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

Abstract

We introduce two sequences of two-variable polynomials $\{L_K^n(t,\ell ){\}}_{n=1}^{\infty }$ and $\{F_K^n(t,\ell ){\}}_{n=1}^{\infty }$, 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 $L_K^n$ and $F_K^n$ are virtual knot invariants containing Kauffman affine index polynomial as a particular case. Using $L_K^n$ we give sufficient conditions when virtual knot does not admit cosmetic crossing change.

21 pages, 23 figure, 6 tables