Biquandles of Small Size and some Invariants of Virtual and Welded Knots

TL;DR

This paper reports a computer search for small biquandles and introduces new invariants for virtual and welded knots, demonstrating their effectiveness in distinguishing non-trivial knots and revalidating known results.

Contribution

It provides the first comprehensive list of small biquandles and develops new invariants that enhance the understanding of virtual and welded knot theory.

Findings

Identified small biquandles and their properties

Introduced new invariants for welded knots

Found examples of non-trivial welded knots indistinguishable by linear biquandles

Abstract

In this paper we give the results of a computer search for biracks of small size and we give various interpretations of these findings. The list includes biquandles, racks and quandles together with new invariants of welded knots and examples of welded knots which are shown to be non-trivial by the new invariants. These can be used to answer various questions concerning virtual and welded knots. As an application we reprove the result that the Burau map from braids to matrices is non injective and give an example of a non-trivial virtual (welded) knot which cannot be distinguished from the unknot by any linear biquandles.