A Polynomial Invariant of Virtual Links

Abstract

In this paper, we define some polynomial invariants for virtual knots and links. In the first part we use Manturov's parity axioms [19] to obtain a new polynomial invariant of virtual knots. This invariant can be regarded as a generalization of the odd writhe polynomial defined by the first author in [1]. The relation between this new polynomial invariant and the affine index polynomial [14,3] is discussed. In the second part we introduce a polynomial invariant for long flat virtual knots. In the third part we define a polynomial invariant for 2-component virtual links. This polynomial invariant can be regarded as a generalization of the linking number.

Writhe polynomials and shell moves for virtual knots and links

“…(3) The concatenation of +S ij (n) and −S ij (n) is S-equivalent to that of n copies of +S i (1) for n > 0 or n copies of canceling pairs for n < 0 by using Lemma 2.3 (2) and Corollary 2.4 (1). See Figure 13, where ε is the sign of n; that is, n = ε|n|.…”

“…This invariant is introduced in several papers [2,8,12] independently. A characterization of W K (t) is given as follows.…”

An unknotting index for virtual links

“…In [1], Z. Cheng and H. Gao defined an invariant, called span, for 2-component virtual links using Gauss diagram. Remark that span is same as the absolute value of wriggle number provided by L. C. Folwaczny and L. H. Kauffman in [3].…”

A note on coverings of virtual knots

“…Later the notion of index is introduced in [2,4,6,11] which assigns an integer to each real crossing such that the parity of the index coincides with the original parity. The n-writhe is defined as a refinement of the odd writhe.…”