Cocycle Enhancements of Psyquandle Counting Invariants

Abstract

We bring cocycle enhancement theory to the case of psyquandles. Analogously to our previous work on virtual biquandle cocycle enhancements, we define enhancements of the psyquandle counting invariant via pairs of a biquandle 2-cocycle and a new function satisfying some conditions. As an application we define new single-variable and two-variable polynomial invariants of oriented pseudoknots and singular knots and links. We provide examples to show that the new invariants are proper enhancements of the counting invariant are are not determined by the Jablan polynomial.

Singquandles, Psyquandles and Singular Knots: A Survey

“…The authors showed that the corresponding notion of psyquandle colorings of an oriented singular link, or pseudolink, determines a counting invariant. Furthermore, in [12], the authors gave a cocycle enhancement theory to the case of psyquandles. Precisely, they defined enhancements of the psyquandle counting invariant via pairs of a biquandle 2-cocycle and a new function satisfying some conditions.…”

“…In this section we will provide an overview of results obtained for signular knots from psyquandles. The following definition was initially introduced in [25], but was reformulated in [12]. We will present the reformulated version of the definition.…”

“…Definition 8.1. [12] Let X be a set. A psyquandle structure on X is set of four binary operations , , • , • : X × X → X satisfying the conditions (I) All four operations are right-invertible,i.e.…”

“…A pI-adaquate psyquandle is an invariant of pseudolinks. For more details and results regarding pI-adaquate psyquandles the reader is referred to [12,25]. In this paper we will focus on the results for singular links obtained from psyquandles.…”

“…In Section 7, we survey recent enhancements to the signquandle counting invariant through the use of the singquandle polynomial and the shadow singuqandle polynomial. We end Section 8 with discussing psyquandle counting invariants of oriented singular links and pseudolinks [25] and enhancements of the psyquandle counting invariant [12].…”