I. Nikonov

Multicrossing complex of knot and secant classes

Igor Nikonov

The multicrossing homology of a knot can be considered as a generalization of the homology of its fundamental quandle. We show that certain sums of knot secants define invariant cl…

Homotopy index polynomials for knotoids

Igor Nikonov

We define homotopy index polynomials and a homotopy type of knotoids in an oriented surface, and consider some basic properties of these polynomials. We show that for planar knotoi…

Exotic 4-manifolds and Khovanov-Lipshitz-Sarkar homotopy type

Louis H. Kauffman, Igor M. Nikonov, Eiji Ogasa

We introduce a new diffeomorphism invariant of smooth compact oriented 4-manifolds $X$ with a framed oriented 1-link $L$ in the boundary, where $L$ may be the empty set, and call i…

A generalisation of the Burau representation and groups $G_{n}^{3}$ for classical braids

Vassily Olegovich Manturov, Igor Mikhailovich Nikonov

We consider a certain modification of the group $G_{n}^{3}$ which describes dynamics of point configurations, in particular braids, and define a representation of the modified $G_{n}^{3}$ .…

math.GT2025

The crossing and the arc from the topological viewpoint

Igor Nikonov

The combinatorial approach to knot theory treats knots as diagrams modulo Reidemeister moves. Many constructions of knot invariants (e.g., index polynomials, quandle colorings, etc…

math.GT2024

On skein invariants

Igor Nikonov

A knot invariant is called skein if it is determined by a finite number of skein relations. In the paper we discuss some basic properties of skein invariants and mention some known…

math.RA2024

Algebra, group, and Hopf Rota-Baxter operators

Valeriy G. Bardakov, Igor M. Nikonov, Viktor N. Zhelaybin

We know definition of Rota--Baxter operators on different algebraic systems. For examples, on groups, on algebras, on Hopf algebras. On some algebraic systems it is possible to def…

math.GT2024

Flat-virtual knot: introduction and some invariants

V. O. Manturov, I. M. Nikonov

The motivation for this work is to construct a map from classical knots to virtual ones. What we get in the paper is a series of maps from knots in the full torus (thickened torus)…

math.GR2024

Lie Rota--Baxter operators on the Sweedler algebra $H_{4}$

Valeriy G. Bardakov, Igor M. Nikonov, Viktor N. Zhelaybin

If $A$ is an associative algebra, then we can define the adjoint Lie algebra $A^{(-)}$ and Jordan algebra $A^{(+)}$ . It is easy to see that any associative Rota--Baxter operator on…

math.GR2023

Relative Rota--Baxter operators on groups and Hopf algebras

Valeriy G. Bardakov, Igor M. Nikonov

M. Goncharov introduced and studied a Rota--Baxter operator on a cocommutative Hopf algebra. In the present paper we define relative Rota--Baxter operators on an arbitrary Hopf alg…

Photography principle, data transmission, and invariants of manifolds

L. Kauffman, V. O. Manturov, I. M. Nikonov +1

In the present paper we develop the techniques suggested in \cite{ManturovNikonov} and the photography principle \cite{ManturovWan} for constructing an invariant of 3-manifolds bas…

The groups $Γ_{n}^{4}$ , braids, and $3$ -manifolds

Vassily Olegovich Manturov, Igor Mikhailovich Nikonov

We introduce a family of groups $Γ_{n}^{k}$ for integer parameters $n > k$ . These groups originate from discussion of braid groups on $2$ -surfaces. On the other hand, they turn out to be…

On an invariant of pure braids

V. O. Manturov, I. M. Nikonov

Using the recoupling theory, we define a representation of the pure braid group and show that it is not trivial.

Local transformations and functorial maps

Igor Nikonov

Picture-valued invariants are the main achievement of parity theory by V.O. Manturov. In the paper we give a general description of such invariants which can be assigned to a parit…

math.GT2022

Maps from knots in the cylinder to flat-virtual knots

V. O. Manturov, I. M. Nikonov

In the present paper, we address the problem how to get a map from knots in the cylinder and on the thickened torus to some (generalisation of) virtual knots called virtual-flat kn…

math.GT2022

Maps from braids to virtual braids and braid representations

V. O. Manturov, I. M. Nikonov

Virtual knot theory has experienced a lot of nice features that did not appear in classical knot theory, e.g., parity and picture-valued invariants. In the present paper we use vir…

math.GT2022

A group-valued invariant of knots in the full torus

Vassily Olegovich Manturov, Igor Mikhailovich Nikonov

Knot concordance plays a crucial role in the low dimensional topology. We propose a very elementary techniques which allows one to construct a lot of sliceness obstructions for kno…

Crossing tribes of tangles in a thickened surface

Igor Nikonov

Crossings of knot diagrams can be divided into classes (tribes) compatible with Reidemeister moves. Tribes can be considered as localization of the notion of weak chord index intro…

Intersection formulas for parities on virtual knots

Igor Nikonov

We prove that parities on virtual knots come from invariant 1-cycles on the arcs of knot diagrams. In turn, the invariant cycles are determined by quasi-indices on the crossings of…

Parity on based matrices

Igor Nikonov

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 introdu…

On universal parity on free two-dimensional knots

Igor Nikonov

We prove that the Gaussian parity on free two-dimensional knots is universal.

Crossing indices, traits and the principle of indistinguishability

Igor Nikonov

A (weak chord) index is a function on the crossings of knot diagrams such that: 1) the index of a crossing does not change under Reidemeister moves; 2) crossings which can be paire…

Parity functors

Igor Nikonov

A parity is a rule to assign labels to the crossings of knot diagrams in a way compatible with Reidemeister moves. Parity functors can be viewed as parities which provide to each k…