Parity biquandle invariants of virtual knots

Abstract

We define counting and cocycle enhancement invariants of virtual knots using parity biquandles. The cocycle invariants are determined by pairs consisting of a biquandle 2-cocycle φ 0 and a map φ 1 with certain compatibility conditions leading to one-variable or two-variable polynomial invariants of virtual knots. We provide examples to show that the parity cocycle invariants can distinguish virtual knots which are not distinguished by the corresponding non-parity invariants.

Multi-tribrackets

“…• Multi-cocycle enhancements. Following [2,7], define a general theory of multi-tribracket cocycle enhancements.…”

“…As with quandles, biquandles and other coloring structures, we can define enhancements of the multi-tribracket counting invariant, e.g. extending the tribracket modules in [7] to the case of multi-tribrackets.…”

Virtual Parity Alexander Polynomial

“…See BDGGHN's work [1] and Kitano's survey paper on Alexander polynomials [15] for reference. See Kaestner and Kauffman's work [10] or KNS [11] for reference on using parity with biquandle structures. Differentiating between even and odd crossings results in a polynomial that gives additional information about the non-planarity of virtual knot.…”