Two-variable polynomial invariants of virtual knots arising from flat virtual knot invariants (1803.05191v1)

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.