}^{f}(K)$ be the counter clockwise rotation by the
angle $\frac{2\pi}{5}$. Let $\gamma$ be the path in $K$ from $z$ to $\Phi(z)$
which has the opposite orientation to that of $K$. Then
$\Phi_{\gamma}(a)=a^{-1}da$, $\Phi_{\gamma}(b)=a^{-1}ea$,
$\Phi_{\gamma}(c)=a$, $\Phi_{\gamma}(d)=a^{-1}ba$,
$\Phi_{\gamma}(e)=a^{-1}ca$. Then $\Phi_{\gamma}(h^{f}(v_{1}))=h^{f}(v_{2})$,
$\Phi_{\gamma}(h^{f}(v_{2}))=h^{f}(v_{3})$,
$\Phi_{\gamma}(h^{f}(v_{3}))=h^{f}(v_{4})$,
$Ad_{\kappa}\Phi_{\gamma}(h^{f}(v_{4}))=h^{f}(v_{5})$ and
$\Phi_{\gamma}(h^{f}(v_{5}))=h^{f}(v_{1})$. We see that the homotopy indices
of all crossings coincide.

## References

  * [1] J. S. Carter, S. Kamada, and M. Saito, Stable equivalence of knots on surfaces and virtual knot cobordisms J. Knot Theory Ramifications 11:3 (2002) 311–322.
  * [2] Z. Cheng, A polynomial invariant of virtual knots, _Proc. Amer. Math. Soc._ 142:2 (2014) 713–725.
  * [3] Z. Cheng, A transcendental function invariant of virtual knots, J. Math. Soc. Japan 69:4 (2017) 1583–1599.
  * [4] Z. Cheng, The chord index, its definitions, applications and generalizations, _Canad. J. Math._ 73:3 (2021) 597–621.
  * [5] Z. Cheng, H. Gao, M. Xu, Some remarks on the chord index, _J. Knot Theory Ramifications_ 29:10 (2020) 2042003.
  * [6] Z. Cheng, D.A. Fedoseev, H. Gao, V.O. Manturov, M. Xu, From chord parity to chord index, _J. Knot Theory Ramifications_ 29:13 (2020) 2043004.
  * [7] H. Dye, Smoothed invariants, _J. Knot Theory Ramifications_ 21:13 (2012) 1240003.
  * [8] L.C. Folwaczny, L.H. Kauffman, A linking number definition of the affine index polynomial and applications, _J. Knot Theory Ramifications_ 22:12 (2013) 1341004.
  * [9] D. Freund, Intersections of virtual multistrings, Thesis (Ph.D.)–Dartmouth College. 2018.
  * [10] A. Henrich, A sequence of degree one vassiliev invariants for virtual knots, J. Knot Theory Ramifications 19:4 (2010) 461–487.
  * [11] Y.H. Im, K. Lee, S.Y. Lee, Index polynomial invariant of virtual links, _J. Knot Theory Ramifications_ 19 (2010) 709–725.
  * [12] D. P. Ilyutko, V. O. Manturov, I. M. Nikonov, Parity in knot theory and graph links, J. Math. Sci. 193:6 (2013) 809–965.
  * [13] M.-J. Jeong, A zero polynomial of virtual knots, J. Knot Theory Ramifications 25:1 (2016) 1550078.
  * [14] N. Kamada and S. Kamada, “Abstract link diagrams and virtual knots”, Knot Theory and Its Ramifications J. 9:1, 93–109 (2000).
  * [15] L. H. Kauffman, An affine index polynomial invariant of virtual knots, _J. Knot Theory Ramifications_ 22:4 (2013) 1340007.
  * [16] K. Kaur, M. Prabhakar, A. Vesnin, Two-variable polynomial invariants of virtual knots arising from flat virtual knot invariants, _J. Knot Theory Ramifications_ 27:13 (2018) 1842015.
  * [17] J. Kim, The affine index polynomial invariant of flat virtual knots, _J. Knot Theory Ramifications_ 23:14 (2014) 1450073.
  * [18] G. Kuperberg, “What is a Virtual Link?”, Algebraic and Geometric Topology 3, 587–591 (2003).
  * [19] I. M. Nikonov, Weak parities and functorial maps, _J. Math. Sci._ 214:5 (2016) 699–717.
  * [20] I. M. Nikonov, Parity functors, arxiv:2109.12230.
  * [21] I. M. Nikonov, Parity on based matrices, arxiv:2110.04915.
  * [22] I. M. Nikonov, An intersection formula for parities on virtual knots, arxiv:2110.08392.
  * [23] V. Turaev, Virtual strings, Ann. Inst. Fourier 54:7 (2004) 2455–2525.
  * [24] V. Turaev, Cobordism of knots on surfaces, J. Topol 1:2 (2008) 285–305.
  * [25] M.Xu, Writhe polynomial for virtual links, arXiv:1812.05234.