On the axioms of singquandles

TL;DR

This paper introduces a new, simplified axiomatization of singquandles, algebraic structures that model Reidemeister moves for singular links, enhancing understanding and applications in knot theory.

Contribution

It provides a novel, streamlined axiomatization of singquandles and reformulates axioms for affine singquandles, especially in the idempotent case.

Findings

Simplified axioms for singquandles

New algebraic insights into singular link invariants

Enhanced framework for affine singquandles

Abstract

In this paper we deal with the notion of singquandles introduced in Indu R. U. Churchill, Mohamed Elhamdadi, Mustafa Hajij, and Sam Nelson, Singular knots and involutive quandles, Journal of Knot Theory and Its Ramifications 26 (2017), no. 14. This is an algebraic structure that naturally axiomatizes Reidemeister moves for singular links, similarly to what happens for ordinary links and quandle structure. We present a new axiomatization that shows different algebraic aspects and simplifies applications. We also reformulate and simplify the axioms for affine singquandles (in particular in the idempotent case).