Band-Passes and Long Virtual Knot Concordance

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 $\mathcal{VC}$. It is shown that for every concordance class $[K]\in \mathcal{VC}$, 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 $v_{2,1}+v_{2,2}(\mathrm{mod}2)$ 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.

added more background information