Band-Passes and Long Virtual Knot Concordance

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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 VC. It is shown that for every concordance class [K] ∈ VC, there is a J ∈ [K] that is not band-pass equivalent to K and an L ∈ [K] that is not band-pass equivalent to either the long unknot or any long trefoil. This is accomplished by proving that v₂,₁+v₂,₂ 2 is a band-pass invariant but not a concordance invariant of long virtual knots, where v₂,₁ and v₂,₂ generate the degree two Polyak group for long virtual knots.