Virtual Yang-Baxter cocycle invariants

Abstract

Abstract. We extend the Yang-Baxter cocycle invariants for virtual knots by augmenting Yang-Baxter 2-cocycles with cocycles from a cohomology theory associated to a virtual biquandle structure. These invariants coincide with the classical Yang-Baxter cocycle invariants for classical knots but provide extra information about virtual knots and links. In particular, they provide a method for detecting non-classicality of virtual knots and links.

Virtual tribrackets

“…In [4] and [12], Boltzmann weights are used to enhance the biquasile counting invariant. A scheme similar to that used in [3] could be applied similarly here with different Boltzmann weights at the classical and virtual crossings to obtain a two-variable polynomial enhancements of the virtual tribracket counting enhancement.…”

“…The pair of 3-tensors below defines a virtual tribracket structure on the set X = {1, 2, 3}:Then we have for instance[1,3,2] = 3 and 2, 3, 3 = 2.…”

Extensions of set-theoretic solutions of the Yang–Baxter equation and a conjecture of Gateva-Ivanova

“…The cocycles of Example 3.4 are related to the homology theory for set-theoretic solutions of the Yang-Baxter equation of Carter, Elhamdadi and Saito. This homology is useful to construct invariants of virtual knots, see for example [2, §2] and [6].…”

Parity biquandle invariants of virtual knots

“…We begin this section with a brief review of biquandle homology; see [3,5,6], etc., for more. Let X be a finite biquandle and A an abelian group.…”

“…In [10], biquandles incorporating the notion of parity were introduced (see also [1]). In this paper we extend the counting invariant to the case of finite parity biquandles and define enhancements of the counting invariant using parity enhanced cocycles, cocycles in the second cohomology of the even part of the parity biquandle with extra information analogous to the virtual cocycles in [6].…”