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

TL;DR

This paper reports a computer search for small biracks, introduces new invariants for welded knots, and demonstrates their effectiveness in distinguishing non-trivial virtual knots, advancing the understanding of virtual and welded knot theory.

Contribution

It provides a comprehensive list of small biquandles, introduces new invariants for welded knots, and offers new insights into virtual knot distinctions.

Findings

Identification of small biquandles, racks, and quandles

New invariants capable of detecting non-trivial welded knots

Reproof of non-injectivity of the Burau map

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. (This is a revised version of an earlier paper of the same name)