# Parity on based matrices

Igor Nikonov

###### Abstract

A parity is a labeling of the crossings of knot diagrams which is compatible
with Reidemeister moves. We define the notion of parity for based matrices —
algebraic objects introduced by V. Turaev [11] in his research of virtual
strings. We present the reduced stable parity on based matrix which gives a
new example of a parity of virtual knots.

## Introduction

V.O. Manturov [8] defined a parity as a rule to assign labels $0$ and $1$
(considered as elements of $\mathbb{Z}_{2}$) to the crossings of knot diagrams
in a way compatible with Reidemeister moves. Capability to distinguish odd and
even crossings allows to treat the crossings differently when transforming
knot diagrams or calculating their invariants. The notion of parity has proved
to be an effective tools in knot theory. It allows to strengthen knot
invariants, to prove minimality theorem and to construct (counter)examples
[5].

Then a natural question is to describe existing parities on virtual knot. So
far the only known parity on virtual knots was the Gaussian parity [8]. On the
other hand, virtual knots can be viewed as knots in surfaces modulo isotopy,
Reidemeister moves and (de)stabilization. The parities for knots in a fixed
surface are well known [5]. They comes from homology of the underlying
surface: the parity of a knot crossing is the homology class of a knot half at
the crossing. This is the point where a link to the based matrices is
established.

Based matrices introduced by V. Turaev [11] in his research of virtual strings
(also known as flat knots). One can assign to a diagram of a flat knot the
matrix whose elements are intersection numbers of diagram halves at the
crossing of the diagram. Reidemeister moves induce transformations of based
matrices. The reduced form of based matrices is a powerful invariant of flat
knots, and some other known invariants can be deduced from it.

Later analogues of based matrices were introduced for virtual knots [12],
singular flat knots [1, 4], framed and long virtual knots [10] and virtual
links [2].

The aim of the present paper is to define parity on based matrices and find
new parities for flat and virtual knots.

The paper is structured as follows. Section 1 recalls definitions of flat
knots, parity and based matrices. In Section 2 we introduce the definition of
parity and parity functor on based matrices and present an example of such a
parity — stable parity. The stable parity determines a new parity on flat and
virtual knots.

##  1 Definitions

###  1.1 Virtual and flat knots

A $4$-graph is any union of four-valent graphs and trivial components, i.e.
circles considered as graphs without vertices and with one (closed) edge. A
virtual diagram is an embedding of a $4$-graph into plane so that each vertex
of the graph is marked as either classical of virtual vertex. At a classical
vertex one a pair of opposite edges (called overcrossing) is chosen. The other
pair of opposite edges at the vertex is called undercrossing. The vertices of
the diagram are called also crossings.

Virtual crossings of a virtual diagram are usually drawn circled. The
undercrossing of a classical vertex is drawn with a broken line whereas the
overcrossing is drawn with a solid line (see Fig. 1). A diagram without
virtual crossings is classical.

Figure 1: Virtual trefoil diagram with two classical and one virtual crossings

######  Definition 1.

A virtual link [7] is an equivalence class of virtual diagrams modulo
Reidemeister moves and detour moves (Fig. 2).

Figure 2: Reidemester moves and the detour move

A unicursal component is a minimal set of diagram edges which is closed under
passing from an edge to its opposite (at some end of the edge) edge. A diagram
with one unicursal component is called a diagram of a virtual knot.

Below we assume that the unicursal components of diagrams are oriented.

There is another description of virtual knots.

A virtual link can be viewed as an equivalence class of pairs $(S,D)$ where
$S$ is a closed oriented surface and $D$ is a diagram in $S$ whose crossings
are all classical [6]. The equivalence relation is generated by diagram
isotopies, classical Reidemeister moves and stabilizations (see Fig. 3) which
change the surface.

Figure 3: Stabilization move

For example, the virtual trefoil can be given by a diagram in the torus (Fig.
4).

Figure 4: Virtual trefoil

If one excludes the stabilization moves she gets a knot theory in a given
surface.

If one admits crossing switch transformations (Fig. 5) of virtual diagrams,
i.e. neglects the over-undercrossing structure, one gets the theory of flat
knots.

Figure 5: Crossing switching

Equivalently, flat knots (links) are equivalence classes of flat diagrams
($4$-valent graphs whose vertices are virtual crossing and classical crossings
without under-overcrossing structure, Fig. 6) modulo flat Reidemeister moves
(Fig. 7).

Figure 6: Flat knot diagram

Figure 7: Flat Reidemeister moves

Flat knots can be also identified with generic immersions of the circle into
surfaces considered up to homotopies, isomorphisms and
stabilizations/destabilizations.

###  1.2 Based matrices

Let $H$ be an abelian group.

######  Definition 2 ([11]).

A based matrix with coefficients in the group $H$ is a triple $(G,s,b)$ where
$G$ is a finite set, $s\in G$ and a map $b\colon G\times G\to H$ which defines
a skew-symmetric matrix, i.e. $b(h,g)=-b(g,h)$ for any $g,h\in G$.

Let us denote $G^{\circ}=G\setminus\\{s\\}$.

The motivating example of based matrices comes from intersection matrices for
diagrams of flat knots. Let $H=\mathbb{Z}$ or $\mathbb{Z}_{2}$.

######  Example 1 (Based matrix of a virtual (flat) knot).

Let $D$ be a diagram of an oriented virtual or flat knot in a surface $S$ and
${\mathcal{V}}(D)$ be the set of its classical crossings. For any crossing
$c\in{\mathcal{V}}(D)$ define the left and right halves of the diagram at the
crossing as shown in Fig. 8.

Figure 8: Halves of the diagram

The halves $D^{l}_{c}$, $c\in{\mathcal{V}}(D)$, and the diagram $D$ correspond
to elements of the homology group $H_{1}(S,H)$.

Let us define the based matrix of the diagram. Let
$G(D)={\mathcal{V}}(D)\sqcup\\{s\\}$ where $s$ is a formal element. The map
$b_{D}$ is given by the formula

$b_{D}(c,c^{\prime})=D^{l}_{c}\cdot D^{l}_{c^{\prime}},\quad
b_{D}(c,s)=D^{l}_{c}\cdot D,\quad b_{D}(s,s)=D\cdot D=0,$ (1)

where $c,c^{\prime}\in{\mathcal{V}}(D)$ and the dot denotes the intersection
map $H_{1}(S,H)\times H_{1}(S,H)\to H_{0}(S,H)=H$. In other word,
$b_{D}(c,c^{\prime})$ is the intersection number of the cycles $D^{l}_{c}$ and
$D^{l}_{c^{\prime}}$. The triple $T(D)=(G(D),s,b_{D})$ is the based matrix of
the diagram $D$.

For example, in case $H=\mathbb{Z}_{2}$ for the diagram $D$ in Fig. 6 with
three classical crossings we have $G(D)=\\{s,1,2,3\\}$ and

$b_{D}=\left(\begin{array}[]{c|ccc}0&1&1&0\\\ \hline\cr 1&0&0&0\\\ 1&0&0&1\\\
0&0&1&0\end{array}\right).$

######  Definition 3.

Let $T=(G,s,b)$ be a based matrix. Then

  * •

element $g\in G^{\circ}$, is called annihilating if $b(g,h)=0$ for any $h\in
G$;

  * •

element $g\in G^{\circ}$, is called core if $b(g,h)=b(s,h)$ for any $h\in G$;

  * •

elements $g_{1},g_{2}\in G^{\circ}$, are called complementary if
$b(g_{1},h)+b(g_{2},h)=b(s,h)$ for any $h\in G$.

######  Definition 4.

Two based matrices are homologous if one can be obtained from the other by a
sequence of the operations:

  * •

$M_{1}$ which transforms $(G,s,b)$ into $(G_{1}=G\sqcup\\{g\\},s,b_{1})$ such
that $b_{1}\colon G_{1}\times G_{1}\to H$ extends $b$ and $b_{1}(g,h)=0$ for
all $h\in G_{1}$;

  * •

$M_{2}$ which transforms $(G,s,b)$ into $(G_{2}=G\sqcup\\{g\\},s,b_{1})$ such
that $b_{2}\colon G_{2}\times G_{2}\to H$ extends $b$ and
$b_{2}(g,h)=b_{2}(s,h)$ for all $h\in G_{2}$;

  * •

$M_{3}$ which transforms $(G,s,b)$ into
$(G_{1}=G\sqcup\\{g,g^{\prime}\\},s,b_{1})$ such that $b_{3}\colon G_{3}\times
G_{3}\to H$ is any skew-symmetric map extending $b$ with
$b_{3}(g,h)+b_{3}(g^{\prime},h)=b_{3}(s,h)$ for all $h\in G_{3}$

and the inverse operations $M_{1}^{-1},M_{2}^{-1},M_{3}^{-1}$.

######  Definition 5.

A based matrix is called primitive if it contains neither annihilating, nor
core, nor complementary elements.

######  Lemma 1 ([11]).

Any based matrix is homologous to a unique up to isomorphism primitive based
matrix.

######  Lemma 2 ([11]).

If two virtual (flat) diagrams are equivalent then the corresponding based
matrices are homologous.

The Lemmas 1 and 2 imply that the unique primitive based matrix
$T_{\bullet}(D)=(G_{\bullet}(D),s,b_{\bullet})$ homologous to the based matrix
$T(D)=(G(D),x,b_{D})$ of a virtual (flat) knot diagram is an invariant of
virtual (flat) knots.

###  1.3 Parity

Let $\mathcal{K}$ be a virtual or flat knot. Consider the set $\mathfrak{K}$
of the diagrams of the knot $K$. The set $\mathfrak{K}$ can be considered as
the objects of a diagram category whose morphisms are compositions of
isotopies and Reidemeister moves.

For any diagrams $D,D^{\prime}\in\mathfrak{K}$ 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 set because some crossing can disappear within the transformation $f$
and other crossing can appear by a first or second Reidemeister move.

Let $A$ be an abelian group.

######  Definition 6.

A parity on the diagrams of the knot $\mathcal{K}$ with coefficients in the
group $A$ is a family of maps $p_{D}\colon{\mathcal{V}}(D)\to A$ from the sets
of crossings ${\mathcal{V}}(D)$ of knot diagrams $D$, such that

  * •

Reidemeister moves do not change the parity value of any crossing (which
survives under the move),

  * •

the sum of parities values of the crossing participating in a Reidemeister
move is equal to zero (Fig. 9).

Figure 9: Parity axioms

The first and the main example of parity on virtual knots is the Gaussian
parity (with coefficients in $\mathbb{Z}_{2}$).

######  Example 2.

The Gaussian parity of a crossing is the parity of the number of classical
crossings that lie on a half of the knot corresponding to the crossing. For
example, a diagram of a virtual eight-knot in Fig. 10 has two odd and one even
crossings.

Figure 10: Gaussian parity on a virtual knot diagram

Another example are homological parities on diagrams of knots in a fixed
surface $S$.

######  Theorem 1 ([5]).

Let $p$ be a parity for diagrams of some (flat) knot $\mathcal{K}$ in a
surface $S$ with coefficients in a group $A$. Then there exists a unique
homomorphism $\phi\colon H_{1}(S,\mathbb{Z}_{2})/[K]\to A$ such that for any
crossing $v$ of a knot diagram $D$ its parity is equal to
$p(v)=\phi[D^{l}_{v}]$. Here $D^{l}_{v}$ is the left half of the diagram at
the vertex $v$ (see Fig. 8).

As a corollary of the theorem we have the statement that any parity on
classical knots is trivial.

Theorem 1 shows that a parity on diagrams of knots in a fixed surface is
determined by the homology of the surface. Since homological information on
the knot is accumulated by the based matrix of the diagram, we can express the
parity in terms of based matrices and use this expression to define a parity
for virtual and flat knots.

##  2 Parity on based matrices

Let $A$ be an abelian group.

######  Definition 7.

A parity on based matrices with coefficients in $A$ is a family of maps
$p_{T}\colon G\to A$ defined for any based matrix $T=(G,s,b)$ such that

  * (P0)

if a based matrix $T^{\prime}$ is obtained from a based matrix $T$ by adding
an annihilating or core element of a pair of complementary elements then for
any $g\in G$ $p_{T}(g)=p_{T^{\prime}}(g)$;

  * (P1)

if $g\in G^{\circ}$ is an annihilating or core elements then $p_{T}(g)=0$,

  * (P2)

if $g_{1},g_{2}\in G^{\circ}$ are complementary then
$p_{T}(g_{1})+p_{T}(g_{2})=0$,

  * (P3)

if $g_{1},g_{2},g_{3}\in G^{\circ}$ such that
$b(g_{1},h)+b(g_{2},h)+b(g_{3},h)=b(s,h)$ for any $h\in G$ then
$p_{T}(g_{1})+p_{T}(g_{2})+p_{T}(g_{3})=0$.

By definition we set $p_{T}(s)=0$.

######  Example 3.

Gaussian parity with coefficients in $A=\mathbb{Z}$ or $\mathbb{Z}_{2}$ on
based matrices with coefficients in $A$ is given by the formula
$p_{T}(g)=b(g,s)\in A$. The parity properties follows from the fact that the
moves $M_{i}$, $i=1,2,3$, extend the form $b$ and in particular keep the value
$b(g,s)$, $g\in G^{\circ}$, and from the equality $b(s,s)=0$.

Below we use a more general notion than parity — parity functor [9]. Parity
functor uses different coefficient groups for different diagrams, and the
coefficient groups are linked by (partial) isomorphism which allow to identify
parity values of a crossing under transformations (isotopies and Reidemeister
moves). Here is the formal definition.

######  Definition 8.

A parity functor on based matrices is a family of maps $P_{T}\colon G\to A(T)$
into abelian groups $A(T)$ defined for any based matrix $T=(G,s,b)$ such that

  * (P0)

if a based matrix $T^{\prime}=(G^{\prime},s,b^{\prime})$ is obtained from a
based matrix $T=(G,s,b)$ by a move $f$ of type $M_{1}$, $M_{2}$ or $M_{3}$
then there is a fixed monomorphism $A(f)\colon A(T)\to A(T^{\prime})$ such
that $P_{T^{\prime}}(g)=A(f)(P_{T}(g))$ for any $g\in G$;

  * (P1)

if $g\in G^{\circ}$ is an annihilating or core element, or $g=s$ then
$p_{T}(g)=0$,

  * (P2)

if $g_{1},g_{2}\in G^{\circ}$ are complementary then
$P_{T}(g_{1})+P_{T}(g_{2})=0$,

  * (P3)

if $g_{1},g_{2},g_{3}\in G^{\circ}$ such that
$b(g_{1},h)+b(g_{2},h)+b(g_{3},h)=b(s,h)$ for any $h\in G$ then
$P_{T}(g_{1})+P_{T}(g_{2})+P_{T}(g_{3})=0$.

Any parity $p$ with coefficients in a group $A$ is a parity functor with the
coefficient groups $A(T)=A$, the maps $P_{T}=p_{T}$ and the monomorphisms
$A(f)=id_{A}$.

######  Definition 9.

Let $T=(G,s,b)$ be a based matrix and $\mathfrak{C}$ be some partition of $G$
into disjoint subsets such that $\\{s\\}\in\mathfrak{C}$. The subgroup

$Ann({\mathfrak{C}})=\\{v\in\mathbb{Z}[G^{\circ}]\,|\,\exists
k\in\mathbb{Z}:\,\forall C\in{\mathfrak{C}}\ b(v,\chi_{C})=k\cdot
b(s,\chi_{C})\\}$ (2)

in $\mathbb{Z}[G^{\circ}]$ where $\chi_{C}=\sum_{g\in C}g$, is called the
annulator of the partition $\mathfrak{C}$.

The derived partition $A\mathfrak{C}$ of $\mathfrak{C}$ is the partition which
consists of $\\{s\\}$ and the equivalence classes of the following relation on
$G^{\circ}$:

$g_{1}\simeq g_{2}\Longleftrightarrow g_{1}-g_{2}\mbox{ or }g_{1}+g_{2}\in
Ann({\mathfrak{C}}).$ (3)

We say that a partition $\mathfrak{C}$ of $G$ is finer than a partition
$\mathfrak{C}^{\prime}$ (${\mathfrak{C}}\succ{\mathfrak{C}}^{\prime}$) if all
classes of $\mathfrak{C}^{\prime}$ are unions of some classes of
$\mathfrak{C}$.

######  Lemma 3.

If ${\mathfrak{C}}\succ{\mathfrak{C}}^{\prime}$ then $A{\mathfrak{C}}\succ
A{\mathfrak{C}}^{\prime}$.

###### Proof.

Let ${\mathfrak{C}}\succ{\mathfrak{C}}^{\prime}$. Then
$Ann(\mathfrak{C})\subset Ann(\mathfrak{C}^{\prime})$. Indeed, if $v\in
Ann(\mathfrak{C})$ then $b(v,\chi_{C})=k\cdot b(s,\chi_{C})$ for any
$C\in\mathfrak{C}$. For any $C^{\prime}\in\mathfrak{C}^{\prime}$ we have
$C^{\prime}=\sqcup_{i}C_{i},C_{i}\in\mathfrak{C}$. Then
$\chi_{C^{\prime}}=\sum_{i}\chi_{C}$ and

$b(v,\chi_{C^{\prime}})=\sum_{i}b(v,\chi_{C_{i}})=\sum_{i}k\cdot
b(s,\chi_{C_{i}})=k\cdot b(s,\chi_{C^{\prime}}).$

For the relation $A{\mathfrak{C}}\succ A{\mathfrak{C}}^{\prime}$ it suffices
to show that for any $C\in A{\mathfrak{C}}$ and $C^{\prime}\in
A{\mathfrak{C}^{\prime}}$ such that $C\cap C^{\prime}\neq\emptyset$ one has
inclusion $C\subset C^{\prime}$. Let $x\in C\cap C^{\prime}$ and $y\in C$.
Then $x\pm y\in Ann(\mathfrak{C})\subset Ann(\mathfrak{C}^{\prime})$. Hence,
$x$ and $y$ belong to the same class in $A{\mathfrak{C}}^{\prime}$, i.e.
$x,y\in C^{\prime}$. Thus, $C\subset C^{\prime}$. ∎

Let ${\mathfrak{C}}_{0}=\bigcup_{g\in G}\\{\\{g\\}\\}$ and
${\mathfrak{C}}_{n}=A^{n}{\mathfrak{C}}_{0}$. Then
${\mathfrak{C}}_{0}\succ{\mathfrak{C}}_{1}\succ{\mathfrak{C}}_{2}\succ\dots$.
Let ${\mathfrak{C}}_{\infty}=\lim_{n\to\infty}{\mathfrak{C}}_{n}$. The
partition ${\mathfrak{C}}_{\infty}$ is called the stable partition. Let
$U=Ann({\mathfrak{C}}_{\infty})$ and $A^{st}=\mathbb{Z}[G^{\circ}]/U$. There
is a natural map $P^{st}\colon G^{\circ}\to A^{st}$ which maps an element $g$
to the coset $gU$.

Repeating this construction for all based matrices $T=(G,s,b)$, we get a
family of stable partitions ${\mathfrak{C}}_{\infty}(T)$, groups
$A^{st}(T)=\mathbb{Z}[G^{\circ}]/Ann({\mathfrak{C}}_{\infty}(T))$ and maps
$P^{st}_{T}\colon G^{\circ}\to A^{st}(T)$. We claim that these maps form a
parity functor on based matrices. We will check the claim in a more general
situation.

######  Definition 10.

A family $\mathfrak{C}$ of partitions $\mathfrak{C}(T)$ of the sets $G$ where
$T=(G,s,b)$ are based matrices, is called a tribal system if

  * •

for any $T=(G,s,b)$ $\\{s\\}\in\mathfrak{C}(T)$;

  * •

if a based matrix $T^{\prime}$ is obtained from a based matrix $T=(G,s,b)$ by
adding an annihilating or core element of a pair of complementary elements
then $\mathfrak{C}(T)=\mathfrak{C}(T^{\prime})\cap G$, i.e. for any
$C\in\mathfrak{C}(T)$ there exists a unique
$C^{\prime}\in\mathfrak{C}(T^{\prime})$ such that $C=C^{\prime}\cap G$;

  * •

if $g_{1},g_{2}\in G^{\circ}$ in a based matrix $T=(G,s,b)$ are complementary
then there exists $C\in\mathfrak{C}(T)$ such that $g_{1},g_{2}\in C$.

The classes $C\in\mathfrak{C}(T)$ are called tribes.

######  Remark 1.

We can reformulate the second and the third properties of tribal system as
follows:

  * •

if two elements belong to one tribe and survive after a transformation of the
base matrix then after the transformation they remain in one tribe;

  * •

complementary vectors belong to one tribe.

######  Theorem 2.

Let $\mathfrak{C}$ be a tribal system. For any based matrix $T=(G,s,b)$, let
$A^{\mathfrak{C}}(T)=\mathbb{Z}[G^{\circ}]/Ann({\mathfrak{C}}(T))$ and
$P^{\mathfrak{C}}_{T}\colon G^{\circ}\to A^{\mathfrak{C}}(T)$ be the natural
maps. Then the maps $P^{\mathfrak{C}}_{T}$ define a parity functor on based
matrices with coefficients $A^{\mathfrak{C}}(T)$.

The parity functor $P^{\mathfrak{C}}_{T}$ is called the parity functor
associated with the tribal system $\mathfrak{C}$.

First we prove an auxiliary lemma.

######  Lemma 4.

Let $T=(G,s,b)$ be a based matrix and $T^{\prime}=(G^{\prime},s,b^{\prime})$
be the result of an operation $M_{1}$, $M_{2}$ or $M_{3}$ applied to $T$. Let
$\mathfrak{C}^{\prime}$ be a partition of $G^{\prime}$ such that
$\\{s\\}\in\mathfrak{C}^{\prime}$ and there exists
$C^{\prime}\in\mathfrak{C}^{\prime}$ such that $G^{\prime}\setminus G\subset
C^{\prime}$. Then

  1. 1.

$Ann(\mathfrak{C}^{\prime}\cap
G)=Ann(\mathfrak{C}^{\prime})\cap\mathbb{Z}[G^{\circ}]$

  2. 2.

$A(\mathfrak{C}^{\prime}\cap G)=A(\mathfrak{C}^{\prime})\cap G$

###### Proof.

1) Since $G^{\prime}\setminus G\subset C^{\prime}$, we have
$\mathfrak{C}^{\prime}\cap
G=\mathfrak{C}^{\prime}\setminus\\{C^{\prime}\\}\cup\\{C\\}$ where
$C=C^{\prime}\cap G$. By definition we have
$b^{\prime}(v,\chi_{C^{\prime}})=b^{\prime}(v,\chi_{C})+l\cdot
b^{\prime}(v,s)$ for any $v\in\mathbb{Z}[G^{\prime\circ}]$ where $l=0$ when
the element in $G^{\prime}\setminus G$ is annihilating, and $l=1$ when the new
element is core or when $G^{\prime}\setminus G$ consists of a pair of
complementary elements. Hence,
$b^{\prime}(s,\chi_{C^{\prime}})=b^{\prime}(s,\chi_{C})$. Then the pair of
conditions for the classes $C^{\prime}$ and $\\{s\\}$ in the definition of
$Ann(\mathfrak{C}^{\prime})$ is eqivalent to the pair of conditions for the
classes $C$ and $\\{s\\}$:

$\left\\{\begin{array}[]{c}b^{\prime}(v,\chi_{C^{\prime}})=k\cdot
b^{\prime}(s,\chi_{C^{\prime}})\\\ b^{\prime}(v,s)=k\cdot
b^{\prime}(s,s)=0\end{array}\right.\Longleftrightarrow\left\\{\begin{array}[]{c}b^{\prime}(v,\chi_{C})=k\cdot
b^{\prime}(s,\chi_{C})\\\ b^{\prime}(v,s)=0\end{array}\right.$

This means that we can replace the tribe $C^{\prime}$ with $C$ in the
definition of $Ann({\mathfrak{C}}(T^{\prime}))$, i.e. replace the partition
$\mathfrak{C}^{\prime}$ with the partition
$\mathfrak{C}^{\prime}\setminus\\{C^{\prime}\\}\cup\\{C\\}=\mathfrak{C}^{\prime}\cap
G$. Thus,

$Ann({\mathfrak{C}^{\prime}})\cap\mathbb{Z}[G^{\circ}]=\\\
\\{v\in\mathbb{Z}[G^{\circ}]\,|\,\exists k\in\mathbb{Z}:\,\forall
C^{\prime\prime}\in{\mathfrak{C}^{\prime}}\
b^{\prime}(v,\chi_{C^{\prime\prime}})=k\cdot
b^{\prime}(s,\chi_{C^{\prime\prime}})\\}=\\\
\\{v\in\mathbb{Z}[G^{\circ}]\,|\,\exists k\in\mathbb{Z}:\,\forall
C^{\prime\prime}\in{\mathfrak{C}^{\prime}\cap G}\ \
b(v,\chi_{C^{\prime\prime}})=k\cdot
b(s,\chi_{C^{\prime\prime}})\\}=Ann({\mathfrak{C}^{\prime}}\cap G).$

2) Let $\sim_{A(\mathfrak{C}^{\prime}\cap G)}$ and
$\sim_{A(\mathfrak{C}^{\prime})}$ be the equivalence relations on $G^{\circ}$
determined by the correspondent partitions. Then for any $g_{1},g_{2}\in
G^{\circ}$ we have a chain of equivalences

$g_{1}\sim_{A(\mathfrak{C}^{\prime}\cap G)}g_{2}\Leftrightarrow g_{1}\pm
g_{2}\in Ann({\mathfrak{C}^{\prime}}\cap G)\Leftrightarrow g_{1}\pm g_{2}\in
Ann({\mathfrak{C}^{\prime}})\Leftrightarrow
g_{1}\sim_{A(\mathfrak{C}^{\prime})}g_{2}.$

Thus, the partitions $A(\mathfrak{C}^{\prime}\cap G)$ and
$A(\mathfrak{C}^{\prime})\cap G$ coincide. ∎

###### Proof of Theorem 2.

The properties (P1),(P2),(P3) follow from the definition of the annulator
$Ann({\mathfrak{C}}(T))$.

Let us check the property (P0). Let a based matrix
$T^{\prime}=(G^{\prime},s,b^{\prime})$ is obtained from a based matrix
$T=(G,s,b)$ by adding an annihilating or core element of a pair of
complementary elements. We need to show that the natural map
$A^{\mathfrak{C}}(T)\to A^{\mathfrak{C}}(T^{\prime})$ which maps $g\in G$ to
$g$, is a well defined monomorphism. This condition is equivalent to the
equality
$Ann({\mathfrak{C}}(T^{\prime}))\cap\mathbb{Z}[G^{\circ}]=Ann({\mathfrak{C}}(T))$.
The last statement holds by Lemma 4.

∎

######  Proposition 1.

The stable partition $\mathfrak{C}_{\infty}$ is a tribal system.

###### Proof.

Let $T=(G,s,b)$ be a based matrix. By the definition of annulator, for any
partition $\mathfrak{C}(T)$ of $G$ any pair of complementary elements belongs
to one tribe of the partition $A\mathfrak{C}(T)$.

Let $T^{\prime}=(G^{\prime},s,b^{\prime})$ is obtained from $T$ by an
operation $M_{1}$, $M_{2}$ or $M_{3}$. Let us show that
$\mathfrak{C}_{\infty}(T^{\prime})\cap G=\mathfrak{C}_{\infty}(T)$.

Let $\tilde{\mathfrak{C}}_{0}(T^{\prime})$ be the partition of $G^{\prime}$
which consists of $\\{s\\}$, one-element sets $\\{g\\}$, $g\in G^{\circ}$, and
the subset $G^{\prime}\setminus G$. Then
$\mathfrak{C}_{0}(T^{\prime})\succ\tilde{\mathfrak{C}}_{0}(T^{\prime})\succ
A\mathfrak{C}_{0}(T^{\prime})=\mathfrak{C}_{1}(T^{\prime})$.

Let
$\tilde{\mathfrak{C}}_{n}(T^{\prime})=A^{n}\tilde{\mathfrak{C}}_{0}(T^{\prime})$
and
$\tilde{\mathfrak{C}}_{\infty}(T^{\prime})=\lim_{n\to\infty}\tilde{\mathfrak{C}}_{n}(T^{\prime})$.
By Lemma 3 we have
$\mathfrak{C}_{n}(T^{\prime})\succ\tilde{\mathfrak{C}}_{n}(T^{\prime})\succ\mathfrak{C}_{n+1}(T^{\prime})$.
Hence,
$\tilde{\mathfrak{C}}_{\infty}(T^{\prime})=\mathfrak{C}_{\infty}(T^{\prime})$.

By definition $\tilde{\mathfrak{C}}_{0}(T^{\prime})\cap
G=\mathfrak{C}_{0}(T)$. Lemma 4 implies that

$\tilde{\mathfrak{C}}_{n}(T^{\prime})\cap
G=A^{n}\tilde{\mathfrak{C}}_{0}(T^{\prime})\cap
G=A^{n}(\tilde{\mathfrak{C}}_{0}(T^{\prime})\cap
G)=A^{n}\mathfrak{C}_{0}(T)=\mathfrak{C}_{n}(T).$

Then $\mathfrak{C}_{\infty}(T)=\tilde{\mathfrak{C}}_{\infty}(T^{\prime})\cap
G=\mathfrak{C}_{\infty}(T^{\prime})\cap G$. Thus, $\mathfrak{C}_{\infty}$ is a
tribal system. ∎

Theorem 2 and Proposition 1 imply the following corollary

######  Corollary 1.

The family of maps $P^{st}_{T}$ is a parity functor (called the stable parity
functor) with coefficients in the groups $A^{st}(T)$.

######  Example 4.

Consider a based matrix $T=(G,s,b)$ with coefficients in $\mathbb{Z}_{2}$
where $G=\\{s,1,2,3,4,5,6,7,8\\}$ and $b$ be given by induces the matrix $B$
($s$ corresponds to the first row and column).

$B=\blockarray{*{9}{c}}s&\mathbf{1}\mathbf{2}\mathbf{3}\mathbf{4}\mathbf{5}\mathbf{6}\mathbf{7}\mathbf{8}\\\
{(c|c|c|c|c|c|c|c|c)}000000000\\\ \BAhline 000101100\\\ \BAhline 000101100\\\
\BAhline 011000110\\\ \BAhline 000000110\\\ \BAhline 011000001\\\ \BAhline
011110001\\\ \BAhline 000110001\\\ \BAhline 000001110\\\ $

Let us find the stable partition. We look for the pairs of rows whose sum is
equal to the first row. Those are the rows $1$ and $2$. This means the
elements $\\{1,2\\}$ form one class in the partition $\mathfrak{C}_{1}$. We
replace the columns with numbers $1$ and $2$ by their sum (it corresponds to
the vector $\chi_{\\{1,2\\}}$) and get the matrix $B_{1}$. We use separators
between rows to distinguish the classes of the partition. Then we look for
rows in the matrix $B_{1}$ as before. Repeating the process, we get the
sequence of the partitions $\mathfrak{C}_{2},\mathfrak{C}_{3}$ and the
corresponding matrices $B_{2},B_{3}$ whose rows correspond to the elements of
$G$ and the columns correspond to the classes of the partition.

$B_{1}=\left(\begin{array}[]{c|c|c|c|c|c|c|c}0&0&0&0&0&0&0&0\\\ \hline\cr
0&0&1&0&1&1&0&0\\\ 0&0&1&0&1&1&0&0\\\ \hline\cr 0&0&0&0&0&1&1&0\\\ \hline\cr
0&0&0&0&0&1&1&0\\\ \hline\cr 0&0&0&0&0&0&0&1\\\ \hline\cr 0&0&1&1&0&0&0&1\\\
\hline\cr 0&0&1&1&0&0&0&1\\\ \hline\cr
0&0&0&0&1&1&1&0\end{array}\right),\qquad
B_{2}=\left(\begin{array}[]{c|c|c|c|c|c}0&0&0&0&0&0\\\ \hline\cr
0&0&1&1&1&0\\\ 0&0&1&1&1&0\\\ \hline\cr 0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\
\hline\cr 0&0&0&0&0&1\\\ \hline\cr 0&0&0&0&0&1\\\ 0&0&0&0&0&1\\\ \hline\cr
0&0&0&1&0&0\end{array}\right),$

$B_{3}=\left(\begin{array}[]{c|c|c|c|c}0&0&0&0&0\\\ \hline\cr 0&0&1&0&0\\\
0&0&1&0&0\\\ \hline\cr 0&0&0&0&0\\\ 0&0&0&0&0\\\ \hline\cr 0&0&0&0&1\\\
0&0&0&0&1\\\ 0&0&0&0&1\\\ \hline\cr 0&0&0&1&0\end{array}\right).$

$\displaystyle{\mathfrak{C}}_{0}=\\{\\{s\\},\\{1\\},\\{2\\},\\{3\\},\\{4\\},\\{5\\},\\{6\\},\\{7\\},\\{8\\}\\},$
$\displaystyle{\mathfrak{C}}_{1}=\\{\\{s\\},\\{1,2\\},\\{3\\},\\{4\\},\\{5\\},\\{6\\},\\{7\\},\\{8\\}\\},$
$\displaystyle{\mathfrak{C}}_{2}=\\{\\{s\\},\\{1,2\\},\\{3,4\\},\\{5\\},\\{6,7\\},\\{8\\}\\},$
$\displaystyle{\mathfrak{C}}_{3}=\\{\\{s\\},\\{1,2\\},\\{3,4\\},\\{5,6,7\\},\\{8\\}\\}={\mathfrak{C}}_{\infty}.$

The partition $\mathfrak{C}_{3}$ stabilises (there are no two rows from
different classes whose sum gives the first row of the table). Thus,
$\mathfrak{C}_{3}=\mathfrak{C}_{\infty}$ is the stable partition.

The annulator of $\mathfrak{C}_{\infty}$ in
$\mathbb{Z}[\\{1,2,3,4,5,6,7,8\\}]$ is generated by the elements
$2\cdot\mathbf{i}$, $i=1,\dots,8$ and $\mathbf{1}+\mathbf{2}$, $\mathbf{3}$,
$\mathbf{4}$, $\mathbf{5}+\mathbf{6}$, $\mathbf{5}+\mathbf{7}$ (we look for
combinations of rows in the table $B_{3}$ that give zero row). Then
$A^{st}=\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$ and the parity
map is the following

$\displaystyle P^{st}(\mathbf{1})=P^{st}(\mathbf{2})=(1,0,0),\quad
P^{st}(\mathbf{3})=P^{st}(\mathbf{4})=(0,0,0),$ $\displaystyle
P^{st}(\mathbf{5})=P^{st}(\mathbf{6})=P^{st}(\mathbf{7})=(0,1,0),\quad
P^{st}(\mathbf{8})=(0,0,1).$

Let us modify the definition of parity functor in Theorem 2 in order to get a
parity on based matrix.

###  2.1 Reduced parity functor

Let $T_{0}=(G_{0},s_{0},b_{0})$ be a based matrix with coefficients in $H$ and
let $\mathfrak{C}$ be a tribal system defined on the based matrices
$T=(G,s,b)$ homologous to $T_{0}$.

For a based matrix $T=(G,s,b)$, consider the map $\hat{P}^{\mathfrak{C}}_{T}$
from $\mathbb{Z}(G^{\circ})$ to the group
$\hat{A}^{\mathfrak{C}}(T)=H[\mathfrak{C}(T)]/\langle\hat{b}(s)\rangle$ where
$\langle\hat{b}(s)\rangle$ is the cyclic subgroup generated by

$\hat{b}(s)=\sum_{C\in\mathfrak{C}(T)}b(s,\chi_{C})\cdot C,$ (4)

defined by the formula

$\hat{P}^{\mathfrak{C}}_{T}(g)=\sum_{C\in\mathfrak{C}(T)}b(g,\chi_{C})\cdot
C.$ (5)

The kernel of the homomorphism $\hat{P}^{\mathfrak{C}}_{T}$ is the annulator
$Ann(\mathfrak{C}(T))$. Hence, the group $A^{\mathfrak{C}}(T)$ embeds in
$\hat{A}^{\mathfrak{C}}(T)$. The family of maps $\hat{P}^{\mathfrak{C}}_{T}$
is a parity functor with coefficients in the groups
$\hat{A}^{\mathfrak{C}}(T)$. For an operation $f\colon T\to T^{\prime}$,
$T=(G,s,b)$, $T^{\prime}=(G^{\prime},s,b^{\prime})$, of type $M_{1},M_{2}$ or
$M_{3}$ the monomorphism
$\hat{A}^{\mathfrak{C}}(f)\colon\hat{A}^{\mathfrak{C}}(T)\to\hat{A}^{\mathfrak{C}}(T^{\prime})$
is given by the formula

$\hat{A}^{\mathfrak{C}}(f)(\sum_{C\in\mathfrak{C}(T)}\lambda_{C}\cdot
C)=\sum_{C\in\mathfrak{C}(T)}\lambda_{C}\cdot f(C)+l\cdot\lambda_{s}\cdot
C^{\prime}_{f}$ (6)

where $f(C)\in\mathfrak{C}(T^{\prime})$ is the unique tribe such that
$f(C)\cap G=C$, $\lambda_{s}=\lambda_{\\{s\\}}$,
$C^{\prime}_{f}\in\mathfrak{C}(T^{\prime})$ is the tribe containing
$G^{\prime}\setminus G$, and $l=0$ when $f$ is of type $M_{1}$ and $l=1$ when
$f$ is of type $M_{2}$ or $M_{3}$.

Let $T_{\bullet}=(G_{\bullet},s,b_{\bullet})$ be a primitive based matrix
which is obtained from $T$ by deleting annihilating, core and complementary
elements. Then $G_{\bullet}\subset G$.

######  Definition 11.

A tribe $C\in\mathfrak{C}(T)$ is called primitive if $C\cap
G_{\bullet}\neq\emptyset$. Denote the subset of primitive tribes in
$\mathfrak{C}(T)$ by $\mathfrak{C}(T)_{prim}$.

Note that by definition $\\{s\\}\in\mathfrak{C}(T)_{prim}$.

######  Proposition 2.

1\. The primitive tribes do not depend on the choice of the primitive based
matrix $T_{\bullet}$, that is if $C$ is primitive with respect to a primitive
based matrix $T_{\bullet}=(G_{\bullet},s,b_{\bullet})$ and
$T^{\prime}_{\bullet}=(G^{\prime}_{\bullet},s,b^{\prime}_{\bullet})$ is
another primitive based matrix such that $G^{\prime}_{\bullet}\subset G$ then
$C\cap G^{\prime}_{\bullet}\neq\emptyset$.

2\. Let $T^{\prime}=(G^{\prime},s,b^{\prime})$ be obtained from $T$ by an
operation $M_{1}$, $M_{2}$ or $M_{3}$. Then there is a bijection between the
primitive tribes in $\mathfrak{C}(T)$ and $\mathfrak{C}(T^{\prime})$: a tribe
$C^{\prime}\in\mathfrak{C}(T^{\prime})_{prim}$ iff the tribe $C^{\prime}\cap
G\in\mathfrak{C}(T)_{prim}$.

3\. The support of the element $\hat{b}(s)\in H[\mathfrak{C}(T)]$ includes
only primitive tribes:
$\hat{b}(s)=\sum_{C\in\mathfrak{C}(T)_{prim}}b(s,\chi_{C})\cdot C$

###### Proof.

1\. Let $T\stackrel{{\scriptstyle
f_{1}}}{{\longrightarrow}}T_{1}\stackrel{{\scriptstyle
f_{2}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle
f_{n}}}{{\longrightarrow}}T_{\bullet}$ and $T\stackrel{{\scriptstyle
f^{\prime}_{1}}}{{\longrightarrow}}T^{\prime}_{1}\stackrel{{\scriptstyle
f^{\prime}_{2}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle
f^{\prime}_{n^{\prime}}}}{{\longrightarrow}}T^{\prime}_{\bullet}$ be sequences
of operations $M_{k}^{-1}$, $k=1,2,3$, that reduce $T$ to the primitive based
matrices. We prove the first statement of the proposition by induction on
$d=|G|-|G_{\bullet}|$. Note that this number does not depend on the choice of
primitive matrix because all homologous primitive matrix are isomorphic and
have the same number of elements.

If $d=0$ then $T_{\bullet}=T=T^{\prime}_{\bullet}$, and the statement is
trivial.

Assume the statement holds for all $k<d$. Consider the operations $f_{1}$ and
$f^{\prime}_{1}$. If $f_{1}=f^{\prime}_{1}$ consider the based matrix
$T^{\prime\prime}=f_{1}(T)$,
$T^{\prime\prime}=(G^{\prime\prime},s,b^{\prime\prime})$. Then
$G_{\bullet}\subset G^{\prime\prime}$ and $G^{\prime}_{\bullet}\subset
G^{\prime\prime}$. If $C\in\mathfrak{C}(T)$ then $C\cap
G_{\bullet}\neq\emptyset$. Hence, $(C\cap G^{\prime\prime})\cap
G_{\bullet}\neq\emptyset$, i.e. the tribe $C^{\prime\prime}=C\cap
G^{\prime\prime}$ is primitive in
$\mathfrak{C}(T^{\prime\prime})=\mathfrak{C}(T)\cap G^{\prime\prime}$. By
induction $C^{\prime\prime}\cap G^{\prime}_{\bullet}\neq\emptyset$. Then
$C\cap G^{\prime}_{\bullet}\neq\emptyset$.

Let $f_{1}\neq f^{\prime}_{1}$ commute. Let $T_{1}=f_{1}(T)$,
$T^{\prime}_{1}=f^{\prime}_{1}(T)$ and
$T^{\prime\prime}=f^{\prime}_{1}(T_{1})=f_{1}(T^{\prime}_{1})$. Then
$G_{\bullet}\subset G_{1}$ and $G^{\prime}_{\bullet}\subset G^{\prime}_{1}$.
By operations $M_{k}^{-1}$, $k=1,2,3$, the based matrix $T^{\prime\prime}$ can
be reduced to a primitive based matrix
$T^{\prime\prime}_{\bullet}=(G^{\prime\prime}_{\bullet},s,b^{\prime\prime}_{\bullet})$.
Then $G^{\prime\prime}_{\bullet}\subset G^{\prime\prime}=G_{1}\cap
G^{\prime}_{1}$.

Let $C\in\mathfrak{C}(T)$ be primitive, i.e. $C\cap G_{\bullet}\neq\emptyset$.
Let $C_{1}=C\cap G_{1}\in\mathfrak{C}(T_{1})$ and $C^{\prime}_{1}=C\cap
G^{\prime}_{1}\in\mathfrak{C}(T^{\prime}_{1})$. By induction, $C_{1}\cap
G_{\bullet}=C\cap G_{\bullet}\neq\emptyset$ implies $C_{1}\cap
G^{\prime\prime}_{\bullet}\neq\emptyset$. But $C_{1}\cap
G^{\prime\prime}_{\bullet}=C\cap G^{\prime\prime}_{\bullet}=C^{\prime}_{1}\cap
G^{\prime\prime}_{\bullet}$. Then $C^{\prime}_{1}\cap
G^{\prime\prime}_{\bullet}\neq\emptyset$. Hence, by induction $C\cap
G^{\prime}_{\bullet}=C^{\prime}_{1}\cap G^{\prime}_{\bullet}\neq\emptyset$.

Let $f_{1}\neq f^{\prime}_{1}$ do not commute. If one of the operations (say,
$f_{1}$) is of type $M_{3}^{-1}$ and the other (say, $f^{\prime}_{1}$) is of
type $M_{1}^{-1}$ or $M_{2}^{-1}$ then $f_{1}$ removes a pair of complementary
core and annihilating elements. Then $f_{1}=f^{\prime\prime}\circ
f^{\prime}_{1}$ for some operation $f^{\prime\prime}$ of type $M_{1}^{-1}$ or
$M_{2}^{-1}$, and we can treat this case like the case $f_{1}=f^{\prime}_{1}$
by considering the based matrix $T^{\prime\prime}=f^{\prime}_{1}(T)$.

The other option is when $f_{1}$ and $f^{\prime}_{1}$ are of type
$M_{3}^{-1}$. Denote $T_{1}=f_{1}(T)$ and $T^{\prime}_{1}=f^{\prime}_{1}(T)$.
The sets $G\setminus G_{1}=\\{g_{0},g^{\prime}_{1}\\}$ and $G\setminus
G^{\prime}_{1}=\\{g_{0},g_{1}\\}$ are intersecting pairs of complementary
elements. Then $b(g,g_{1})=b(g,g^{\prime}_{1})$ for all $g\in G$, and the map
$h\colon G_{1}\to G^{\prime}_{1}$, which replaces the element $g_{1}\in G_{1}$
with the element $g^{\prime}_{1}\in G^{\prime}_{1}$, is an isomorphism of the
based matrices $T_{1}$ and $T^{\prime}_{1}$. Hence, the based matrix
$T^{\prime\prime}_{\bullet}=h(T_{\bullet})$ is a primitive submatrix of
$T^{\prime}_{1}$ which is homologous to $T^{\prime}_{1}$.

Let $C\in\mathfrak{C}(T)$ be primitive, i.e. $C\cap G_{\bullet}\neq\emptyset$.
If there exists $g^{\prime\prime}\neq g_{1}$ such that $g^{\prime\prime}\in
C\cap G_{\bullet}$ then $g^{\prime\prime}\in C\cap
G^{\prime\prime}_{\bullet}=C^{\prime}_{1}\cap G^{\prime\prime}_{\bullet}$
where $C^{\prime}_{1}=C\cap G^{\prime}_{1}$. Then by induction $C\cap
G^{\prime}_{\bullet}=C^{\prime}_{1}\cap G^{\prime}_{\bullet}\neq\emptyset$.

Assume that $C\cap G_{\bullet}=\\{g_{1}\\}$. Since $g_{0}$ and $g_{1}$ as well
as $g_{0}$ and $g^{\prime}_{1}$ are complementary in $T$, they all belong to
one tribe, i.e. $g_{0},g_{1},g^{\prime}_{1}\in C$. Then $g^{\prime}_{1}\in
C\cap G^{\prime\prime}_{\bullet}$, so $C^{\prime}_{1}\cap
G^{\prime\prime}_{\bullet}\neq\emptyset$ where $C^{\prime}_{1}=C\cap
G^{\prime}_{1}$. Then by induction $C\cap
G^{\prime}_{\bullet}=C^{\prime}_{1}\cap G^{\prime}_{\bullet}\neq\emptyset$.

2\. Let $T_{\bullet}=(G_{\bullet},s,b_{\bullet})$ be a primitive based matrix
obtained from $T$ by operations $M_{i}^{-1},i=1,2,3$. Then $G_{\bullet}\subset
G$. Then we have the equivalences

$C^{\prime}\in\mathfrak{(}T^{\prime})_{prim}\Longleftrightarrow C^{\prime}\cap
G_{\bullet}=(C^{\prime}\cap G)\cap
G_{\bullet}\neq\emptyset\Longleftrightarrow(C^{\prime}\cap
G)\in\mathfrak{(}T)_{prim}.$

3\. By the proof of Lemma 4, for any reducing operation $f\colon T\to
T^{\prime}$ of type $M_{i}^{-1}$, $i=1,2,3$, and any tribe
$C\in\mathfrak{C}(T)$ we have $b(s,\chi_{C\cap G^{\prime}})=b(s,\chi_{C})$
where $T=(G,s,b)$ and $T^{\prime}=(G^{\prime},s,b|_{G^{\prime}})$. If
$C\in\mathfrak{C}(T)$ is not primitive then it reduces to an empty tribe on
the way to a primitive based matrix. Then
$b(s,\chi_{C})=b(s,\chi_{\emptyset})=0$. Thus,

$\hat{b}(s)=\sum_{C\in\mathfrak{C}(T)}b(s,\chi_{C})\cdot
C=\sum_{C\in\mathfrak{C}(T)_{prim}}b(s,\chi_{C})\cdot C.$

∎

Thus, for any based matrix $T$ a distinguished set $\mathfrak{(}T)_{prim}$ of
the primitive tribes is defined, and any transformation $f\colon T\to
T^{\prime}$ to a homologous based matrix $T^{\prime}$ identifies the sets
$\mathfrak{(}T)_{prim}$ and $\mathfrak{(}T^{\prime})_{prim}$.

Let us consider another distinguished tribe in $\mathfrak{C}(T)$.

######  Proposition 3.

Let $T_{1}=(G_{1},s,b_{1})$ and $T_{2}=(G_{2},s,b_{2})$ be homologous to $T$
and $g_{i}\in G_{i}\cap G$, $i=1,2$ be an element which is annihilating or
core in $G_{i}$. Then $g_{1}$ and $g_{2}$ belong to one tribe in
$\mathfrak{C}(T)$.

###### Proof.

Let $f_{i}\colon T_{i}\to T$ be the sequence of operations that transform
$T_{i}$ to $T$. Add $T$, $T_{1}$, $T_{2}$ and all the intermediate diagrams an
element $g^{\prime}$ which is annihilating for all the diagrams $\tilde{T}$,
$\tilde{T}_{1}$, $\tilde{T}_{2}$ and the intermediate ones. Then $g_{i}$ and
$g^{\prime}$ belong to one tribe in $\mathfrak{C}(T_{i})$, $i=1,2$. Hence,
they belong to one tribe in $\mathfrak{C}(T)$. Then by transitivity $g_{1}$
and $g_{2}$ belong to one tribe in $\mathfrak{C}(T)$. ∎

Let $C_{0}((T)\in\mathfrak{C}(T)$ be the tribe which contains all core and
annihilating elements. We call it the zero tribe.

The zero tribe $C_{0}(T)$ can be empty. On the other hand, $C_{0}(T)$ can be
primitive since it may include other elements besides the core and
annihilating ones.

######  Corollary 2.

Let $T^{\prime}=(G^{\prime},s,b^{\prime})$ be obtained from $T$ by an
operation $M_{1}$, $M_{2}$ or $M_{3}$ and $C_{0}=C_{0}(T)$ and
$C^{\prime}_{0}=C_{0}(T^{\prime})$ be the zero tribes in $\mathfrak{C}(T)$ and
$\mathfrak{C}(T^{\prime})$ correspondingly. Then $C_{0}=C^{\prime}_{0}\cap G$.

Corollary 2 means the zero tribe is equivariant, that any transformation
$f\colon T\to T^{\prime}$ from $T$ to a homologous based matrix $T^{\prime}$
maps elements of the zero tribes $\mathfrak{(}T)_{prim}$ to the zero tribe
$\mathfrak{(}T^{\prime})_{prim}$.

Consider a based matrix $T=(G,s,b)$. Denote

$\mathfrak{C}(T)_{prim}^{*}=\mathfrak{C}(T)_{prim}\setminus\\{C_{0}(T)\\}.$

Let
$\tilde{A}^{\mathfrak{C}}(T)=H[\mathfrak{C}(T)_{prim}^{*}]/\langle\tilde{b}(s)\rangle$
where

$\tilde{b}(s)=\sum_{C\in\mathfrak{C}(T)_{prim}^{*}}b(s,\chi_{C})\cdot C.$ (7)

Define the maps $\tilde{P}^{\mathfrak{C}}_{T}\colon
G\to\tilde{A}^{\mathfrak{C}}(T)$ by the formula

$\tilde{P}^{\mathfrak{C}}_{T}(g)=\sum_{C\in\mathfrak{C}(T)_{prim}^{*}}\left(b(g,\chi_{C})-\left\lfloor\frac{|C|}{2}\right\rfloor\cdot
b(g,s)\right)\cdot C$ (8)

where $\lfloor\cdot\rfloor$ is the floor function.

For an operation $f\colon T\to T^{\prime}$, $T=(G,s,b)$,
$T^{\prime}=(G^{\prime},s,b^{\prime})$, of type $M_{1},M_{2}$ or $M_{3}$ we
define a homomorphism
$\tilde{A}^{\mathfrak{C}}(f)\colon\tilde{A}^{\mathfrak{C}}(T)\to\tilde{A}^{\mathfrak{C}}(T^{\prime})$
by the formula

$\tilde{A}^{\mathfrak{C}}(f)\left(\sum_{C\in\mathfrak{C}(T)_{prim}^{*}}\lambda_{C}\cdot
C\right)=\sum_{C\in\mathfrak{C}(T)_{prim}^{*}}\lambda_{C}\cdot f(C)$ (9)

where $f(C)\in\mathfrak{C}(T^{\prime})_{prim}$ is the unique primitive tribe
such that $f(C)\cap G=C$.

######  Theorem 3.

The maps $\tilde{P}^{\mathfrak{C}}_{T}$ define a parity functor on based
matrices.

The parity functor is called the reduced parity functor associated with the
tribal system $\mathfrak{C}$.

###### Proof.

Since $\tilde{P}^{\mathfrak{C}}_{T}$ factors through
$\hat{P}^{\mathfrak{C}}_{T}$, the conditions (P1),(P2) and (P3) are valid.

Let us check the property (P0). Let $T=(G,s,b)$ be a based matrix and
$T^{\prime}=f(T)$ where $f$ is of type $M_{i}$, $i=1,2,3$. Let
$T^{\prime}=(G^{\prime},s,b^{\prime})$.

First we show that $\tilde{A}^{\mathfrak{C}}(f)$ is a well defined
isomorphism. By Proposition 2 the map $f$ establishes a bijection
$f\colon\mathfrak{C}(T)_{prim}\to\mathfrak{C}(T^{\prime})_{prim}$. By
Corollary 2 $f(C_{0}(T))=C_{0}(T^{\prime})$ if
$C_{0}(T)\in\mathfrak{C}(T)_{prim}$. Then $f$ defines an isomorphism $f\colon
H[\mathfrak{C}(T)_{prim}^{*}]\to H[\mathfrak{C}(T^{\prime})_{prim}^{*}]$.

For any tribe $C^{\prime}\in\mathfrak{C}(T^{\prime})$ we have
$b^{\prime}(s,\chi_{C})=b(s,\chi_{C^{\prime}\cap G})$. Then
$f(\tilde{b}(s))=\tilde{b}(s)$. Hence, the isomorphism $f$ induces an
isomorphism
$\tilde{A}^{\mathfrak{C}}(T)\to\tilde{A}^{\mathfrak{C}}(T^{\prime})$ that
coincides with $\tilde{A}^{\mathfrak{C}}(f)$.

Let us check the equality
$\tilde{P}^{\mathfrak{C}}_{T^{\prime}}|G=\tilde{A}^{\mathfrak{C}}(f)\circ\tilde{P}^{\mathfrak{C}}_{T}$.
It is enough to verify that the terms in the left and the right part of the
equality which correspond to some tribe $C\in\mathfrak{C}(T)_{prim}^{*}$, are
equal.

Let $C^{\prime}\in\mathfrak{C}(T^{\prime})_{prime}^{*}$ and $C=C^{\prime}\cap
G$. If $C^{\prime}\not\supset G^{\prime}\setminus G$ then $C^{\prime}=C$,
hence, $\lfloor\frac{|C|}{2}\rfloor=\lfloor\frac{|C^{\prime}|}{2}\rfloor$ and
$b(g,\chi_{C})=b^{\prime}(g,\chi_{C^{\prime}})$ for any $g\in G$. Thus, the
correspondent terms of the tribes $C$ and $C^{\prime}$ in
$\tilde{P}^{\mathfrak{C}}_{T}(g)$ and
$\tilde{P}^{\mathfrak{C}}_{T^{\prime}}(g)$ are equal.

Let $C^{\prime}\supset G^{\prime}\setminus G$. Since $C^{\prime}\neq
C_{0}(T^{\prime})$, the operation $f$ is of type $M_{3}$. Then
$|C^{\prime}|=|C|+2$ and for any $g\in G$
$b^{\prime}(g,\chi_{C^{\prime}})=b(g,\chi_{C})+b(g,s)$. Hence, for any $g\in
G$

$b^{\prime}(g,\chi_{C^{\prime}})-\left\lfloor\frac{|C^{\prime}|}{2}\right\rfloor\cdot
b^{\prime}(g,s)=\\\
b(g,\chi_{C})+b(g,s)-\left\lfloor\frac{|C|}{2}\right\rfloor\cdot
b(g,s)-b(g,s)=\\\ b(g,\chi_{C})-\left\lfloor\frac{|C|}{2}\right\rfloor\cdot
b(g,s).$

Thus, the correspondent terms of the tribes $C$ and $C^{\prime}$ in
$\tilde{P}^{\mathfrak{C}}_{T}(g)$ and
$\tilde{P}^{\mathfrak{C}}_{T^{\prime}}(g)$ are equal. ∎

######  Definition 12.

The reduces parity functor $\tilde{P}^{st}=\tilde{P}^{\mathfrak{C}_{\infty}}$
associated with the stable tribal system $\mathfrak{C}_{\infty}$ is called the
reduced stable parity functor.

######  Example 5.

Consider the based matrix $T$ with coefficients in $\mathbb{Z}_{2}$ from
Example 4. One of primitive based matrices homologus to $T$ is
$T_{\bullet}=(\\{s,7,8\\},s,b_{\bullet})$ where $b_{\bullet}$ is defined by
the matrix $B_{\bullet}$:

$B_{\bullet}=\blockarray{ccc}s&\mathbf{7}\mathbf{8}\\\ {(c|c|c)}000\\\
\BAhline 001\\\ \BAhline 010\\\ .$

Thus, the primitive tribes in $\mathfrak{C}_{\infty}(T)$ are $\\{s\\}$,
$C_{1}=\\{5,6,7\\}$ and $C_{2}=\\{8\\}$. The zero tribe $C_{0}=\\{3,4\\}$ is
not primitive. The element $\tilde{b}(s)$ is $0$. Then the coefficient group
$\tilde{A}^{st}(T)$ of the reduced stable parity functor is isomorphic to
$\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$ (the first summand is
for $\\{s\\}$). The parity values correspond to the elements in the first and
the last two colums of the matrix $B_{3}$. We extracted those columns to the
matrix $\tilde{B}_{3}$.

$\tilde{B}_{3}=\left(\begin{array}[]{c|c|c}0&0&0\\\ \hline\cr 0&0&0\\\
0&0&0\\\ \hline\cr 0&0&0\\\ 0&0&0\\\ \hline\cr 0&0&1\\\ 0&0&1\\\ 0&0&1\\\
\hline\cr 0&1&0\end{array}\right).$

Thus, the parities of the elements are

$\displaystyle\tilde{P}^{st}(\mathbf{1})=\tilde{P}^{st}(\mathbf{2})=\tilde{P}^{st}(\mathbf{3})=\tilde{P}^{st}(\mathbf{4})=(0,0,0),$
$\displaystyle\tilde{P}^{st}(\mathbf{5})=\tilde{P}^{st}(\mathbf{6})=\tilde{P}^{st}(\mathbf{7})=(0,0,1),\quad\tilde{P}^{st}(\mathbf{8})=(0,1,0).$

######  Remark 2.

Although the coefficient groups $\tilde{A}^{\mathfrak{C}}(T)$ of the reduced
parity functor $\tilde{P}^{\mathfrak{C}}$ are all isomorphic we can not state
that the parity functor defines a parity with coefficients in
$\tilde{A}^{\mathfrak{C}}(T)$ for some fixed $T$.

Consider the following example. Let $T=({s,x,y},s,b)$ be a based matrix with
coefficients in $\mathbb{Z}_{2}$ where $b$ is determined be the matrix

$B(T)=\blockarray{ccc}s&xy\\\ {(c|cc)}000\\\ \BAhline 001\\\ 010\\\ .$

The based matrix $T$ is primitive and has two primitive stable tribes
$\\{x\\},\\{y\\}\in\mathfrak{C}_{\infty}(T)_{prim}$.

Replace the element $x$ with an element $u$ by applying operations $M_{3}$ and
$M_{3}^{-1}$ as shown below. Thus, we get a based matrix
$T^{\prime}=(\\{s,u,y\\},s,b^{\prime})$.

$\blockarray{ccc}s&xy\\\ {(c|cc)}000\\\ \BAhline 001\\\ 010\\\
\quad\Longrightarrow\quad\blockarray{ccccc}s&xpuy\\\ {(c|cccc)}00000\\\
\BAhline 00001\\\ 00001\\\ 00001\\\ 01110\\\
\quad\Longrightarrow\quad\blockarray{ccc}s&uy\\\ {(c|cc)}000\\\ \BAhline
001\\\ 010\\\ $

Analogously, replace $y$ with $v$. Then we get a sequence of transformations
$f_{1}\colon T\to T_{1}=(\\{s,u,v\\},s,b_{1})$. The primitive stable tribes of
$T_{1}$ are $\\{u\\}$ and $\\{v\\}$. The transformation $f_{1}$ identifies $x$
with $u$ and $y$ with $v$.

On the other hand, we can replace $x$ with $v$ and $y$ with $u$ and get a
based matrix $T_{2}=(\\{s,u,v\\},s,b_{2})$. The transformation $f_{2}$ from
$T$ to $T_{2}$ identifies the primitive tribes $x$ with $v$ and $y$ with $u$.

But the based matrices $T_{1}$ and $T_{2}$ coincide because they have the same
underlying set $\\{s,u,v\\}$ and $b_{1}=b_{2}$. Thus, there is no canonical
way to identify primitive tribes of homologous based matrices. The reason is
we have a nontrivial automorphism of the based matrix $T$. In general case we
can assign a monodromy, i.e. a sequence of transformations from a based matrix
to itself, which interchanges the primitive tribes according to any given
automorphism of the based matrix. On the other hand, any monodromy of a
primitive based matrix induces an automorphism of the based matrix [11].

Thus, we need to factorise by isomorphisms if we want to get an invariant
coefficient group for a parity.

###  2.2 Reduced parity

Let $T_{0}=(G_{0},s_{0},b_{0})$ be a based matrix with coefficients in $H$ and
let $\mathfrak{C}$ be a tribal system defined on the based matrices
$T=(G,s,b)$ homologous to $T_{0}$.

Choose a primitive based matrix $T_{\bullet}=(G_{\bullet},s,b_{\bullet})$
homologous to $T_{0}$. The tribes in $G_{\bullet}$ correspond to the primitive
tribes of any based matrix $T=(G,s,b)$ homologous to $T_{\bullet}$. Let
$Aut(T_{\bullet})$ be the automorphism group of the based matrix
$T_{\bullet}$. Note that the group $Aut(T_{\bullet})$ considered up to
isomorphism does not depend on the choice of a primitive based matrix in the
homology class.

Let us define $\bar{\mathfrak{C}}(T_{\bullet})$ to be the finest partition
such that $\mathfrak{C}(T_{\bullet})\succ\bar{\mathfrak{C}}(T_{\bullet})$ and
$\phi({\mathfrak{C}}(T_{\bullet}))=\bar{\mathfrak{C}}(T_{\bullet})$ for any
$\phi\in Aut(T_{\bullet})$. Note that
$\\{s\\}\in\bar{\mathfrak{C}}(T_{\bullet})$. The partition
$\bar{\mathfrak{C}}(T_{\bullet})$ defines a partition on the set
${\mathfrak{C}}(T_{\bullet})_{prim}={\mathfrak{C}}(T_{\bullet})$. By
Proposition 2 this partition can be lifted to some partition of the set
$\bar{\mathfrak{C}}(T)_{prim}$ for any based matrix $T$ homologous to
$T_{\bullet}$, and the partition does not depend on the sequence of
transformations connecting $T$ and $T_{\bullet}$.

For a based matrix $T=(G,s,b)$ denote
$\bar{\mathfrak{C}}(T)_{prim}^{*}=\bar{\mathfrak{C}}(T)_{prim}\setminus\\{\bar{C}_{0}(T)\\}$
where $\bar{C}_{0}(T)$ is the element of the partition
$\bar{\mathfrak{C}}(T)_{prim}$ which includes the zero tribe $C_{0}(T)$ if it
is primitive, and $\bar{C}_{0}(T)=\emptyset$ if $C_{0}(T)$ is not primitive.
Let
$\bar{A}^{\mathfrak{C}}(T)=H[\bar{\mathfrak{C}}(T)_{prim}^{*}]/\langle\bar{b}(s)\rangle$
where

$\bar{b}(s)=\sum_{\bar{C}\in\bar{\mathfrak{C}}(T)_{prim}^{*}}b(s,\chi_{\bar{C}})\cdot\bar{C}.$
(10)

Define the maps $\bar{P}^{\mathfrak{C}}_{T}\colon
G\to\bar{A}^{\mathfrak{C}}(T)$ by the formula

$\bar{P}^{\mathfrak{C}}_{T}(g)=\sum_{\bar{C}\in\bar{\mathfrak{C}}(T)_{prim}^{*}}\left(b(g,\chi_{\bar{C}})-\left\lfloor\frac{|\bar{C}|}{2}\right\rfloor\cdot
b(g,s)\right)\cdot\bar{C}.$ (11)

For an operation $f\colon T\to T^{\prime}$, $T=(G,s,b)$,
$T^{\prime}=(G^{\prime},s,b^{\prime})$, of type $M_{1},M_{2}$ or $M_{3}$ we
define a homomorphism
$\bar{A}^{\mathfrak{C}}(f)\colon\bar{A}^{\mathfrak{C}}(T)\to\bar{A}^{\mathfrak{C}}(T^{\prime})$
by the formula

$\bar{A}^{\mathfrak{C}}(f)\left(\sum_{\bar{C}\in\bar{\mathfrak{C}}(T)_{prim}^{*}}\lambda_{\bar{C}}\cdot\bar{C}\right)=\sum_{\bar{C}\in\bar{\mathfrak{C}}(T)_{prim}^{*}}\lambda_{\bar{C}}\cdot
f(\bar{C})$ (12)

where $f(\bar{C})\in\bar{\mathfrak{C}}(T^{\prime})_{prim}^{*}$ is the unique
class such that $f(\bar{C})\cap G=\bar{C}$.

######  Theorem 4.

  1. 1.

Let $T=(G,s,b)$ be a based matrix homologous to $T_{\bullet}$ and $f\colon
T\to T_{\bullet}$ be a transformation chain connecting these based matrices.
Then the map
$\bar{A}^{\mathfrak{C}}(f)\colon\bar{A}^{\mathfrak{C}}(T)\to\bar{A}^{\mathfrak{C}}(T_{\bullet})$
is an isomorphism which does not depend on the choice of the transformation
$f$.

  2. 2.

The family of maps
$\bar{p}^{\mathfrak{C}}_{T}=\bar{A}^{\mathfrak{C}}(f)\circ\bar{P}^{\mathfrak{C}}_{T}\colon
G\to\bar{A}^{\mathfrak{C}}(T_{\bullet})$ is a parity with coefficients in
$\bar{A}^{\mathfrak{C}}(T_{\bullet})$.

The parity $\bar{p}^{\mathfrak{C}}$ is called the reduced parity associated
with the tribal system $\mathfrak{C}$.

###### Proof.

1\. We can prove that $\bar{A}^{\mathfrak{C}}(f)$ using the arguments of
Theorem 3. Let $f_{i}\colon T\to T_{\bullet}$ be two sequences connecting $T$
and $T_{\bullet}$. Then $\phi=f_{2}\circ f_{1}^{-1}$ is a sequence of
transformations of $T_{\bullet}$ to itself. Since $T_{\bullet}$ is primitive,
$\phi$ is an automorphism of $T_{\bullet}$. Hence, $\phi$ acts trivially on
$\bar{\mathfrak{C}}(T_{\bullet})_{prim}^{*}$. Therefore $\phi$ induces the
identity automorphism of $\bar{A}^{\mathfrak{C}}(T_{\bullet})$, i.e.
$\bar{A}^{\mathfrak{C}}(f_{2}\circ f_{1}^{-1})=id$. Hence,
$\bar{A}^{\mathfrak{C}}(f)=\bar{A}^{\mathfrak{C}}(f_{2}\circ
f_{1}^{-1})\cdot\bar{A}^{\mathfrak{C}}(f_{1})=\bar{A}^{\mathfrak{C}}(f_{1})$.

2\. Following the arguments of Theorem 3 we can show that
$\bar{P}^{\mathfrak{C}}$ is a parity functor. We need only to check the
equality $\bar{p}^{\mathfrak{C}}_{T^{\prime}}|_{G}=\bar{p}^{\mathfrak{C}}_{T}$
for any based matrices $T=(G,s,b)$ and $T^{\prime}=(G^{\prime},s,b^{\prime})$
connected by an operation $f\colon T\to T^{\prime}$ of type $M_{i}$,
$i=1,2,3$.

Let $f^{\prime}\colon T^{\prime}\to T_{\bullet}$ be a sequence of
transformations $M_{i}^{\pm 1}$, $i=1,2,3$ from $T^{\prime}$ to $T_{\bullet}$.
By definition

$\bar{p}^{\mathfrak{C}}_{T^{\prime}}|_{G}=\bar{A}^{\mathfrak{C}}(f^{\prime})\circ\bar{P}^{\mathfrak{C}}_{T^{\prime}}|_{G}=\bar{A}^{\mathfrak{C}}(f^{\prime})\circ\bar{A}^{\mathfrak{C}}(f)\circ\bar{P}^{\mathfrak{C}}_{T}=\bar{A}^{\mathfrak{C}}(f^{\prime}\circ
f)\circ\bar{P}^{\mathfrak{C}}_{T}=\bar{p}^{\mathfrak{C}}_{T}$

where the second equality is the parity functor property. ∎

######  Definition 13.

The reduced parity $\hat{p}^{st}=\hat{p}^{\mathfrak{C}_{\infty}}$ associated
with the stable tribal system $\mathfrak{C}_{\infty}$ is called the reduced
stable parity.

######  Example 6.

Consider the based matrix $T=(\\{s,1,2,3,4,5,6,7,8\\},s,b)$ from Example 4. A
primitive based matrix $T_{\bullet}=(\\{s,7,8\\},s,b_{\bullet})$ is given in
Example 5. It has three primitive stable tribes $\\{s\\}$, $\\{7\\}$ and
$\\{8\\}$. The automorphism group of $T_{\bullet}$ is
$Aut(T_{\bullet})=\mathbb{Z}_{2}$, it interchanges the primitive tribes $7$
and $8$. Then the partition $\bar{\mathfrak{C}}_{\infty}(T_{\bullet})_{prim}$
consists of the element $\\{7,8\\}$ and $\\{s\\}$. The coefficient group
$\bar{A}^{st}$ is equal to $\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$ since the zero
tribe is not primitive and $\bar{b}(s)=0$.

The induced partition for $T$ will be
$\bar{\mathfrak{C}}_{\infty}(T)_{prim}=\\{\\{s\\},\\{5,6,7,8\\}\\}$. It
corresponds to the matrix $\bar{B}_{3}$ which is obtained from $\tilde{B}_{3}$
by taking the sum of the last two columns.

$\bar{B}_{3}=\left(\begin{array}[]{c|c}0&0\\\ \hline\cr 0&0\\\ 0&0\\\
\hline\cr 0&0\\\ 0&0\\\ \hline\cr 0&1\\\ 0&1\\\ 0&1\\\ \hline\cr
0&1\end{array}\right).$

Then the reduced stable parity on $T$ is equal to

$\displaystyle\bar{p}^{st}(\mathbf{1})=\bar{p}^{st}(\mathbf{2})=\bar{p}^{st}(\mathbf{3})=\bar{p}^{st}(\mathbf{4})=(0,0),$
$\displaystyle\bar{p}^{st}(\mathbf{5})=\bar{p}^{st}(\mathbf{6})=\bar{p}^{st}(\mathbf{7})=\bar{p}^{st}(\mathbf{8})=(0,1).$

###  2.3 Reduced stable parity for virtual and flat knot

Let $\mathcal{K}$ be a virtual or flat knot. Then we can define reduced stable
parity on the diagrams of the knot $\mathcal{K}$ as follows. Let $D$ be a
diagram of $\mathcal{K}$. Consider the based matrix $T(D)$ of the diagram $D$
and a primitive based matrix $T_{\bullet}(D)$ homologous to $T(D)$. Denote the
group $\bar{A}^{st}(T_{\bullet}(D))$ by $\bar{A}^{st}(\mathcal{K})$.

######  Proposition 4.

The family of the maps
$\bar{p}^{st}_{T(D)}\colon{\mathcal{V}}(D)\to\bar{A}^{st}(\mathcal{K})$ is a
parity on diagrams of the knot $\mathcal{K}$ with coefficients in the group
$\bar{A}^{st}(\mathcal{K})$.

Proposition 4 follows from Lemma 2 and Theorem 4.

######  Definition 14.

The parity $\bar{p}^{st}$ from Proposition 4 is called the reduced stable
parity on virtual (flat) knots.

######  Example 7.

Consider the flat knot diagram $D$ in Fig. 11. Then $T(D)=(\\{s,1,2,3\\},s,b)$
where $b$ is defined by the matrix $B$ with coefficients in $\mathbb{Z}_{2}$.

$B=\left(\begin{array}[]{c|ccc}0&1&1&0\\\ \hline\cr 1&0&0&0\\\ 1&0&0&1\\\
0&0&1&0\end{array}\right)$

Figure 11: Flat knot diagram and the corresponding based matrix

The based matrix $T(D)$ is primitive. Its stable tribes are

${\mathfrak{C}}_{\infty}=\\{\\{s\\},\\{1\\},\\{2\\},\\{3\\}\\}.$

Since the automorphism group $Aut(T(D)$ is trivial and the zero tribe is
empty, the partition $\bar{\mathfrak{C}}_{\infty}(T(D))_{prim}$ coincides with
${\mathfrak{C}}_{\infty}$. Thus, we have four reduced stable classes. The
element $\bar{b}(s)=(0,1,1,0)\in\mathbb{Z}_{2}[{\mathfrak{C}}_{\infty}]$.
Hence,
$\bar{A}^{st}=\mathbb{Z}_{2}[{\mathfrak{C}}_{\infty}]/\langle\bar{b}(s)\rangle\simeq\mathbb{Z}_{2}^{3}$
where the isomorphism is induced by the map

$\lambda_{s}\cdot\mathbf{s}+\lambda_{1}\cdot\mathbf{1}+\lambda_{2}\cdot\mathbf{2}+\lambda_{3}\cdot\mathbf{3}\mapsto(\lambda_{s},\lambda_{1}+\lambda_{2},\lambda_{3}).$

The parity values are

$\displaystyle\bar{p}^{st}(1)=\left(1,\left(0-\left\lfloor\frac{1}{2}\right\rfloor
1\right)+\left(0-\left\lfloor\frac{1}{2}\right\rfloor
1\right),0-\left\lfloor\frac{1}{2}\right\rfloor 1\right)=(1,0,0),$
$\displaystyle\bar{p}^{st}(2)=\left(1,\left(0-\left\lfloor\frac{1}{2}\right\rfloor
1\right)+\left(0-\left\lfloor\frac{1}{2}\right\rfloor
1\right),1-\left\lfloor\frac{1}{2}\right\rfloor 1\right)=(1,0,1),$
$\displaystyle\bar{p}^{st}(3)=\left(0,\left(0-\left\lfloor\frac{1}{2}\right\rfloor
0\right)+\left(1-\left\lfloor\frac{1}{2}\right\rfloor
0\right),0-\left\lfloor\frac{1}{2}\right\rfloor 0\right)=(0,1,0).$

##  Appendix A Reduced stable parity of knots of complexity $\leq 4$

Using a Mathematica program, we calculated the reduced stable parity for the
diagrams of Green’s table [3] with the number of crossings less or equal $4$.

It turns out that the based matrices of all these diagrams are either
primitive or reduce to the trivial matrix $(0)$.

In the last case there are no primitive tribes and the reduced stable parity
$\bar{p}^{st}$ coincides with the index parity $ip$. This situation occurs for
the knots

$\displaystyle 2.1,3.2,3.5,3.6,3.7,4.3,4.6,4.12,4.25,4.27,4.36,4.37,4.40,$
$\displaystyle 4.41,4.43,4.44,4.46,4.53,4.54,4.61,4.64,4.65,4.68,4.73,$
$\displaystyle 4.74,4.75,4.82,4.84,4.86,4.91,4.92,4.94,4.95,4.96,4.99,$
$\displaystyle 4.100,4.101,4.102,4.104,4.105,4.106,4.108.$

The knots $3.1,3.3,3.4$ have three primitive non zero tribes corresponding to
the crossings and the coefficient group $\bar{A}^{st}$ is isomorphic to
$\mathbb{Z}^{3}$.

For the knots

$\displaystyle
4.2,4.5,4.7,4.10,4.11,4.14,4.15,4.17,4.19,4.20,4.21,4.22,4.23,4.24,$
$\displaystyle
4.26,4.28,4.29,4.30,4.32,4.34,4.35,4.38,4.39,4.42,4.45,4.47,4.48,$
$\displaystyle
4.49,4.50,4.56,4.57,4.59,4.62,4.63,4.66,4.67,4.70,4.71,4.76,4.78,$
$\displaystyle 4.79,4.80,4.81,4.83,4.87,4.88,4.93,4.97,4.103,$

each of 4 crossings determines a primitive non zero tribe, and the coefficient
group $\bar{A}^{st}$ of the reduced stable parity is isomorphic to
$\mathbb{Z}^{4}$.

The knots $4.9,4.16,4.33,4.52,4.58,4.72$ have 3 primitive tribes, and
$\bar{A}^{st}=\mathbb{Z}^{4}$ for them.

For example, the based matrix of the knot $4.9$ is equal to

$\left(\begin{array}[]{c|cccc}0&1&0&0&-1\\\ \hline\cr-1&0&-1&-1&0\\\
0&1&0&-1&-1\\\ 0&1&1&0&-1\\\ 1&0&1&1&0\\\ \end{array}\right).$

The crossings $1$ and $4$ belongs to one tribe because the corresponding rows
(or columns) are proportional. Note that $1$ and $4$ are not complementary.
The parity matrix is given below.

$\left(\begin{array}[]{c|ccc}0&0&0&0\\\ \hline\cr-1&1&-1&-1\\\ 0&0&0&-1\\\
0&0&1&0\\\ 1&-1&1&1\\\ \end{array}\right)$

The first column corresponds to $s$ (and the index parity), the other columns
correspond to non zero primitive tribes. The first row presents the vector
$\bar{b}(s)$, the other rows present the parity values of the crossings.

The knots $4.85,4.89,4.90,4.98,4.107$ have 4 primitive tribes, and
$\bar{A}^{st}=\mathbb{Z}^{4}\oplus\mathbb{Z}_{2}$ for them.

For example, the knot $4.85$ has the based matrix

$\left(\begin{array}[]{c|cccc}0&2&-2&-2&2\\\ \hline\cr-2&0&-2&-1&0\\\
2&2&0&0&3\\\ 2&1&0&0&2\\\ -2&0&-3&-2&0\\\ \end{array}\right)$

which coincides with the parity matrix. Each vertex defines a primitive tribe.
The vector $\bar{b}(s)=(0,2,-2,-2,2)$ is divisible by $2$. Hence, the
coefficient group is equal to

$\bar{A}^{st}=\mathbb{Z}[\bar{\mathfrak{C}}_{\infty}(T(D))_{prim}]/\left<\bar{b}(s)\right>=\mathbb{Z}^{5}/\left<(0,2,-2,-2,2)\right>=\mathbb{Z}^{4}\oplus\mathbb{Z}_{2}.$

The knots $4.13,4.18,4.31,4.51,4.60,4.69$ have 3 primitive tribes, the
symmetry group $Aut(T_{\bullet})=\mathbb{Z}_{2}$ which acts trivially on the
primitive tribes, and $\bar{A}^{st}=\mathbb{Z}^{3}$.

For example, the knot $4.13$ has the based matrix

$\left(\begin{array}[]{c|cccc}0&-1&0&0&1\\\ \hline\cr 1&0&1&1&1\\\
0&-1&0&0&0\\\ 0&-1&0&0&0\\\ -1&-1&0&0&0\\\ \end{array}\right)$

The primitive tribes are $\\{1\\},\\{2,3\\}$ and $\\{4\\}$ (the rows the
second and third crossings coincide). The symmetry group acts by transposition
of the second and the third crossings, and induces trivial action on the
primitive tribes. The parity matrix is

$\left(\begin{array}[]{c|ccc}0&-1&0&1\\\ \hline\cr 1&0&1&1\\\ 0&-1&0&0\\\
0&-1&0&0\\\ -1&-1&1&0\\\ \end{array}\right).$

Since the vector $\bar{b}(s)\neq 0$, the coefficient group is
$\bar{A}^{st}=\mathbb{Z}^{4}/\mathbb{Z}=\mathbb{Z}^{3}$.

Finally, the knots $4.1,4.4,4.8,4.55,4.77$ have 4 primitive tribes and the
symmetry group $Aut(T_{\bullet})=\mathbb{Z}_{2}$ which interchange pairs of
crossings. The coefficient group is equal to
$\bar{A}^{st}=\mathbb{Z}^{2}\oplus\mathbb{Z}_{2}$.

For example, the knot $4.1$ has the based matrix

$\left(\begin{array}[]{c|cccc}0&1&-1&1&-1\\\ \hline\cr-1&0&-1&0&0\\\
1&1&0&0&0\\\ -1&0&0&0&-1\\\ 1&0&0&1&0\\\ \end{array}\right).$

Each crossing defines a primitive tribe. There is a unique nontrivial symmetry
$(13)(24)$ of the based matrix. Hence,
$\bar{\mathfrak{C}}(T)_{prim}=\\{\\{1,3\\},\\{2,4\\}\\}$. The parity matrix
looks like

$\left(\begin{array}[]{c|cc}0&2&-2\\\ \hline\cr-1&1&0\\\ 1&0&-1\\\ -1&1&0\\\
1&0&-1\\\ \end{array}\right).$

The coefficient groups is equal
$\bar{A}^{st}=\mathbb{Z}^{3}/\left<(0,2,-2)\right>=\mathbb{Z}^{2}\oplus\mathbb{Z}_{2}$.

## References

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https://www.math.toronto.edu/drorbn/Students/GreenJ/

  * [4] A. Henrich, A sequence of degree one vassiliev invariants for virtual knots, J. Knot Theory Ramifications 19:4 (2010) 461–487.
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  * [6] N. Kamada, S. Kamada, Abstract link diagrams and virtual knots, J. Knot Theory Ramifications 9:1 (2000) 93–106.
  * [7] L.H. Kauffman, Virtual knot theory, Europ. J. Combinatorics 20:7 (1999) 663–690.
  * [8] V.O. Manturov, Parity in knot theory, Sb. Math. 201:5 (2009) 693–733.
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*e-mail<EMAIL_ADDRESS>
# Deconvolution in Fluorescence Lifetime imaging microscopy (FLIM)

Varun Mannam∗ Department of Electrical Engineering, University of Notre Dame,
Notre Dame, IN 46556, USA  Xiaotong Yuan  Department of Electrical
Engineering, University of Notre Dame, Notre Dame, IN 46556, USA  Scott Howard
Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN
46556, USA

###### Abstract

Fluorescence lifetime imaging microscopy (FLIM) is an important technique to
understand the chemical micro-environment in cells and tissues since it
provides additional contrast compared to conventional fluorescence imaging.
When two fluorophores within a diffraction limit are excited, the resulting
emission leads to non-linear spatial distortion and localization effects in
intensity (magnitude) and lifetime (phase) components. To address this issue,
in this work, we provide a theoretical model for convolution in FLIM to
describe how the resulting behavior differs from conventional fluorescence
microscopy. We then present a Richardson-Lucy (RL) based deconvolution
including total variation (TV) regularization method to correct for the
distortions in FLIM measurements due to optical convolution, and
experimentally demonstrate this FLIM deconvolution method on a multi-photon
microscopy (MPM)-FLIM images of fluorescent-labeled fixed bovine pulmonary
arterial endothelial (BPAE) cells.

###### keywords:

Fluorescence lifetime imaging microscopy, FLIM, fluorescence lifetime,
deconvolution, localization, super-resolution, multi-photon microscopy,
Richardson-Lucy (RL), total variation (TV).

##  1 Introduction

Fluorescence lifetime imaging microscopy (FLIM) is a powerful technique in
biomedical research that provides enhanced molecular contrast as fluorescence
lifetime [1] information in addition to the intensity information that is
usually obtained from a conventional fluorescence microscopy [2]. Fluorescence
microscopy illuminates the samples to excite fluorophores within cells and
collect the emitted light, whereas, in FLIM, additional measurements estimate
the intrinsic property of the fluorophore as fluorescence lifetime. In FLIM
measurements, the lifetime information can be extracted using either TD-FLIM
[3, 4] that measures the time delay between excited and emitted pulsed laser
using an exponential fit function or FD-FLIM [5, 6] that measures the relative
change in magnitude and phase of the periodically modulated laser pulse. While
FLIM has been leveraged in several biological applications, several crucial
techniques from fluorescence microscopy has not been translated to FLIM. For
example, deconvolution techniques used to compensate for system optical
response (point spread function: PSF) [7] and perform denoising are
commonplace (and often essential) steps in fluorescence microscopy but have
not been rigorously applied to FLIM data. FLIM data extraction and
interpretation is often a complicated and time-consuming process, as the raw
FLIM data are multi-dimensional, and the FLIM algorithms are computationally
intensive. Recently, machine learning (ML), particularly deep learning [8],
has gained more attention for its great performance in various FLIM image
processing tasks such as image denoising [9, 10], super-resolution [11, 12],
estimation of lifetime from raw FLIM data using convolutional neural networks
(CNNs) [13, 14], however, it requires training dataset [8]. To overcome large
training data problems, self-supervised machine learning methods are proposed,
but they provide limited accuracy with huge computational time [15]. Also,
unsupervised machine learning methods are proposed in the literature; however,
these methods are useful to form lifetime clusters, and segmentation [16, 17].
This paper presents the analytical, theoretical description of optical
convolution in FLIM and how the resulting behavior is different from
traditional fluorescence microscopy. We then present the theoretical grounding
and experimental demonstration of FLIM deconvolution on multi-photon FLIM
images of fluorescent-labeled fixed bovine pulmonary arterial endothelial
(BPAE) cells captured using our custom-built setup [18].

##  2 Methodology

When two fluorophores within a diffraction limit are excited in fluorescence
microscopy, the resulting emission intensity in the convolution of PSF with
emitters yields a linear response. In FLIM measurements, however the lifetime
information convolution is highly non-linear. For example, in frequency domain
FLIM, complex exponentials are summed together instead of purely real-values
in fluorescence microscopy. Summation of complex exponentials in FLIM results
in an effective blurring of the boundaries between two fluorophores (as seen
in fluorescence microscopy) as well as a shifting of the measured interface.
This is represented in Fig. 1. Fig. 1 shows this behavior from a simulated
two-fluorophore model [19] and the effect of convolution on the lifetime for
two different scenarios. First, if the amplitude of two fluorophores are same
as shown in Fig. 1 (a) and the second, if the magnitudes (magnitude is the
product of concentration $C$ and cross-section $\sigma$ of a fluorophore) of
the two-fluorophores are different as shown in Fig. 1 (b).

Figure 1: Simulation results on the 1D data for two different fluorophores
with different lifetime values. The effect of deconvolution of estimated
lifetime in finding the ideal boundary between the two fluorophores with equal
magnitudes (a) and unequal magnitudes (b).

In Fig. 1 (a), the magenta line shows the ideal lifetime of two fluorophores
on a single 1D line where the left fluorophore has a lifetime of
$1\text{\,}\mathrm{n}\mathrm{s}$ ($x<0$) and the right fluorophore has a
lifetime of $2\text{\,}\mathrm{n}\mathrm{s}$ ($x>0$). The orange line
indicates the estimated lifetime after applying system optical response (PSF)
on this two-fluorophore model. Clearly, the orange line shows the effect of a
smooth transition between lifetime values (missing ideal boundaries). One way
to extract the ideal boundary is by applying a lifetime threshold (mean of two
lifetime values). If the magnitudes of the fluorophores are equal, the
threshold method can identify the ideal boundary between two fluorophores.
However, this method doesn’t work when the magnitudes are different, as
illustrated in Fig. 1 (b) orange curve. This is due to the unequal magnitudes
leading to a shifting effect on the estimated lifetime. To solve this problem,
we performed the Richardson-Lucy (RL) deconvolution [20] on the 1D data using
known PSF (used in the previous convolution step), and the results are shown
in the blue line. Here RL method is an iterative approach for the maximum
likelihood estimation for Poisson noise. Fig. 1 (a) blue line indicates the
estimated lifetime after applying deconvolution on the two-fluorophore FLIM
data, which shows the ideal boundary. Similarly, Fig. 1 (b) blue line corrects
the shift (shift towards the ideal boundary) caused by the PSF convolution and
can make it similar to the ideal boundary.

Method: The Richardson-Lucy (RL) deconvolution [20] method applied in the
previous step is explained here. Consider the estimated fluorophores is $o(s)$
at a position $s$ and PSF is $h(s)$. Measured FLIM data (either real or
imaginary values of FLIM data or G and S of the phasor [8]) follows the
Poisson distribution, represented is $i(s)$. Next the original fluorophores
real/imaginary values are estimated using the iterative approach using the
measured values $i(s)$ and $h(s)$.

$o_{k+1}(s)=\Bigg{\\{}\frac{i(s)}{(o_{k}*h(s))}*h(-s)\Bigg{\\}}.o_{k}(s)$ (1)

Here the “$*$”, indicates convolution operation, “.” indicates point-wise
multiplication, “$k$” indicates the iteration index, respectively. However,
the deconvolution operation leads to overshooting at the boundaries. The
total-variation (TV) method is used as a regularization method to mitigate
this issue. With the TV regularization, the final equation is modified as
given below.

$o_{k+1}(s)=\Bigg{\\{}\frac{i(s)}{(o_{k}*h(s))}*h(-s)\Bigg{\\}}.\frac{o_{k}(s)}{1-\lambda
div(\frac{\Delta o_{k}(s)}{|\Delta o_{k}(s)|})}$ (2)

Here $\lambda$ indicates the regularization factor, typically small such as
0.005. The deconvolution operation, as shown in Eq.2 is applied on real and
imaginary values of the measured FLIM data, and lifetime is extracted from the
deconvolved real and imaginary FLIM measurements. If the PSF is unknown, blind
deconvolution methods can estimate deconvolved results, including estimated
PSF values. Recently, image processing tools such as ImageJ plugins are
helpful to process the data, and the deconvolution can be applied to the FLIM
data using the plugin [21].

##  3 Results and discussion

In this section, the above-mentioned deconvolution method is extended to the
2D FLIM data and analysis.

###  3.1 Simulated results

Figure 2: Simulation results on the 2D data with two fluorophores with
different lifetime values. (a) indicates the original objects with two
fluorophore model. When the magnitude is similar in two fluorophores,
convolved lifetime and deconvolved lifetime, line plots of the dotted line are
given in (b), (c), and (d), respectively. Similarly, for the unequal
magnitudes of two fluorophores, convolved lifetime and deconvolved lifetime,
line plots of the dotted line are given in (e), (f), and (g), respectively.
Scale bar: 1 $\mu$m.

Fig. 2(a) show the original lifetime of two-fluorophore model in frequency
domain FLIM system with lifetime values of $1\text{\,}\mathrm{n}\mathrm{s}$
(left fluorophore: $\tau_{1}$) and $2\text{\,}\mathrm{n}\mathrm{s}$ (right
fluorophore: $\tau_{2}$) respectively. In the first scenario, if the magnitude
of the emitted signal is similar ($A_{1}\approx A_{2}$) for each fluorophore,
then the effect of convolution on lifetime is shown in Fig. 2 (b). In the
alternate scenario, consider that the magnitude of the emitted signal for one
of the fluorophores is stronger compared to the other ($A_{1}\approx 5A_{2}$),
and the corresponding lifetime after convolution with PSF is shown in Fig. 2
(c). Clearly, in the second scenario, the ideal boundary between the two
fluorophores is shifted towards the right by the stronger fluorophore, hence
appears as inaccurate localization. Hence, the localization-based SR methods
using lifetime information cannot be applied directly since it provides
inaccurate results.

We demonstrate a RL-TV based deconvolution method that can correct the shift
and identify the actual boundary between the two fluorophores to address this
problem. First, we perform Richardson-Lucy (RL) deconvolution on each lifetime
imaging component (phasor G and S, or real and imaginary). The RL
deconvolution finds the maximum likelihood estimation of the original sample
assuming Poisson noise. However, the real and imaginary (or phasor G and S)
components do not follow Poisson statistics (more accurately follow a Skellam
distribution [22]) and experimentally appear Gaussian-like. Similarly, the RL
deconvolution method can be applied to the Gaussian noise distribution to
extract the lifetime information. In addition, total variation (TV) based
regularization is added to RL deconvolution to avoid additional artifacts, as
commonly employed in fluorescence microscopy.

Fig. 2 (c) and (d) shows the deconvolved lifetime when the magnitude of both
fluorescence emission is equal. From this 1D data, deconvolved lifetime
accurately separates two fluorophores along the intersection. Similarly,
deconvolved lifetime for the unequal magnitudes scenario is shown in Fig. 2
(f) and corresponding line plots are shown in Fig. 2 (g). Clearly, from the
unequal magnitudes scenario, the lifetime with deconvolution brings back the
real boundary (shift towards the dominating fluorophore, see orange curve to
blue curve). In Fig. 2, we show the convolution and deconvolution lifetime
results after taking an average over 100 Gaussian noisy iterations with a
standard deviation of $0.05$ (corresponding signal-to-noise ratio (SNR) is
$26$ dB).

###  3.2 Experimental results

The proposed deconvolution approach is demonstrated experimentally in Fig. 3.
FLIM images of fixed BPAE cells [labeled with MitoTracker Red CMXRos
(mitochondria), Alexa Fluor 488 phalloidin (F-actin), and DAPI (nuclei)] from
Invitrogen FluoCells F36924 is presented before and after deconvolution. The
lifetime images are captured with our custom-built two-photon FLIM setup [18]
with an excitation wavelength of 800 nm, single-channel (captured all the
fluorophores simultaneously using a bandpass filter (300-700nm), an image
dimension of 260 $\times$ 260, the pixel dwell time of 12 µs, total
acquisition time is $1\text{\,}\mathrm{s}$, and pixel width of 300 nm.

Figure 3: Fixed BPAE sample composite lifetime before and after deconvolution
on the lifetime information. Image acquired with custom two-photon FLIM
microscope (see text). Scale bar: 10 $\mu$m.

Fig. 3 (a) shows the experimentally captured FLIM lifetime information of BPAE
cells with poor resolution. In contrast, after applying RL deconvolution on
the real and imaginary values of FLIM data, the lifetime is extracted as shown
in Fig. 3 (b) shows a better resolution of nucleus and mitochondria
fluorophores. The methods used in this manuscript are provided as open-source,
accessible via GitHub111https://github.com/ND-
HowardGroup/Deconvolution_lifetime_imaging.git.

##  4 Conclusions

In Fluorescence lifetime imaging microscopy (FLIM), when two fluorophores
within a diffraction limit are excited, the resulting emission leads to
different spatial distortion and localization effects in intensity (magnitude)
and lifetime (phase) components. To address this issue, in this work, we
provide a theoretical model to demonstrate a Richardson-Lucy (RL) based
deconvolution, including the TV-based regularization method to correct for the
distortions in FLIM measurements due to optical convolution and provide an
accurate boundary of the two-fluorophore model. The technique is
experimentally demonstrated on MPM-FLIM images of fluorescent-labeled fixed
bovine pulmonary arterial endothelial (BPAE) cells. Finally, this deconvolved
lifetime information can effectively enhance biological structures of interest
in biomedical imaging applications.

## Disclosures

The authors declare no conflicts of interest.

## Funding.

This material is based upon work supported by the National Science Foundation
(NSF) under Grant No. CBET-1554516.

###### Acknowledgements.

The authors further acknowledge the Notre Dame Center for Research Computing
(CRC) for providing the Nvidia GeForce GTX 1080-Ti GPU resources for
performing the deconvolution on the fluorescence lifetime images and data
analysis in MatLAB and Python.

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