Psyquandles, Singular Knots and Pseudoknots

TL;DR

This paper introduces psyquandles, a new algebraic structure extending biquandles, to define invariants for singular knots and pseudolinks, including Alexander psyquandle invariants and the Jablan polynomial, with computations for small cases.

Contribution

It generalizes biquandles to psyquandles and develops new invariants for singular links and pseudolinks, including Alexander and Jablan polynomials.

Findings

Defined Alexander psyquandle polynomials and invariants.

Computed the Jablan polynomial for small pseudoknots and graphs.

Established relationships between colorings of pseudolinks.

Abstract

We generalize the notion of biquandles to psyquandles and use these to define invariants of oriented singular links and pseudolinks. In addition to psyquandle counting invariants, we introduce Alexander psyquandles and corresponding invariants such as Alexander psyquandle polynomials and Alexander-Gr\"obner psyquandle invariants of oriented singular knots and links. We consider the relationship between Alexander psyquandle colorings of pseudolinks and p-colorings of pseudolinks. As a special case we define a generalization of the Alexander polynomial for oriented singular links and pseudolinks we call the Jablan polynomial and compute the invariant for all pseudoknots with up to five crossings and all 2-bouquet graphs with up to 6 classical crossings.