Cocycle Enhancements of Psyquandle Counting Invariants

TL;DR

This paper extends cocycle enhancement theory to psyquandles, introducing new polynomial invariants for pseudoknots and singular knots that improve upon existing counting invariants.

Contribution

It develops a novel cocycle enhancement framework for psyquandles and defines new polynomial invariants that are not reducible to Jablan polynomial.

Findings

New polynomial invariants for pseudoknots and singular knots.

Enhancements are proper and provide additional distinguishing power.

Invariants are not determined by Jablan polynomial.

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.