Virtual knot cobordism and the affine index polynomial

Abstract

This paper studies cobordism and concordance for virtual knots. We define the affine index polynomial, prove that it is a concordance invariant for knots and links (explaining when it is defined for links), show that it is also invariant under certain forms of labeled cobordism and study a number of examples in relation to these phenomena. Information on determinations of the four-ball genus of some virtual knots is obtained by via the affine index polynomial in conjunction with results on the genus of positive virtual knots using joint work with Dye and Kaestner.

Recurrent Generalization of F-Polynomials for Virtual Knots and Links

“…For properties and applications of P K (t) see [3,10,11,12]. For each n ∈ Z \ {0} the n-th writhe J n (D) of a virtual knot diagram D is defined as the number of positive sign crossings minus number of negative sign crossings of D with index value n. The n-th writhe is a virtual knot invariant.…”

“…Polynomial invariants based on the index values in classical crossings were introduced by Cheng and Gao [3], known as th writhe polynomial, and by Kauffman [10], known as the affine index polynomial. The connection of the affine index polynomial with the virtual knot cobordism is described in [11]. For related polynomial invariants and their properties see [12,15,16,17].…”

The Affine Index Polynomial and the Sawollek Polynomial

“…References for this invariant are [2,6,13,14]. We define the Affine Index Polynomial invariant of virtual knots by first describing how to calculate the polynomial.…”

“…The purpose of this paper is to give a new basis for examining the relationships of the Affine Index Polynomial [13,14] and the Sawollek Polynomial [17]. Blake Mellor [16] has written a pioneering paper showing how the Affine Index Polynomial may be extracted from the Sawollek Polynomial.…”

Recurrent Generalization of F-Polynomials for Virtual Knots and Links

“…Polynomial invariants for virtual knots, based on the index values in classical crossings, were introduced by Cheng and Gao [9], known as the writhe polynomial, and by Kauffman [10], known as the affine index polynomial. The connection of the affine index polynomial with the virtual knot cobordism is described in [11]. For related polynomial invariants and their properties, see [12][13][14][15][16].…”