# Recurrent Generalization of F-Polynomials for Virtual Knots and Links

Amrendra Gill  Department of Mathematics, Indian Institute of Technology
Ropar; 140001 Punjab, India<EMAIL_ADDRESS>,  Maxim Ivanov
Laboratory of Topology and Dynamics, Novosibirsk State University, 630090
Novosibirsk, Russia<EMAIL_ADDRESS>,  Madeti Prabhakar  Department of
Mathematics, Indian Institute of Technology Ropar; 140001 Punjab, India
<EMAIL_ADDRESS>and  Andrei Vesnin  Tomsk State University, Tomsk,
Russia; Higher School of Economics, Moscow, Russia; Sobolev Institute of
Mathematics, Siberian Branch of the Russian Academy of Sciences, 630090
Novosibirsk, Russia<EMAIL_ADDRESS>
###### Abstract.

F-polynomials for virtual knots were defined by Kaur, Prabhakar and Vesnin in
2018 using flat virtual knot invariants. These polynomials naturally
generalize Kauffman’s affine index polynomial and use smoothing in classical
crossing of a virtual knot diagram. In this paper we introduce weight
functions for ordered orientable virtual and flat virtual link. A flat virtual
link is an equivalence class of virtual links in respect to a local symmetry
changing type of classical crossing in a diagram. By considering three types
of smoothings in classical crossings of a virtual link diagram and suitable
weight functions, we provide a recurrent construction for new invariants. We
demonstrate by providing explicit examples, that newly defined polynomial
invariants are stronger than F-polynomials.

###### Key words and phrases:

difference writhe; virtual knot invariant; flat virtual knot invariant.

###### 2010 Mathematics Subject Classification:

57K12

A. Gill and M. Prabhakar were supported by DST (project number
DST/INT/RUS/RSF/P-02), M. Ivanov was supported by RFBR (grant number
19-01-00569), A. Vesnin was supported by RSF (grant number 20-61-46005).

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Weight functions
  4. 4 I-functions
  5. 5 Smoothings in classical crossings and invariants
  6. 6 Recurrent construction of a sequence of invariants
  7. 7 (n,m)-difference writhe and invariants of virtual knots
  8. 8 Flat span and invariants of 2-component virtual links

##  1\. Introduction

Theory of virtual knots and links was introduced by Kauffman in [9] as a
generalization of classical knot theory. It was observed by Kuperberg in [14]
that the study of virtual knots is naturally related to the study of knots and
links embedded in 3-manifolds which are thickened surfaces. Following the
Kauffman’s approach we consider virtual links as equivalence classes of
virtual link diagrams up the equivalence relations corresponding to
generalized Reidemeister moves. Recall that a virtual knot or link diagram is
4-regular planar graph, where each vertex is indicated as either classical or
virtual crossing. We will consider oriented virtual links, where each virtual
crossing is depicted by placing a small circle at the vertex. In Figure 1,
there are presented two types of classical crossings and the virtual crossing
of an oriented virtual link diagram.

Polynomial invariants based on the index values in classical crossings were
introduced by Cheng and Gao [3], known as th writhe polynomial, and by
Kauffman [10], known as the affine index polynomial. The connection of the
affine index polynomial with the virtual knot cobordism is described in [11].
For related polynomial invariants and their properties see [12, 15, 16, 17].
Three kinds of invariants of a virtual knot called the first, second and third
intersection polynomials were introduced recently in [7]. F-polynomials for
oriented virtual knots were constructed in [13] as a generalization of the
affine index polynomial. The construction of F-polynomials is based on
invariants of flat virtual knots. Flat virtual link is an equivalence class of
virtual links in respect to a local symmetry changing type of classical
crossing in a diagram. Two types of classical crossings are presented in
Figure 1. F-polynomials were calculated for tabulated virtual knots in [8] and
[19], and successfully used to distinguish some oriented virtual knots in [6].
Another approach to construct invariants of flat virtual knots can be based on
representation of flat virtual braids by automorphisms of free groups, see,
for example, [1].

Figure 1. Two classical and one virtual crossings.

The paper is organized as the following. Section 2 contains some preliminary
information about generalized Reidemeister moves, definitions of sign and
index of a classical crossings, orientation revising smoothing of an oriented
virtual knot diagram, and $F$-polynomials introduced in [13]. In Section 3 we
introduce weight functions associated with classical crossings (see
Definitions 1 and 2). In Section 4 we use weight functions to define
I-function and flat I-function (see Definitions 5 and 6. We prove in Theorem 1
that these functions are invariants of ordered oriented virtual links and
ordered oriented flat virtual links, respectively. All introduced notions are
illustrated by Examples 1–4. In Section 5 we consider two more types of
smoothings in classical crossings and construct corresponding weight functions
taking values in the free $\mathbb{Z}$-module, generated by ordered oriented
flat virtual links (see Theorem 2). Such invariants, corresponding to type-2
smoothings, are constructed in Corollary 1 and used in Example 5 to
demonstrate that the virtual Kishino knot, a famous connected sum of two
trivial virtual knots, is non-trivial. In Section 6 we presents a recurrent
construction of a sequences of invariants, see Proposition 1 and realize this
method to define a multi-variable generalization of the $F$-polynomial in
Theorem 3. In Section 7 we define $(n,m)$-difference writhe of a virtual knot
diagram and use it to construct an invariant
$F_{K}^{n,m,k}(t,\ell_{1},\ell_{2})$ of an oriented virtual knots in Theorem
4. Also, we demonstrate that this invariant is stronger than $F$-polynomials.
In Section 8 we introduce an ordered virtual link invariant denoted by
$\operatorname{span}_{n,k}(L)$ and its flat version
$\operatorname{fspan}_{n,k}(L)$, that is an ordered flat virtual link
invariant, see Theorem 5. Then we use $\operatorname{fspan}_{n,k}(L)$ to
construct a family of oriented virtual knot invariants
$\widetilde{F}^{n,k,m}(t,\ell,v)$ on variables $t$, $\ell$ and $v$ in Theorem
6 and demonstrate that these 3-variable polynomials are stronger than
F-polynomials.

##  2\. Preliminaries

We will say that $n$-component link $L$ is _ordered_ , if its components are
labelled by different integers from $1$ to $n$. Analogously, ordered virtual
links and ordered flat virtual links can be defined. Ordered knots are the
particular 1-component case.

In this paper we consider ordered oriented virtual links and ordered flat
virtual links. Forgetting about ordering, we will get usual oriented virtual
links and oriented flat virtual links. When ordering is not important in our
considerations, we don’t present labels of components in figures.

Two diagrams of ordered oriented virtual links are _equivalent_ if and only if
one can be obtained from another by a finite sequence of generalized
Reidmeister moves. By generalized Reidemeister moves we mean the union of
classical Reidemeister moves and virtual Reidemeister moves. The non-oriented
versions of these moves are presented in Figure 2, and oriented versions can
be obtained by considering all possible orientations of link components.

(a) Classical Reidemeister moves.

(b) Virtual Reidemeister moves.

Figure 2. Reidemeister moves.

An ordered oriented virtual link is defined as an equivalence class of link
diagrams modulo generalized Reidemeister moves. For a link diagram $D$ we
denote the set of its classical crossings by $C(D)$. By $D^{*}$ we denote a
_mirror image_ of $D$ that is a diagram obtained from $D$ by changing all
classical crossings, and for $c\in C(D)$ we denote by $c^{*}$ a crossing
corresponding to $c$ in $C(D^{*})$. An arc of a diagram is an edge of a
corresponding 4-regular graph. An object (or a quantity) associated to a
diagram which remains invariant under all generalized Reidemeister moves is
called an _ordered oriented virtual link invariant_.

The _sign_ of a classical crossing $c\in C(D)$, denoted by
$\operatorname{sgn}(c)$, is defined as in Figure 3.

$\operatorname{sgn}(c)=+1$$\operatorname{sgn}(c)=-1$ Figure 3. Crossing signs.

Next we assign an integer value to each arc in $D$ in such a way that the
labeling around each crossing point of $D$ follows the rule as shown in Figure
4.

$b+1$$a$$b$$a-1$$b+1$$a$$b$$a-1$$b$$a$$b$$a$ Figure 4. Labeling around
crossing.

The _index_ value for a classical crossing $c\in C(D)$, denoted by
$\operatorname{Ind}(c)$, is defined as

(1) $\operatorname{Ind}(c)=\operatorname{sgn}(c)(a-b-1).$

Then the _affine index polynomial_ of a knot $K$ is defined via its diagram
$D$ as

$P_{K}(t)=\sum_{c\in C(D)}\operatorname{sgn}(c)(t^{\operatorname{Ind}(c)}-1).$

For properties and applications of $P_{K}(t)$ see [3, 10, 11, 12].

For each $n\in\mathbb{Z}\setminus\\{0\\}$ the _$n$ -th writhe $J_{n}(D)$_ of a
virtual knot diagram $D$ is defined as the number of positive sign crossings
minus number of negative sign crossings of $D$ with index value $n$. The
$n$-th writhe is a virtual knot invariant. Using $n$-th writhe, a new
invariant _$n$ -th d-writhe_ (difference writhe) of $D$ denoted by $\nabla
J_{n}(D)$ was defined in [13] as

(2) $\nabla J_{n}(D)=J_{n}(D)-J_{-n}(D).$

Thus $\nabla J_{n}(D)$ is a flat virtual knot invariant. For a classical
crossing $c\in C(D)$ denote by $D_{c}$ a diagram, obtained from $D$ by the
_type-1 smoothing_ of the diagram $D$ at the crossing $c$, see Figure 5. This
smoothing was used in [13] and called a _smoothing against orientation_.

Figure 5. Type-1 smoothing (smoothing against orientation).

Values $\nabla J_{n}(D_{c})$ for $c\in C(D)$ were used in [13] to construct a
family of polynomial invariants $F^{n}_{K}(t,\ell)$. An _$n$ -th
$F$-polynomial_ of a knot $K$ is defined via its diagram $D$ as

$F_{K}^{n}(t,\ell)=\sum_{c\in
C(D)}\operatorname{sgn}(c)t^{\text{Ind}(c)}\ell^{\nabla
J_{n}(D_{c})}-\sum_{c\in T_{n}(D)}\operatorname{sgn}(c)\ell^{\nabla
J_{n}(D_{c})}-\sum_{c\notin T_{n}(D)}\operatorname{sgn}(c)\ell^{\nabla
J_{n}(D)},$

where $T_{n}(D)$ consists of classical crossings of the diagram $D$ with the
following property:

$T_{n}(D)=\\{c\in C(D)\,\mid\,\nabla J_{n}(D_{c})=\pm\nabla J_{n}(D)\\}.$

For properties and applications of $F_{K}^{n}(t,\ell)$ see [8, 13, 19].

##  3\. Weight functions

Let $\mathcal{D}$ be a subset of all ordered oriented virtual link diagrams
with the following property: if $D\in\mathcal{D}$ then all diagrams obtained
from $D$ by generalized Reidemeister moves, crossing change operation,
reversing orientation and reordering of components also belong to
$\mathcal{D}$. We call $\mathcal{D}$ a _regular set of diagrams_. The
corresponding set of links is said to be a _regular set of ordered oriented
virtual links_. By forgetting ordering of components we obtain a _regular set
of unordered oriented virtual links_ , and similarly, by forgetting types of
classical crossings we obtain a _regular set of ordered oriented flat virtual
links_.

Denote by $C(\mathcal{D})$ the set of all classical crossings of diagrams
$D\in\mathcal{D}$.

######  Definition 1.

Let $G$ be an abelian group and $w:C(\mathcal{D})\to G$ be a function which
assigns a _value_ $w(c)\in G$ to a classical crossing $c\in C(D)$ for all
diagrams $D\in\mathcal{D}$. Function $w$ is said to be a _weight function_ ,
write $w\in W_{G}$, if it satisfies _weight function conditions_ (C1)–(C3):

  * (C1)

$w$ is _local_ , i.e. if $D^{\prime}$ is obtained from $D$ by a generalized
Reidemeister move such that a crossing $c\in D$ is not involved in this move
and $c^{\prime}\in D^{\prime}$ is the corresponding crossing, then
$w(c^{\prime})=w(c)$;

  * (C2)

if diagram $D^{\prime}$ is obtained from $D$ by RIII-move and involved
classical crossings $a,b,c\in D$ have weights $w(a)$, $w(b)$ and $w(c)$, as
well as involved crossings of $a^{\prime},b^{\prime},c^{\prime}\in D^{\prime}$
have weights $w(a^{\prime})$, $w(b^{\prime})$ and $w(c^{\prime})$, see Figure
6, then $w(a^{\prime})=w(a)$, $w(b^{\prime})=w(b)$ and $w(c^{\prime})=w(c)$.

$a$$b$$c$$\longleftrightarrow$$a^{\prime}$$b^{\prime}$$c^{\prime}$ Figure 6.
RIII move.

  * (C3)

if diagram $D^{\prime}$ is obtained from $D$ by SV-move and involved classical
crossing $c\in D$ has weight $w(c)$, as well as involved classical crossing
$c^{\prime}\in D^{\prime}$ has weight $w^{\prime}(c^{\prime})$, see Figure 7,
then $w^{\prime}(c^{\prime})=w(c)$.

$c$$\longleftrightarrow$$c^{\prime}$ Figure 7. SV move.

Definition 1 may be thought of as a generalization of the Chord Index axioms
in [2].

######  Definition 2.

Let $w:C(\mathcal{D})\to G$ be a weight function. Assume that diagram
$D^{\prime}$ is obtained from $D$ by RII-move and $\alpha$, $\beta$ are
crossings involved. If $w(\beta)=-w(\alpha)$, then we say that _$w$ is an odd
weight function_ and write $w\in W^{odd}_{G}$. If $w(\beta)=w(\alpha)$, then
we say that _$w$ is an even weight function_ and write $w\in W^{even}_{G}$.

######  Example 1.

Let $\alpha$ and $\beta$ be classical crossings involved in RII move as in
Figure 8.

Figure 8. RII move involving crossings $\alpha$ and $\beta$.

Consider two functions $w_{1},w_{2}:C(D)\to\mathbb{Z}$, where
$w_{1}(c)=\operatorname{sgn}(c)$, is the sign of crossing, defined in Figure
3, and $w_{2}(c)=\operatorname{Ind}(c)$, is the index of crossing, defined by
(1). Both of them are weight functions. Since $w_{1}(\alpha)=1$ and
$w_{1}(\beta)=-1$, we get $w_{1}(\beta)=-w_{1}(\alpha)$ and hence $w_{1}\in
W^{odd}_{\mathbb{Z}}$. Since
$w_{2}(\alpha)=\operatorname{sgn}(\alpha)\cdot(a-b-1)=a-b-1$ and
$w_{2}(\beta)=\operatorname{sgn}(\beta)\cdot(b+1-(a-1)-1)=a-b-1$, we get
$w_{2}(\beta)=w_{2}(\alpha)$ and hence $w_{2}\in W^{even}_{\mathbb{Z}}$.

For two weight function $u,v:C(D)\to G$, where $G$ is an abelian group, we can
define a sum and a product (if the codomain $G$ is a ring) as follows:

(3) $(u+v)(c)=u(c)+v(c)\qquad\text{and}\qquad(u*v)(c)=u(c)v(c).$

Therefore $W_{G}$ is an abelian group with $W_{G}^{odd}$ and $W_{G}^{even}$ as
subgroups. The set $W_{\mathbb{Z}}$ with operations (3) forms a ring, and it
may be convenient to regard $W_{G}$ as a $W_{\mathbb{Z}}$-module with the
module multiplication denoted by the same symbol ”$*$”.

For any $w\in W_{G_{1}}$ and any homomorphism $\phi:G_{1}\to G_{2}$ of abelian
groups the composition $w^{\prime}=\phi\circ w$ is a weight $w^{\prime}\in
W_{G_{2}}$.

We also admit cases when weight functions may be not defined for some
crossings of a diagram. In these cases we will use the following approach.

######  Definition 3.

A subset $C^{\prime}(\mathcal{D})\subset C(\mathcal{D})$ is said to be
_consistent_ if the characteristic function
$1_{C^{\prime}(\mathcal{D})}:C(\mathcal{D})\to\\{0,1\\}\subset\mathbb{Z}$ of
the set $C^{\prime}(\mathcal{D})$ is an even weight function.

######  Example 2.

For positive integer $i$ and $j$ consider the regular set $\mathcal{D}L_{n}$
of all diagrams of ordered oriented virtual links with at least
$n$-components, where $n=\max\\{i,j\\}$. Let $C_{i}(\mathcal{D}L_{n})$ be the
set containing crossings only of the $i$-th component, and let
$C_{ij}(\mathcal{D}L_{n})$ be the set containing only crossings, that belong
to both $i$-th and $j$-th components. The characteristic functions of sets
$C_{i}(\mathcal{D}L_{n})$ and $C_{ij}(\mathcal{D}L_{n})$ are even weight
functions.

Weight functions for consistent subsets of $C(\mathcal{D})$ can be defined in
the following way.

######  Definition 4.

Let $C^{\prime}(\mathcal{D})\subset C(\mathcal{D})$ be consistent. Then
$w^{\prime}:C^{\prime}(\mathcal{D})\to G$ is said to be _a weight function
defined for $C^{\prime}(\mathcal{D})$_ if $w^{\prime}$ satisfies weight
function conditions (C1) - (C3) for all crossings in
$C^{\prime}(\mathcal{D})$.

######  Remark 1.

If $C^{\prime}(\mathcal{D})\subset C(\mathcal{D})$ is consistent, and
$w^{\prime}:C^{\prime}(\mathcal{D})\to G$ is a weight function, then
$w^{\prime}$ can be extended to $w:C(\mathcal{D})\to G$ by defining

$w(c)=\begin{cases}w^{\prime}(c),&c\in C^{\prime}(\mathcal{D}),\\\
0,&\text{otherwise}.\end{cases}$

##  4\. I-functions

Let $C^{\prime}(\mathcal{D})$ and $C^{\prime\prime}(\mathcal{D})$ be
consistent subsets of $C(\mathcal{D})$. Suppose there are weight functions
$v:C^{\prime}(\mathcal{D})\to G_{1}\text{ and
}w:C^{\prime\prime}(\mathcal{D})\to G_{2}$ are such that $v\in
W^{odd}_{G_{1}}$ and $w\in W^{even}_{G_{2}}$. We can assume, that
$C^{\prime\prime}(\mathcal{D})$ is a subset of $C^{\prime}(\mathcal{D})$.
Otherwise we can replace $v$ by its extension on $C(\mathcal{D})$ as in Remark
1 and take $C^{\prime}(\mathcal{D})=C(\mathcal{D})$.

Let $\mathcal{D}(L)\subset\mathcal{D}$ be the set of all regular diagrams of
an ordered oriented virtual link $L$, and $w:C^{\prime}(\mathcal{D})\to G$ be
a weight function. For a diagram $D\in\mathcal{D}(L)$ denote by $R(w,D)$ the
set of weights $w(c)$, where $c$ is a classical crossing in $D$ that may be
reduced by a single RI-move. Then take a union over all diagrams of $L$:

$R(w,L)=\bigcup\limits_{D\in\mathcal{D}(L)}R(w,D).$

For some weight functions the set $R(w,L)$ can be easy described as in Example
3.

######  Example 3.

(i) Consider the weight function $w(c)=\operatorname{Ind}(c)$. If $c\in C(D)$
can be reduced by RI-move, then by Figure 4, the labelling around $c$ is such
that $a=b-1$, hence by (1), $\operatorname{Ind}(c)=0$. Therefore, for any
oriented virtual knot $K$ we get $R(\operatorname{Ind},K)=\\{0\\}$.
(ii) Consider the weight function $w(c)=\nabla J_{n}(D_{c})$. As above, if
$c\in C(D)$ can be reduced by RI-move, then $\operatorname{Ind}(c)=0$, whence
either $\nabla J_{n}(D_{c})=\nabla J_{n}(D)$ or $\nabla J_{n}(D_{c})=-\nabla
J_{n}(D)$. Therefore, for any oriented virtual knot $K$ we get
$R(w,K)=\\{\pm\nabla J_{n}(K)\\}$.

######  Definition 5.

Let $D$ be a diagram of an ordered oriented virtual link $L$ and in above
notations $g\in G_{2}$ be such that $g\not\in R(w,L)$ or $R(v,L)=\\{0\\}$.
Define $I$-function $I:C^{\prime}(D)\to G_{1}$ by

(4) $I(D;v,w,g)=\sum_{w(c)=g}v(c).$

Hereafter we will require that all our weight functions are local with respect
to crossing change operation. Then to every weight function
$w:C^{\prime}(\mathcal{D})\to G$ one can associate a weight function
$w^{*}:C^{\prime}(\mathcal{D})\to G$ induced by taking a mirror image, i.e.
$w^{*}(c)=w(c^{*})$.

######  Definition 6.

Let $D$ be a diagram of an ordered oriented virtual link $L$ and in above
notations $g\in G_{2}$ is such that $g\not\in R(w,L)$ or $R(v,L)=\\{0\\}$.
Define a flat $I$-function $I_{f}:C^{\prime}(D)\to G_{1}$ by

(5) $I_{f}(D;v,w,g)=\sum_{w(c)=g}v(c)+\sum_{w^{*}(c)=g}v^{*}(c).$

######  Theorem 1.

In above notations,

  1. (i)

$I(D;v,w,g)$ is an ordered oriented virtual link invariant,

  2. (ii)

$I_{f}(D;v,w,g)$ is an ordered oriented flat virtual link invariant.

###### Proof.

(i) Since $I(D;v,w,g)$ is a sum over classical crossings, it is invariant
under moves VRI, VRII and VRIII which involve virtual crossings only.
Moreover, it is invariant under RI-move since the assumption $g\not\in R(w,L)$
or $R(v,L)=\\{0\\}$ implies that any crossing involved in RI-move does not
participate in the sum. The invariance under RII-move follows from the
assumption $v\in W^{odd}$. Indeed, if two crossings $c_{1}$ and $c_{2}$ in the
sum are involved in RII-move, then $v(c_{1})=-v(c_{2})$. Invariance under
moves RIII and SV follows from assumptions (C2) and (C3) of Definition 1.

(ii) Note that $v\in W^{odd}_{G_{1}}$ implies $v^{*}\in W^{odd}_{G_{1}}$, and
$w\in W^{even}_{G_{2}}$ implies $w^{*}\in W^{even}_{G_{2}}$. Therefore, by
(i), $I(D;v^{*},w^{*},g)$ is also a virtual link invariant, and hence
$I_{f}(D;v,w,g)$ is a virtual link invariant being a sum of invariants.

To prove that $I_{f}(D;v,w,g)$ is an invariant of flat virtual links, we need
to show, that it is invariant under crossing change. Let $D$ be a diagram of
$L$ and $c_{0}\in C(D)$. Let $D^{\prime}$ be a diagram obtained from $D$ by a
crossing change in $c_{0}$, and $c_{0}^{\prime}$ be a crossing in $D^{\prime}$
corresponding to $c_{0}$. Then we can write

(6) $\displaystyle I_{f}(D;v,w,g)=\sum_{w(c)=g,\,c\neq
c_{0}}v(c)+\sum_{w^{*}(c)=g,\,c\neq c_{0}}v^{*}(c)+S,$

where

$S=\begin{cases}0,&\text{if}\quad w(c_{0})\neq g\text{ and }w^{*}(c_{0})\neq
g,\\\ v(c_{0}),&\text{if}\quad w(c_{0})=g\text{ and }w^{*}(c_{0})\neq g,\\\
v^{*}(c_{0}),&\text{if}\quad w(c_{0})\neq g\text{ and }w^{*}(c_{0})=g,\\\
v(c_{0})+v^{*}(c_{0}),&\text{if}\quad w(c_{0})=w^{*}(c_{0})=g.\end{cases}$

Analogously,

(7) $I_{f}(D^{\prime};v,w,g)=\sum_{w(c^{\prime})=g,\,c^{\prime}\neq
c_{0}^{\prime}}v(c^{\prime})+\sum_{w^{*}(c^{\prime})=g,\,c^{\prime}\neq
c_{0}^{\prime}}v^{*}(c^{\prime})+S^{\prime},$

where

$S^{\prime}=\begin{cases}0,&\text{if}\quad w(c_{0}^{\prime})\neq g\text{ and
}w^{*}(c_{0}^{\prime})\neq g,\\\ v(c_{0}^{\prime}),&\text{if}\quad
w(c_{0}^{\prime})=g\text{ and }w^{*}(c_{0}^{\prime})\neq g,\\\
v^{*}(c_{0}^{\prime}),&\text{if}\quad w(c_{0}^{\prime})\neq g\text{ and
}w^{*}(c_{0}^{\prime})=g,\\\
v(c_{0}^{\prime})+v^{*}(c_{0}^{\prime}),&\text{if}\quad
w(c_{0}^{\prime})=w^{*}(c_{0}^{\prime})=g.\end{cases}$

Since weight functions $v$ and $w$ are local by (C1) of Definition 1, then
after crossing change in $c_{0}$, in expressions (6) and (7) only $S$ and
$S^{\prime}$ may differ. Now let us change all crossings in $D^{\prime}$
except $c_{0}^{\prime}$. Then we obtain a mirror image $D^{*}$ of a diagram
$D$. Denote by $c^{*}_{0}$ the crossing in $D^{*}$ which corresponds to
$c_{0}$ Hence, $v(c_{0}^{\prime})=v(c_{0}^{*})$ and
$w(c_{0}^{\prime})=w(c_{0}^{*})$. Since by definition $v^{*}(c)=v(c^{*})$ and
$w^{*}(c)=w(c^{*})$, we conclude $v(c_{0}^{\prime})=v^{*}(c_{0})$ and
$w(c_{0}^{\prime})=w^{*}(c_{0})$. Similarly we acquire
$v^{*}(c_{0}^{\prime})=v^{*}(c_{0}^{*})=v(c_{0})$ and
$w^{*}(c_{0}^{\prime})=w^{*}(c_{0}^{*})=w(c_{0})$. Applying those equalities
to $S^{\prime}$ we get imply $S^{\prime}=S$. Therefore, (7) is equal to (6),
hence $I_{f}$ is a flat virtual link invariant. ∎

By Theorem 1 we can use notations $I(L;v,w,g)$ and $I_{f}(L;v,w,g)$ instead of
$I(D;v,w,g)$ and $I_{f}(D;v,w,g)$, where $D$ is a diagram of a virtual link
$L$.

######  Example 4.

Consider weight functions $v(c)={\operatorname{sgn}}(c)\in
W^{odd}_{\mathbb{Z}}$ and $w(c)={\operatorname{Ind}}(c)\in
W^{even}_{\mathbb{Z}}$. Then

(8)
$I(D;v,w,n)=\sum\limits_{w(c)=n}v(c)=\sum\limits_{{\operatorname{Ind}}(c)=n}{\operatorname{sgn}}(c)=J_{n}(D)$

is the defined above the $n$-th writhe number and

(9) $\displaystyle I_{f}(D;v,w,n)$ $\displaystyle=$
$\displaystyle\sum\limits_{w(c)=n}v(c)+\sum\limits_{w^{*}(c)=n}v^{*}(c)$
$\displaystyle=$ $\displaystyle
J_{n}(D)+\sum\limits_{{\operatorname{Ind}}(c)=-n}(-{\operatorname{sgn}}(c))$
$\displaystyle=$ $\displaystyle J_{k}(D)-J_{-k}(D)$ $\displaystyle=$
$\displaystyle\nabla J_{k}(D),$

that is the $n$-th difference writhe number defined by (2).

##  5\. Smoothings in classical crossings and invariants

Above we used the type-1 smoothing, presented in Figure 5, to construct
$F$-polynomials. Applying the type-1 smoothing to a classical crossing $c\in
D$, which belongs to a single connected component, we obtain a link diagram
with one less classical crossing and the same number of components.

Let us consider another types of smoothings of virtual link diagrams in
classical crossings as below. The _type-2 smoothing_ is presented in Figure 9.
Consider a diagram of an $n$-component ordered oriented virtual link and
assume that in classical crossing $c$ two meeting arcs belong to the same say,
$i$-th, component. After the presented smoothing we will obtain a diagram of
an ordered $(n+1)$-component link. The smoothing induces the order change: one
arc will preserve orientation and the corresponding component will be the
$i$-th, but another arc will get reverse orientation and the corresponding
component will be the $(n+1)$-th.

Figure 9. Type-2 smoothing.

.

The _type-3 smoothing_ is presented in Figure 10. Consider a diagram of
2-component ordered oriented virtual link, and assume that crossing $c$ belong
to both components. Then after smoothing we will get a diagram of knot. Note
that type-3 smoothing can be generalized to $n$-component ordered oriented
virtual links, and we obtain an $(n-1)$-component link as a result.

Figure 10. Type-3 smoothing.

Further for a classical crossing $c\in D$ we denote by $D_{c}$, $D^{c}$ and
$\prescript{c}{}{D}$ a diagram obtained from $D$ by smoothing of type-1,
type-2 and type-3, respectively.

Let us denote by ${\mathcal{VL}_{\text{flat}}}$ a free $\mathbb{Z}$-module
generated by ordered oriented flat virtual links. For a virtual link diagram
$D$ denote by $[D]$ a flat virtual link whose diagram is obtained from $D$ by
replacing all classical crossings by flat crossings. Then
$[D]\in{\mathcal{VL}_{\text{flat}}}$.

######  Theorem 2.

Functions $S_{i}:C(\mathcal{D})\to{\mathcal{VL}_{\text{flat}}}$, for
$i=1,2,3$, defined by

$S_{1}(c)=[D_{c}],\qquad S_{2}(c)=[D^{c}],\qquad\text{and}\quad
S_{3}(c)=[\prescript{c}{}{D}],$

are even weight functions. Moreover, if crossing $c\in D$ can be reduced by
RI-move, then

  * •

$[D_{c}]$ is equivalent to either $[D]$ or $[D^{\prime}]$, where $D^{\prime}$
is obtained from $D$ by reversing orientation on the component, containing
$c$;

  * •

$[D^{c}]$ is equivalent to $[D]$ with one unknot added, where ordering and
orientation of components are induced by type-2 smoothing;

  * •

type-3 smoothing can not be applied at crossing $c$.

###### Proof.

The proof is a straightforward check of Reidemeister moves with all possible
orientations. For $S_{1}$ it was done in [13, Theorem 3.3]. Now we give a
proof for $S_{2}$.

RI-move: Let $D$ be a diagram of an ordered oriented $n$-component virtual
link. Consider crossing $c\in D$ that can be reduced by RI-move on $i$-th
component, see Figure 11. After type-2 smoothing we get a new component which
is an unknot. If after smoothing the $i$-th component remains $i$-th, then its
orientation is preserved. If after smoothing the $i$-th component becomes
$(n+1)$-th component, then its orientation is reversed.

Figure 11. All possible RI-moves and corresponding smoothings.

RII-move: Let $D$ be a diagram of an ordered oriented $n$-component virtual
link. Consider crossings $c_{1},c_{2}\in D$ that belong to $i$-th component
and can be reduced by RII-move. Depending on orientation, there are two cases,
presented in Figure 12 and 13. For each case we have two possibilities of
type-2 smoothing: since $i$-th component splits into two components, one of
them will be either $i$-th or $(n+1)$-th, and, respectively, another will be
$(n+1)$-th or $i$-th. In the case, presented in Figure 12, two diagrams,
obtained by smoothings, are equivalent under one crossing change, so they are
equivalent as flat diagrams. In the case, presented in Figure 13, two
diagrams, obtained by smoothings, are equivalents under RI-moves, so they are
equivalent.

Figure 12. 1-st case of RII-move and corresponding smoothings. Figure 13. 2-nd
case of RII-move and corresponding smoothings.

RIII-move: Consider RIII-move presented in Figure 14 with components numerated
by $i$, $j$ and $k$, where some of these numbers may coincide.

Figure 14. RIII-move.

Let us apply type-2 smoothings in three crossings of the initial (before RIII-
move) diagram and in three crossings of the terminal (after RIII-move)
diagram. Thus, we get three smoothed diagrams of virtual links on the left
side in Figure 15 and three smoothed diagrams of virtual links on the right
side in the same figure. It is clear from Figure 15 that smoothed diagrams for
corresponding crossing are flat equivalent (for crossing of components $i$ and
$k$ ) or equivalent (for crossing of components $i$ and $j$ and crossing of
components $j$ and $k$).

Figure 15. Smoothings before and after RIII move.

SV-move: Consider SV-move presented in Figure 16, were link components are
numerated by $i$, $j$, and $k$, where some of these numbers may coincide.

Figure 16. SV-move.

Let us apply type-3 smoothing in a crossing of the initial (before SV-move)
diagram and a crossing of the terminal (after SV-move) diagram, see Figure 17.
It is clear that smoothed diagrams are equivalent.

Figure 17. Smoothing before and after SV-move.

Thus, we have shown that $S_{2}$ satisfies weight function conditions.

For $S_{3}$ the proof follows by analogous considerations. ∎

Since by Theorem 2 function $S_{2}$ is an even weight function, according to
Theorem 1 we can construct the following invariants.

######  Corollary 1.

Consider function $B^{i}:D\to{\mathcal{VL}_{\text{flat}}}$, defined by

$B^{i}=\sum\limits_{c\in C_{i}(D)}{\operatorname{sgn}}(c)[K^{c}],$

and flat virtual link invariant
$B^{i}_{\text{flat}}:D\to{\mathcal{VL}_{\text{flat}}}$, defined by

$B^{i}_{\text{flat}}=\sum\limits_{c\in
C_{i}(D)}{\operatorname{sgn}}(c)([K^{c}]-[K^{c^{*}}]),$

where $c\in C_{i}(D)$ means that in the crossing $c$ both arcs belong to the
$i$-th component of a link. Then $B^{i}$ is an ordered oriented virtual link
invariant and $B^{i}_{\text{flat}}$ is an ordered oriented flat virtual link
invariant.

Invariants $B^{i}$ and $B^{i}_{\text{flat}}$ appear to be useful for studying
connected sums of virtual knots. We will demonstrate in Example 5 that these
invariants can be used to prove that a Kishino knot, the famous connected sum
of two trivial virtual knots [4, p. 23], is non-trivial.

######  Example 5.

Let $K$ be an oriented virtual Kishino knot presented in Figure 18. We will
show that $B^{1}(K)\neq 0$, hence $K$ is distinguished from the unknot by
$B^{1}$.

Figure 18. Kishino knot.

Denote the classical crossings by $a$,$b$,$c$, and $d$ as shown in Figure 18.
To calculate $B^{1}(K)$ we find signs of these crossings

${\operatorname{sgn}}(a)={\operatorname{sgn}}(d)=-1\quad
and\quad{\operatorname{sgn}}(b)={\operatorname{sgn}}(c)=1.$

Since $K$ is a knot, according to the definition of $B^{1}$, we consider
smoothings in all crossings:

$B^{1}(K)=-[K^{a}]+[K^{b}]+[K^{c}]-[K^{d}].$

Each smoothing provide an ordered oriented 2-component virtual link. One can
check, that

$[K^{a}]=[K^{d}]\quad and\quad[K^{b}]=[K^{c}],$

so we only need to prove, that $[K^{a}]$ and $[K^{b}]$, presented in Figure
19, are distinct.

Figure 19. Diagrams of $K_{1}=K^{a}$ and $K_{2}=K^{b}$.

To simplify notations we change notations, as $K_{1}=K^{a}$ and $K_{2}=K^{b}$.
It is easy to see that only crossings $c$ and $d$ corresponds to $2$-nd
component of $K_{1}$. Then

$B^{2}_{\text{flat}}(K_{1})=[K_{1}^{c}]-[K_{1}^{c^{*}}]-[K_{1}^{d}]+[K_{1}^{d^{*}}]\quad
and\quad B^{2}_{\text{flat}}(K_{2})=0,$

where $c^{*}$ and $d^{*}$ are crossing changes of $c$ and $d$, respectively.
Smoothed diagrams of $K_{1}$ are presented in Figure 20, and smoothed
diagrams, corresponding to $c^{*}$ and $d^{*}$, are presented in Figure 21.

Figure 20. Diagrams of $K_{1}^{c}$ and $K_{1}^{d}$. Figure 21. Diagrams of
$K_{1}^{c^{*}}$ and $K_{1}^{d^{*}}$.

In $B^{2}_{\text{flat}}(K_{1})$ there are two summands with ”+” and two with
”-”. Note that in $K_{1}^{c}$ and in $K_{1}^{d^{*}}$ the $3$-rd component is
nontrivially linked with two other components, but for $K_{1}^{c^{*}}$ and
$K_{1}^{d}$ it is not true. Hence, there are no cancelations and
$B^{2}_{\text{flat}}(K_{1})\neq 0$. Therefore, Kishino knot is not equivalent
to the unknot.

##  6\. Recurrent construction of a sequence of invariants

In this section we will describe a recurrent construction of invariants based
on Theorem 1 and using recurrently defined odd and even weight functions.
Below we consider the case of ordered links, and analogous statements holds
for the case of unordered links.

As well as above, ${\mathcal{VL}_{\text{flat}}}$ denotes a free
$\mathbb{Z}$-module generated by ordered oriented flat virtual links. Any
invariant $A_{1}$ of ordered oriented flat virtual links taking values in a
group $G$ may be extended to a homomorphism
$A_{1}:{\mathcal{VL}_{\text{flat}}}\to G$. Similarly, any even weight function
$F_{1}\in W^{even}_{{\mathcal{VL}_{\text{flat}}}}$ defines a weight function
$A_{1}\circ F_{1}\in W^{even}_{G}$.

Suppose that there is given an even weight function $w_{1}\in
W^{even}_{H_{1}}$ for some abelian group $H_{1}$. Consider an odd weight
function $u_{1}$ such that $u_{1}\in W^{odd}_{Z}$ if $G$ is a group, and
$u_{1}\in W^{odd}_{G}$ if $G$ is a ring. Then according to Definitions 5 and 6
we can define $I$-function (see formula (4)) and flat $I_{f}$-function (see
formula (5)) by using even weight $w_{1}$ and odd weight
$v_{1}=u_{1}*(A_{1}\circ F_{1})$, where the product $*$ of weights is defined
by formula (3):

$I(D;v_{1},w_{1},h_{1})=\sum_{w_{1}(c)=h_{1}}v_{1}(c)$

and

$I_{f}(D;v_{1},w_{1},h_{1})=\sum_{w_{1}(c)=h_{1}}v_{1}(c)+\sum_{w_{1}(c^{*})=h_{1}}v_{1}(c^{*}),$

for $h_{1}\in H_{1}$ such that $h_{1}\not\in R(w_{1},L)$ or
$R(v_{1},L)=\\{0\\}$, where $D$ is a diagram of an ordered oriented virtual
link $L$.

The procedure can be repeated by taking odd weights $u_{i}\in W^{odd}_{G}$,
even weights $F_{i}\in W^{even}_{{\mathcal{VL}_{\text{flat}}}}$, and even
weights $w_{i}\in W^{even}_{H_{i}}$ for abelian groups $H_{i}$, where
$i=2,3,\ldots$. By Theorem 1, we get a family of ordered oriented flat virtual
link invariants $A_{i}:{\mathcal{VL}_{\text{flat}}}\to G$ defined by

$A_{i}(D)=I_{f}(D;v_{i-1},w_{i-1},h_{i-1}),\qquad\textrm{for }\quad i>1,$

where $v_{i-1}=u_{i-1}*(A_{i-1}\circ F_{i-1})$. Continuing the process, for
$i>1$ we will get I-functions

$I(D;v_{i},w_{i},h_{i})=\sum_{w_{i}(c)=h_{i}}v_{i}(c)$

and flat I-functions

$I_{f}(D;v_{i},w_{i},h_{i})=\sum_{w_{i}(c)=h_{i}}v_{i}(c)+\sum_{w_{i}(c^{*})=h_{i}}v_{i}(c^{*}).$

Note, that in general, $h_{i}$ cannot be arbitrary elements of $H_{i}$, since

$I_{f}(D;u_{i}*(A_{i}\circ F_{i}),w_{i},h_{i})$

is guaranteed to be an invariant only for $h_{i}\notin R(w_{i},D)$. Finding
proper elements $h_{i}$ to make $A_{i+1}\circ F_{i+1}$ a well-defined weight
function seems a difficult problem. To avoid this problem we require
$R(u_{i},D)=\\{0\\}$ for all $D$. Then we can take $h_{i}$ equals to any
element of the group $H_{i}$ and we obtain the following proposition.

######  Proposition 1.

Assume that there is an ordered oriented flat virtual link invariant $A_{1}$
taking values in a group $G$ and a sequence
$\\{(F_{i},w_{i},h_{i},u_{i})\\}_{i\in\mathbb{N}}$ such that $F_{i}\in
W^{even}_{{\mathcal{VL}_{\text{flat}}}}$, $w_{i}\in W^{even}_{H_{i}}$,
$h_{i}\in H_{i}$, and $u_{i}\in W^{odd}_{\mathbb{Z}}$ (or, $u_{i}\in
W^{odd}_{\mathbb{G}}$ if $G$ is a ring) such that $R(u_{i},D)=\\{0\\}$ for all
link diagrams. Then there are even weight functions $A_{i}\circ F_{i}$ and
corresponding sequences of invariants

$I(D;u_{i}*(A_{i}\circ F_{i}),w_{i},h_{i})\quad\text{and}\quad
A_{i+1}=I_{f}(D;u_{i}*(A_{i}\circ F_{i}),w_{i},h_{i}).$

The sequence we obtain is useful when
$(F_{i},w_{i},h_{i},u_{i})=(F_{1},w_{1},h_{1},u_{1})$ for all $i$.

######  Corollary 2.

Given three weight functions $v\in W^{odd}_{H}$, $w\in W^{even}_{G}$ and $F\in
W^{even}_{{\mathcal{VL}_{\text{flat}}}}$ such that $R(w,L)=\\{0\\}$ for all
links $L$ and a sequence $\\{g_{i}\in G\\}_{i\in\mathbb{N}}$ there is an
infinite sequence of weight functions $\\{v_{i}\\}_{i\in\mathbb{N}}$,
generated by them.

###### Proof.

Let $A_{1}(D)=I_{f}(D,v,w,g)$. By taking $F_{i}=F$, $w_{i}=w$,
$u_{i}={\operatorname{sgn}}*w$ and applying Proposition 1 we obtain the
desired sequence. ∎

######  Remark 2.

On each step we may substitute ${\mathcal{VL}_{\text{flat}}}$ by a module
generated by a regular set of flat links $\mathcal{D}_{\text{flat}}$ and
consider weight functions and invariants that are defined for those links.
Proposition 1 and Corollary 2 will remain true with some obvious changes.

Further we will restrict ourselves to the case where $F_{i}$ are smoothings
and provide some examples, generalizing already known invariants. Before we
continue with examples let us define a polynomial invariants, associated to
those sequences. For simplicity suppose that $G=H_{i}=\mathbb{Z}$.

######  Theorem 3.

Let $S=\\{s_{1},\dots,s_{k}\\}$ be a finite set of weight functions where
$s_{i}=A_{m_{i}}\circ F_{m_{i}}\,$ for some $m_{i}\in\mathbb{N}$, $w\in
W^{odd}_{\mathbb{Z}}$ and $v\in W^{even}_{\mathbb{Z}}$ such that
$R(v,D)=\\{0\\}$ for all $D$. Then

$\begin{gathered}F(t,\ell_{1},\ldots,\ell_{k})=\sum_{c\in
C(D)}w(c)t^{v(c)}\ell_{1}^{s_{1}(c)}\cdots\ell_{k}^{s_{k}(c)}-\sum_{c\in
T(D)}w(c)\ell_{1}^{s_{1}(c)}\cdots\ell_{k}^{s_{k}(c)}\\\ -\sum_{c\not\in
T(D)}w(c)\ell_{1}^{A_{m_{1}}(D)}\cdots\ell_{k}^{A_{m_{k}}(D)}\end{gathered}$

is a link invariant, where $T(D)=\\{c\in C(D)\ |\ s_{i}(c)\in R(s_{i},L)\text{
for all }i\\}$.

###### Proof.

The proof is analogous to the proof of invariance of F-polynomials in [13,
Theorem 3.3]. ∎

######  Example 6.

Let $v={\operatorname{Ind}}$, $w={\operatorname{sgn}}$, $A_{1}=\nabla J_{n}$
and $s_{1}=\nabla J_{n}(D_{c})=A_{1}\circ F$, where F is a type-1 smoothing
presented in Figure 5. Then we get F-polynomials from [13]:

$F_{D}^{n}(t,\ell)=\sum_{c\in C(D)}w(c)t^{v(c)}\ell^{s_{1}(c)}-\sum_{c\in
T(D)}w(c)\ell^{s_{1}(c)}-\sum_{c\notin T(D)}w(c)\ell^{A_{1}(D)},$

where $T(D)$ is a set of crossings of $D$ having the following property:

$T_{n}(D)=\\{c\in D\,\mid\,\nabla J_{n}(D_{c})=\pm\nabla J_{n}(D)\\}.$

##  7\. (n,m)-difference writhe and invariants of virtual knots

Now we shall continue the recursive procedure and define new invariants
generalizing $F$-polynomials. Take $F_{1}$ to be a type-1 smooting, and let
$F_{i}=F_{1}$ for $i\geq 2$. Recall that ${\operatorname{sgn}}\in
W^{odd}_{\mathbb{Z}}$ and ${\operatorname{Ind}}\in W^{even}_{\mathbb{Z}}$, and
similar to Example 4 (formular (9)) define
$A_{1}=I_{f}(D;{\operatorname{sgn}},{\operatorname{Ind}},n)=\nabla J_{n}$.
Denoting $u_{i}={\operatorname{sgn}}*{\operatorname{Ind}}$ for $i\geq 1$ we
define

$\displaystyle\nabla J_{n,m}(D)$ $\displaystyle=$ $\displaystyle
A_{2}(D)=I_{f}(D;u_{2}*(A_{1}\circ F_{1}),w_{1},m)=$ $\displaystyle=$
$\displaystyle\sum_{{\operatorname{Ind}}(c)=m}{\operatorname{sgn}}(c){\operatorname{Ind}}(c){\nabla
J}_{n}(D_{c})-\sum_{{\operatorname{Ind}}(c)=-m}{\operatorname{sgn}}(c){\operatorname{Ind}}(c){\nabla
J}_{n}(D_{c})$ $\displaystyle=$ $\displaystyle
m\sum_{{\operatorname{Ind}}(c)=m}{\operatorname{sgn}}(c){\nabla
J}_{n}(D_{c}).$

By Theorem 1, $\nabla J_{n,m}(D)$ is an oriented flat virtual knot invariant,
and we call it the _(n,m)-dwrithe_ (difference writhe) of an oriented virtual
knot $K$ whose diagram is $D$.

######  Theorem 4.

(i) The polynomial

$\displaystyle F^{n,m,k}_{D}(t,\ell_{1},\ell_{2})$ $\displaystyle=$
$\displaystyle\sum_{c\in
C(D)}{\operatorname{sgn}}(c)t^{{\operatorname{Ind}}(c)}\ell_{1}^{\nabla
J_{n}(D_{c})}\ell_{2}^{\nabla J_{m,k}(D_{c})}$ $\displaystyle-\sum_{c\in
T(D)}{\operatorname{sgn}}(c)\ell_{1}^{\nabla J_{n}(D_{c})}\ell_{2}^{\nabla
J_{m,k}(D_{c})}-\sum_{c\not\in T(D)}{\operatorname{sgn}}(c)\ell_{1}^{\nabla
J_{n}(D)}\ell_{2}^{\nabla J_{m,k}(D)},$

where

$T(D)=\\{c\in C(D)\ |A_{2}\circ F_{2}\in R(A_{2}\circ F_{2},L)\text{ and }\
A_{3}\circ F_{3}\in R(A_{3}\circ F_{3},L)\\},$

is an oriented virtual knot invariant.
(ii) The family of invariants $F_{D}^{n,m,k}(t,\ell_{1},\ell_{2})$ is stronger
than invariants $F^{n}_{D}(t,\ell)$.

###### Proof.

(i) By letting $S=\\{A_{2}\circ F_{2},A_{3}\circ F_{3}\\}$,
$w={\operatorname{sgn}}\in W^{odd}_{\mathbb{Z}}$ and
$v={\operatorname{Ind}}\in W^{even}_{\mathbb{Z}}$ in Theorem 3 we obtain the
desired polynomial invariant.

(ii) It follows from the definition that $F_{D}^{n,m,k}$ coincide with
$F^{n}_{D}$ for large enough $k$. Thus it suffices to find a knot which cannot
be distinguished from the unknot by $F$–polynomials, but can be distinguished
by $F_{D}^{n,m,k}$–polynomials.

Firstly, let us consider an oriented virtual knot $K=4.31$ from the Green’s
table (see also [8]) as presented in Figure 22. The diagram has four classical
crossings: $\alpha$, $\beta$, $\gamma$, and $\delta$. Denote by $K_{\alpha}$
and $K_{\beta}$ oriented virtual knots obtained by type-1 smoothing at
classical crossings $\alpha$ and $\beta$, respectively. Figure 22 presents
labelings of arcs of diagrams of $K$, $K_{\alpha}$ and $K_{\beta}$ satisfying
the rule from Figure 4.

Figure 22. Diagrams of knots $K$, $K_{\alpha}$ and $K_{\beta}$.

Values of sign and index for classical crossings, and of $n$-dwrithe for knots
$K$, $K_{\alpha}$ and $K_{\beta}$ are presented in Table 1.

Table 1. Values for $K$, $K_{\alpha}$ and $K_{\beta}$.  $K$ | $K_{\alpha}$ | $K_{\beta}$
---|---|---
${\operatorname{sgn}}(\alpha)=-1$ | ${\operatorname{sgn}}(\beta)=1$ | ${\operatorname{sgn}}(\alpha)=-1$
${\operatorname{sgn}}(\beta)=1$ | ${\operatorname{sgn}}(\gamma)=-1$ | ${\operatorname{sgn}}(\gamma)=1$
${\operatorname{sgn}}(\gamma)=-1$ | ${\operatorname{sgn}}(\delta)=1$ | ${\operatorname{sgn}}(\delta)=-1$
${\operatorname{sgn}}(\delta)=-1$ |  |
${\operatorname{Ind}}(\alpha)=1$ | ${\operatorname{Ind}}(\beta)=1$ | ${\operatorname{Ind}}(\alpha)=1$
${\operatorname{Ind}}(\beta)=1$ | ${\operatorname{Ind}}(\gamma)=2$ | ${\operatorname{Ind}}(\gamma)=-1$
${\operatorname{Ind}}(\gamma)=0$ | ${\operatorname{Ind}}(\delta)=1$ | ${\operatorname{Ind}}(\delta)=-2$
${\operatorname{Ind}}(\delta)=0$ |  |
${\nabla J}_{1}(K)=0$ | ${\nabla J}_{1}(K_{\alpha})=2$ | ${\nabla J}_{1}(K_{\beta})=-2$
${\nabla J}_{2}(K)=0$ | ${\nabla J}_{2}(K_{\alpha})=-1$ | ${\nabla J}_{2}(K_{\beta})=1$

To calculate the (1,1)-dwrithe of $K$ we observe that the index equals to $1$
for only two classical crossings: ${\operatorname{Ind}}(\alpha)=1$ and
${\operatorname{Ind}}(\beta)=1$. Therefore,

$\displaystyle{\nabla J}_{1,1}(K)$ $\displaystyle=$ $\displaystyle
1\cdot\sum_{{\operatorname{Ind}}(c)=1}{\operatorname{sgn}}(c)\nabla
J_{1}(D_{c})$ $\displaystyle=$
$\displaystyle{\operatorname{sgn}}(\alpha){\nabla
J}_{1}(K_{\alpha})+{\operatorname{sgn}}(\beta){\nabla
J}_{1}(K_{\beta})=(-1)\cdot 2+1\cdot(-2)=-4.$

Now we consider an oriented virtual knot $K^{\prime}$ with diagram presented
in Figure 23.

Figure 23. Diagram of $K^{\prime}$.

Virtual knot $K^{\prime}$ has five classical crossings: $\alpha$, $\beta$,
$\gamma$, $\delta_{1}$ and $\delta_{2}$. By direct calculations we get

$\begin{gathered}{\operatorname{sgn}}(\beta)={\operatorname{sgn}}(\delta_{1})={\operatorname{sgn}}(\delta_{2})=-1,\qquad{\operatorname{sgn}}(\alpha)={\operatorname{sgn}}(\gamma)=1,\\\
{\operatorname{Ind}}(\gamma)={\operatorname{Ind}}(\delta_{1})={\operatorname{Ind}}(\delta_{2})=0,\qquad{\operatorname{Ind}}(\alpha)={\operatorname{Ind}}(\beta)=1.\end{gathered}$

Therefore, ${\nabla J}_{1}(K^{\prime})={\nabla J}_{2}(K^{\prime})=0$.

It is easy to see, that $K^{\prime}_{\delta_{1}}$ is flat equivalent to the
above considered $K=4.31$ and $K^{\prime}_{\delta_{2}}$ is flat equivalent to
$K^{-}$. Therefore, ${\nabla J}_{n}(K^{\prime}_{\delta_{1}})={\nabla
J}_{2}(K^{\prime}_{\delta_{1}})=0$ for $n=1,2$. Moreover, all other
smoothings, $K^{\prime}_{\alpha}$, $K^{\prime}_{\beta}$ and
$K^{\prime}_{\gamma}$, are flat equivalent to the unknot. Hence, we have
${\nabla J}_{1}(K^{\prime}_{c})={\nabla J}_{2}(K^{\prime}_{c})=0$ for all
classical crossings $c$ in the diagram. Since ${\nabla
J}_{1}(K^{\prime})={\nabla J}_{2}(K^{\prime})=0$, we get sets
$T_{1}(K^{\prime})=T_{2}(K^{\prime})=\\{\alpha,\beta,\gamma,\delta_{1},\delta_{2}\\}$.
Therefore,

$\displaystyle F_{K^{\prime}}^{n}(t,\ell)$ $\displaystyle=$
$\displaystyle\sum_{c\in
C(D)}\operatorname{sgn}(c)\left(t^{\text{Ind}(c)}-1\right)\ell^{\nabla
J_{n}(K_{c})}$ $\displaystyle=$ $\displaystyle
P_{K^{\prime}}(t)={\operatorname{sgn}}(\alpha)(t-1)+{\operatorname{sgn}}(\beta)(t-1)=0$

for any $n\in\mathbb{N}$. Thus, polynomials $F^{n}$ cannot distinguish
$K^{\prime}$ from the unknot.

At the same time, we have ${\nabla J}_{1,1}(K^{\prime})=0$ by straightforward
calculations, and ${\nabla J}_{1,1}(K^{\prime}_{\delta_{1}})=-4$ and ${\nabla
J}_{1,1}(K^{\prime}_{\delta_{2}})=4$ since $K^{\prime}_{\delta_{1}}$ is flat
equivalent to $K$ and $K^{\prime}_{\delta_{2}}$ is flat equivalent to $K^{-}$.
Therefore, $T_{1,1}(D)=\\{\alpha,\beta,\gamma\\}$ and

$\displaystyle F^{1,1,1}_{K^{\prime}}(t,\ell_{1},\ell_{2})$ $\displaystyle=$
$\displaystyle\sum_{c\in\\{\alpha,\beta,\gamma\\}}{\operatorname{sgn}}(c)\left(t^{{\operatorname{Ind}}(c)}-1\right)\ell_{2}^{{\nabla
J}_{1,1}(K^{\prime}_{c})}$
$\displaystyle\qquad+\sum_{c\in\\{\delta_{1},\delta_{2}\\}}{\operatorname{sgn}}(c)\left(t^{{\operatorname{Ind}}(c)}\ell_{2}^{{\nabla
J}_{1,1}(K^{\prime}_{c})}-\ell_{2}^{{\nabla J}_{1,1}(K^{\prime})}\right)$
$\displaystyle=$
$\displaystyle(-1)(t^{0}\ell_{2}^{-4}-1)+(-1)(t^{0}\ell_{2}^{4}-1)=-\ell_{2}^{-4}-\ell_{2}^{4}+2.$

Thus, the polynomial $F^{1,1,1}$ does distinguish virtual knot $K^{\prime}$
from the unknot. ∎

##  8\. Flat span and invariants of 2-component virtual links

Let $L=K_{1}\cup K_{2}$ be an ordered oriented virtual 2-component link, where
$K_{1}$ is the first component, and $K_{2}$ is the second component. Suppose
that $L$ is presented by its diagram. Denote by $O(L)$ the set of crossings
where $K_{1}$ passes over $K_{2}$. Define the _over linking number_ $O_{\ell
k}$ for $L$ as follows:

$O_{\ell k}(L)=\sum_{c\in O(L)}\operatorname{sgn}(c).$

Analogously, denote by $U(L)$ the set of crossings where $K_{1}$ passes under
$K_{2}$ and define the _under linking number_ $U_{\ell k}$ for $L$ as follows:

$U_{\ell k}(L)=\sum_{c\in U(L)}\operatorname{sgn}(c).$

In virtual links the two linking numbers $O_{\ell k}(L)$ and $U_{\ell k}(L)$
may be not equal, as can be seen for the ordered oriented virtual Hopf link
$\mathcal{H}$ shown in Figure 24. It is easy to see that $O_{\ell
k}(\mathcal{H})=-1$ and $U_{\ell k}(\mathcal{H})=0$. Also note that with
reversing order of components in $\mathcal{H}$, the two linking numbers will
exchange.

Figure 24. Virtual Hopf link $\mathcal{H}$.

######  Definition 7.

For an ordered oriented virtual 2-component link $L$ define its _span_ by

$\operatorname{span}(L)=O_{\ell k}(L)-U_{\ell k}(L).$

######  Lemma 1.

$\operatorname{span}(L)$ is an invariant of ordered oriented 2-component
virtual link $L$.

###### Proof.

Since $O_{\ell k}(L)$ and $U_{\ell k}(L)$ are invariants,
$\operatorname{span}(L)$ is an invariant as their difference. ∎

######  Lemma 2.

Let $L$ be an ordered oriented virtual link, and $L^{\prime}$ is obtained from
$L$ by reversing orientation on both component and preserving the order of its
components. Then $\operatorname{span}(L^{\prime})=\operatorname{span}(L)$.

###### Proof.

Changing orientation on both components does not change the signs of classical
crossings, hence $\operatorname{span}(L)$ does not change. ∎

######  Definition 8.

Let $D$ be a diagram of an ordered oriented virtual 2-component link
$L=K_{1}\cup K_{2}$, and $C_{12}(D)$ be the set of all classical crossings in
$D$ in which $K_{1}$ and $K_{2}$ meet. For $c\in C_{12}(D)$ let
$\prescript{c}{}{D}$ be a knot diagram obtained by type-3 smoothing at $c\in
D$ (see Figure 10). For $n,k\in\mathbb{Z}$ consider set

$I_{n,k}=\\{c\in D:\nabla J_{n}(\prescript{c}{}{D})=k\\}.$

Define _(n,k)-span_ for $L$ as follows:

$\operatorname{span}_{n,k}(L)=\sum\limits_{c\in O(L)\cap
I_{n,k}}\operatorname{sgn}(c)-\sum\limits_{c\in U(L)\cap
I_{n,k}}\operatorname{sgn}(c).$

######  Definition 9.

For a 2-component link $L$ define _flat span_ of $L$ as follows:

$\operatorname{fspan}_{n,k}(L)=\operatorname{span}_{n,k}(L)+\operatorname{span}_{n,-k}(L).$

######  Theorem 5.

Let $D$ be a diagram of an ordered oriented virtual 2-component link. Then
$\operatorname{span}_{n,k}(D)$ is a link invariant and
$\operatorname{fspan}_{n,k}(D)$ is a flat link invariant.

###### Proof.

Let us consider two weight functions

$w_{1}(c)=\begin{cases}{\operatorname{sgn}}(c),&c\in O(D),\\\
-{\operatorname{sgn}}(c),&c\in U(D),\\\ 0,&\text{otherwise,}\end{cases}$

and

$w_{2,n}(c)=\begin{cases}\nabla J_{n}(\prescript{c}{}{D}),&c\in O(D),\\\
0,&\text{otherwise.}\end{cases}$

Then $w_{1}\in W^{odd}_{\mathbb{Z}}$, $w_{2,n}\in W^{even}_{\mathbb{Z}}$, and
by Theorem 1 (formular (8)) the function

$\displaystyle I(D;w_{1},w_{2,n},k)$ $\displaystyle=$
$\displaystyle\sum_{w_{2,n}(c)=k}w_{1}(c)$ $\displaystyle=$
$\displaystyle\sum_{c\in O(D)\cap I_{n,k}}{\operatorname{sgn}}(c)+\sum_{c\in
U(D)\cap I_{n,k}}(-{\operatorname{sgn}}(c))$ $\displaystyle=$
$\displaystyle\operatorname{span}_{n,k}(D),$

where $I_{n,k}=\\{c\in D:w_{2,n}(c)=\nabla J_{n}(\prescript{c}{}{D})=k\\}$ is
an ordered oriented virtual link invariant.

Note, that $w_{2,n}^{*}(c)=w_{2,n}(c^{*})=-w_{2,n}(c)$ since crossing change
leads to change of orientation induced after smoothing, and
$w_{1}^{*}(c)=w_{1}(c^{*})=w_{1}(c)$ since
${\operatorname{sgn}}(c)=-{\operatorname{sgn}}(c^{*})$. Moreover, $c\in O(D)$
if and only if $c^{*}\in U(D)$. Therefore, the value

$\displaystyle I_{f}(K;w_{1},w_{2,n},k)$ $\displaystyle=$
$\displaystyle\sum_{w_{2,n}(c)=k}w_{1}(c)+\sum_{w_{2,n}^{*}(c)=k}w_{1}^{*}(c)$
$\displaystyle=$
$\displaystyle\sum_{w_{2,n}(c)=k}w_{1}(c)+\sum_{w_{2,n}(c)=-k}w_{1}(c)$
$\displaystyle=$
$\displaystyle\operatorname{span}_{n,k}(D)+\operatorname{span}_{n,-k}(D)$
$\displaystyle=$ $\displaystyle\operatorname{fspan}_{n,k}(L),$

is an ordered oriented flat virtual link invariant by Theorem 1. ∎

######  Lemma 3.

Let $L=K_{1}\cup K_{2}$ be a 2-component link, and $L^{\prime}=K_{2}\cup
K_{1}$ be obtained from $L$ by exchanging the order of its components. Then
$\operatorname{fspan}_{n,k}(L^{\prime})=-\operatorname{fspan}_{n,k}(L)$.

The following theorem gives a 3-parameter family of 3-variable polynomials
which are invariants of oriented virtual knots.

######  Theorem 6.

(i) The polynomial

$\displaystyle\widetilde{F}_{K}^{n,k,m}(t,\ell,v)$ $\displaystyle=$
$\displaystyle\sum_{c\in
C(D)}\operatorname{sgn}(c)t^{{\operatorname{Ind}}(c)}\ell^{\nabla
J_{n}\left(D_{c}\right)}v^{\operatorname{fspan}_{k,m}\left(D^{c}\right)}$
$\displaystyle-\sum_{c\in T(D)}\operatorname{sgn}(c)\ell^{\nabla
J_{n}\left(D_{c}\right)}v^{\operatorname{fspan}_{k,m}\left(D^{c}\right)}-\sum_{c\notin
T(D)}\operatorname{sgn}(c)\ell^{\nabla
J_{n}(D)}v^{\operatorname{fspan}_{k,m}\left(D^{c}\right)},$

where

$T(D)=\\{c\in C(D)\ |\ \nabla J_{n}(D_{c})=\pm\nabla J_{n}(D)\text{ and
}\operatorname{fspan}_{k,m}(D^{c})=0\\}$

is an oriented virtual knot invariant.
(ii) The family of polynomial invariants $\widetilde{F}_{D}^{n,k,m}(t,\ell,v)$
is stronger than polynomial invariants $F^{n}_{D}(t,\ell)$.

###### Proof.

(i) Consider weight functions $s_{1}=\nabla J_{n}\circ F_{1}$ and
$s_{2}=\operatorname{fspan}_{k,m}\circ F_{2}$, where $F_{1}$ is a type-1
smoothing and $F_{2}$ is a type-2 smoothing. Suppose
$w={\operatorname{sgn}}\in W^{odd}_{\mathbb{Z}}$ and
$v={\operatorname{Ind}}\in W^{even}_{\mathbb{Z}}$. Then the statement follows
from Theorem 3.

(ii) It is easy to see from the definition of polynomials
$\widetilde{F}_{K}^{n,k,m}(t,\ell,v)$ that substituting $v=1$ gives back
polynomials $F_{K}^{n}(t,\ell)$. Therefore, if two virtual knots $K$ and
$K^{\prime}$ can be distinguished by $F_{K}^{n}(t,\ell)$ for some $n$, then
$K$ and $K^{\prime}$ can also be distinguished by
$\widetilde{F}_{K}^{n,k,m}(t,\ell,v)$ for any values $k$ and $m$.

However in the case when $F_{K}^{n}(t,\ell)$ fails to distinguish virtual
knots $K$ and $K^{\prime}$, it may be possible to find a pair $(k,m)$ such
that $K$ and $K^{\prime}$ can be distinguished by polynomials
$\widetilde{F}_{K}^{n,k,m}(t,\ell,v)$.

Indeed, consider an infinite family of virtual knots $VK_{q}$, for positive
integer $q$, presented in Figure 25, where $c_{j}$, $j=1,\ldots,q$, are
repeating blocks separated by dashed lines. These virtual knots were
constructed in [6] to provide examples of $q$-simplexes in the Gordian complex
of virtual knots corresponding to the arc shift move defined in [5]. It was
shown in [5] that the arc shift move together with generalized Reidemeister
moves is an unknotting operation for virtual knots. By using the Kauffman
$f$-polynomial it was shown in [6] that virtual knots $VK_{q}$ are distinct
for distinct $q$.

$c_{0}$$c_{1}$$c_{2}$$\cdots$$\cdots$$\cdots$$\cdots$$c_{q}$ Figure 25.
Virtual knot $VK_{q}$.

For $q\leqslant 10$ polynomials $F_{VK_{q}}^{n}(t,\ell)$ were computed in [6].
It turns out that each of polynomials $F^{2}(t,\ell)$, $F^{4}(t,\ell)$ and
$F^{6}(t,\ell)$ is able to distinguish virtual knots $VK_{1}$, $VK_{2}$ and
$VK_{3}$, but not able to distinguish virtual knots $VK_{3},\ldots,VK_{10}$.

It is shown in Table 2 that virtual knots $VK_{3}$ and $VK_{4}$ can be
distinguished by each of polynomials $\widetilde{F}^{2,k,m}(t,\ell,v)$ for
$(k,m)=(2,0)$ and $(k,m)=(2,2)$.

Table 2. Polynomials $\widetilde{F}_{VK_{3}}^{2,k,m}(t,\ell,v)$ and $\widetilde{F}_{VK_{4}}^{2,k,m}(t,\ell,v)$.  knot | $(k,m)$ | $\widetilde{F}^{2,k,m}(t,\ell,v)$
---|---|---
$VK_{3}$ | $(2,0)$ | $\begin{gathered}\ell^{-1}-t^{-2}v^{-8}-\ell^{2}v^{6}+\ell^{-2}+3\ell^{2}+v^{4}+2v^{6}+v^{-8}\cr-v^{8}-t^{2}v^{4}\ell^{-2}-t^{2}v^{6}\ell^{-4}+t^{2}v^{8}\ell^{-3}-5\end{gathered}$
$VK_{3}$ | $(2,2)$ | $\begin{gathered}\ell^{-1}-v^{2}t^{-2}-\ell^{2}v^{-2}+\ell^{-2}+3\ell^{2}+v^{-2}\cr+v^{2}-t^{2}\ell^{-2}+t^{2}\ell^{-3}v^{-2}-t^{2}\ell^{-4}v^{-2}-4\end{gathered}$
$VK_{4}$ | $(2,0)$ | $\begin{gathered}\ell^{-1}-t^{-2}v^{-8}-\ell^{2}v^{6}+\ell^{-2}+3\ell^{2}+v^{4}+2v^{6}+v^{-8}\cr-v^{8}-2t^{2}v^{4}\ell^{-2}+t^{2}v^{4}\ell^{-3}+t^{2}v^{8}\ell^{-2}-t^{2}v^{6}\ell^{-4}-5\end{gathered}$
$VK_{4}$ | $(2,2)$ | $\begin{gathered}\ell^{-1}-v^{2}t^{-2}-\ell^{2}v^{-2}+\ell^{-2}+3\ell^{2}+v^{-2}\cr+v^{2}-2t^{2}\ell^{-2}+t^{2}\ell^{-3}+t^{2}\ell^{-2}v^{-2}-t^{2}\ell^{-4}v^{-2}-4\end{gathered}$

This observation completes the proof. ∎

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