Band-Passes and Long Virtual Knot Concordance

TL;DR

This paper introduces the long virtual knot concordance group and demonstrates that band-pass equivalence is not a complete invariant within concordance classes, using new invariants derived from the Polyak group.

Contribution

It defines the long virtual knot concordance group and shows that band-pass equivalence does not fully classify knots within this group, introducing a new invariant based on the Polyak group.

Findings

Band-pass equivalence is not a complete concordance invariant.

Existence of knots not band-pass equivalent to the unknot or trefoil within the same concordance class.

The invariant v_{2,1}+v_{2,2} mod 2 distinguishes certain virtual knots.

Abstract

Every classical knot is band-pass equivalent to the unknot or the trefoil. The band-pass class of a knot is a concordance invariant. Every ribbon knot, for example, is band-pass equivalent to the unknot. Here we introduce the long virtual knot concordance group $\mathscr{VC}$. It is shown that for every concordance class $[K] \in \mathscr{VC}$, there is a $J \in [K]$ that is not band-pass equivalent to $K$ and an $L \in [K]$ that is not band-pass equivalent to either the long unknot or any long trefoil. This is accomplished by proving that $v_{2,1}+v_{2,2} \pmod 2$ is a band-pass invariant but not a concordance invariant of long virtual knots, where $v_{2,1}$ and $v_{2,2}$ generate the degree two Polyak group for long virtual knots.