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 [Formula: see text]. It is shown that for every concordance class [Formula: see text], there is a [Formula: see text] that is not band-pass equivalent to [Formula: see text] and an [Formula: see text] that is not band-pass equivalent to either the long unknot or any long trefoil. This is accomplished by proving that [Formula: see text] is a band-pass invariant but not a concordance invariant of long virtual knots, where [Formula: see text] and [Formula: see text] generate the degree two Polyak group for long virtual knots.