Two-variable polynomial invariants of virtual knots arising from flat virtual knot invariants

TL;DR

This paper introduces new two-variable polynomial invariants for virtual knots derived from flat virtual knot invariants, providing tools to analyze knot properties and conditions for cosmetic crossing changes.

Contribution

It presents two novel sequences of polynomial invariants based on index and dwrithe values, extending the Kauffman affine index polynomial and applying them to knot symmetry analysis.

Findings

$L^n_K$ and $F^n_K$ are invariants for virtual knots.

$L^n_K$ helps identify knots without cosmetic crossing changes.

The polynomials generalize the Kauffman affine index polynomial.

Abstract

We introduce two sequences of two-variable polynomials $\{ L^n_K (t, \ell)\}_{n=1}^{\infty}$ and $\{ F^n_K (t, \ell)\}_{n=1}^{\infty}$, expressed in terms of index value of a crossing and $n$-dwrithe value of a virtual knot $K$, where $t$ and $\ell$ are variables. Basing on the fact that $n$-dwrithe is a flat virtual knot invariant we prove that $L^n_K$ and $F^n_K$ are virtual knot invariants containing Kauffman affine index polynomial as a particular case. Using $L^n_K$ we give sufficient conditions when virtual knot does not admit cosmetic crossing change.