Virtual Knot Cobordism and the Affine Index Polynomial

Louis H Kauffman

Published 2018-04-07, updated 2018-04-17Version 2

This paper defines a theory of cobordism for virtual knots and studies the cobordism invariance of the affine index polynomial, denoted $P_{K}(t).$ This invariant is also called the writhe polynomial, $W_{K}(t),$ in the context of Gauss diagrams. In this paper we work in the context of virtual knot and link diagrams with affine labelings and extend the definitions in our previous work to an affine index polynomial for links. We prove that this generalized invariant is a concordance invariant of knots and links (where concordance of virtual links is defined in the body of the paper).