On ambiguity in knot polynomials for virtual knots

TL;DR

This paper explores additional parameters in HOMFLY polynomials for virtual knots, revealing new topological invariants that do not appear in classical knots, and discusses their implications and limitations.

Contribution

It introduces a new set of parameters in virtual knot HOMFLY polynomials, expanding the understanding of topological invariants beyond classical knots.

Findings

New parameters in virtual HOMFLY polynomials grow rapidly with crossings.

These parameters are topologically invariant and do not appear in classical knots.

The Kishino unknot remains indistinguishable from the trivial unknot despite the new invariants.

Abstract

We claim that HOMFLY polynomials for virtual knots, defined with the help of the matrix-model recursion relations, contain more parameters, than just the usual $q$ and $A = q^N$. These parameters preserve topological invariance and do not show up in the case of ordinary (non-virtual) knots and links. They are most conveniently observed in the hypercube formalism: then they substitute $q$-dimensions of certain fat graphs, which are not constrained by recursion and can be chosen arbitrarily. The number of these new topological invariants seems to grow fast with the number of non-virtual crossings: 0, 1, 1, 5, 15, 91, 784, 9160, ... This number can be decreased by imposing the factorization requirement for composites, in addition to topological invariance -- still freedom remains. None of these new parameters, however, appear in HOMFLY for Kishino unknot, which thus remains unseparated from the ordinary unknots even by this enriched set of knot invariants.