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Denis P. Ilyutko

Publications and source records attributed to Denis P. Ilyutko.

6 recordsLinked to original sources
arXiv2019

Picture-valued parity-biquandle bracket II. Examples

In [3] we constructed the parity-biquandle bracket valued in {\em pictures} (linear combinations of $4$-valent graphs). We gave no example of classical links such that the parity-biquandle bracket of which is not trivial. In the present paper we slightly change the notation of the parity-biquandle bracket and give examples of knots and links having a non-trivial parity-biquandle bracket. As a result we get the minimality theorem. This is the first evidence that graphs (link shadows) appear as invariants of link diagrams instead of just polynomials groups and other tractable objects.

math.GT↗
arXiv2017

Picture-valued biquandle bracket

In [14], the second named author constructed the bracket invariant [.] of virtual knots valued in pictures (linear combinations of virtual knot diagrams with some crossing information omitted), such that for many diagrams K, the following formula holds: [K]=K', where K' is the underlying graph of the diagram, i.e., the value of the invariant on a diagram equals the diagram itself with some crossing information omitted. This phenomenon allows one to reduce many questions about virtual knots to questions about their diagrams. In [25], the authors discovered the following phenomenon: having a biquandle colouring of a certain knot, one can enhance various state-sum invariants (say, Kauffman bracket) by using various coefficients depending on colours. Taking into account that the parity can be treated in terms of biquandles, we bring together the two ideas from these papers and construct the the picture-valued parity biquandle bracket for classical and virtual knots. This is an invariant of virtual knots valued in pictures. Both the parity bracket and Nelson-Orrison-Rivera invariants are partial cases of this invariants, hence this invariant enjoys many properties of various kinds.

math.GT↗
arXiv2009

Framed 4-Valent Graphs: Euler Tours, Gauss Circuits and Rotating Circuits

In the present paper we give an explicit formula which allows us immediately to describe a unique Gauss circuit on a framed 4-valent graph (a graph with a structure of opposite edges) from an arbitrary Euler tour on the graph whenever the Gauss circuit exists. This formula only depends on the adjacency matrix of an Euler tour and also tells us whether there exists a Gauss tour on a framed 4-valent graph or not. It turns out that the results are also valid for all symmetric matrices (not just realisable by a chord diagram).

math.CO↗
arXiv2008

Introduction to Graph-Link Theory

The present paper is an introduction to a combinatorial theory arising as a natural generalisation of classical and virtual knot theory. There is a way to encode links by a class of `realisable' graphs. When passing to generic graphs with the same equivalence relations we get `graph-links'. On one hand graph-links generalise the notion of virtual link, on the other hand they do not feel link mutations. We define the Jones polynomial for graph-links and prove its invariance. We also prove some a generalisation of the Kauffman-Murasugi-Thistlethwaite theorem on `minmal diagrams' for graph-links

math.GT↗