Bilinear Enhancements of Quandle Invariants

TL;DR

This paper introduces new enhanced invariants for classical and virtual knots and links by applying bilinear forms inspired by symplectic quandle operations, improving the ability to distinguish knots.

Contribution

It develops a novel method to enhance quandle invariants using bilinear forms, expanding the toolkit for knot distinction beyond traditional quandle counting invariants.

Findings

The new invariants can distinguish knots that traditional invariants cannot.

Examples demonstrate the effectiveness and proper enhancement of the invariants.

The method applies to both classical and virtual knots and links.

Abstract

We enhance the quandle counting invariants of oriented classical and virtual knots and links using a construction similar to quandle modules but inspired by symplectic quandle operations rather than Alexander quandle operations. Given a finite quandle $X$ and a vector space $V$ over a field, sets of bilinear forms on $V$ indexed by pairs of elements of $X$ satisfying certain conditions yield new enhanced multiset- and polynomial-valued invariants of oriented classical and virtual knots and links. We provide examples to illustrate the computation of the invariants and to show that the enhancement is proper.