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

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.

Erratum: Biquandles of small size and some invariants of virtual and welded knots

“…Analysis of the results presented in the first version of this paper, [1], revealed an ambiguity in the utilization of the theory. This led to an example of a welded knot which was claimed to be nontrivial, but was in fact trivial.…”

Unsolved problems in virtual knot theory and combinatorial knot theory

“…Virtual Biquandle: Construct presentations of the virtual biquandle with the a linear (non-commutative) representation at classical crossings and some interesting structure at virtual crossings. A start has been made by Bartholomew and Fenn [6]. In this paper various biquandle decorations are made at classical and virtual crossings which were found by a computer search.…”

“…The action defined by linear biquandles is not faithful. This almost certainly means that the corresponding linear invariants of virtual knots and links are not faithful [44] (see also [6]).…”

“…Example 8.1. Let the chord diagram consist of 2n chords, where the the following pairs of chords are linked: (1,2), (3,4), (5,6), . .…”

Biquandles With Structures Related to Virtual Links and Twisted Links

“…The concept of a coloring by (X, R, V ) in Definition 1.1 and the following theorem (Theorem 1.2) were considered in [2]. Refer to [2] for examples. This is generalized to twisted link diagrams.…”