Narrow Results

We prove that the two conditions from the definition of a biquandle by Fenn, Jordan-Santana, Kauffman [1] are equivalent and thus answer a question posed in the paper. We also construct a weak biquandle, which is not a biquandle.

This paper studies the biquandle as an invariant of virtual knots and links, analyzing its structure via composition of biquandle morphisms where elementary morphisms are associated with the crossings and virtual crossings in the diagram.

In this paper, we define and give examples of a family of polynomial invariants of virtual knots and links. They arise by considering certain 2 × 2 matrices with entries in a possibly non-commutative ring, for example, the quaternions. These polynomials are sufficiently powerful to distinguish the Kishino knot from any classical knot, including the unknot.

In this paper, we show how generalized quaternions including some 2 × 2 matrices, can be used to find solutions of the equation

Let A, B be invertible, non-commuting elements of a ring R. Suppose that A - 1 is also invertible and that the equation

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.

The fundamental biquandle is an invariant of an oriented surface link, which is defined by a presentation obtained from a surface diagram of the surface link: The generating set consists of labels of the semi-sheets and the relator set consists of relations defined at the double point curves. Any surface link can be presented by a link diagram with some markers which is called a ch-diagram. Using this fact, we give a method for calculating the fundamental biquandle of a surface link from its ch-diagram directly.

The Alexander biquandle of a virtual knot or link is a module over a 2-variable Laurent polynomial ring which is an invariant of virtual knots and links. The elementary ideals of this module are then invariants of virtual isotopy which determine both the generalized Alexander polynomial (also known as the Sawollek polynomial) for virtual knots and the classical Alexander polynomial for classical knots. For a fixed monomial ordering <, the Gröbner bases for these ideals are computable, comparable invariants which fully determine the elementary ideals and which generalize and unify the classical and generalized Alexander polynomials. We provide examples to illustrate the usefulness of these invariants and propose questions for future work.

The fundamental quandles and biquandles are invariants of classical knots and surface knots. It is unknown whether there exist classical or surface knots which have isomorphic fundamental quandles and distinct fundamental biquandles. We show that ribbon 2-knots or ribbon torus-knots with isomorphic fundamental quandles have isomorphic fundamental biquandles. For this purpose, we give a method for obtaining a presentation of the fundamental biquandle of a ribbon 2-knot/torus-knot from its fundamental quandle.

We investigate some algebraic structures called quasi-trivial quandles and we use them to study link-homotopy of pretzel links. Precisely, a necessary and sufficient condition for a pretzel link with at least two components being trivial under link-homotopy is given. We also generalize the quasi-trivial quandle idea to the case of biquandles and consider enhancement of the quasi-trivial biquandle cocycle counting invariant by quasi-trivial biquandle cocycles, obtaining invariants of link-homotopy type of links analogous to the quasi-trivial quandle cocycle invariants in Inoue’s paper [A. Inoue, Quasi-triviality of quandles for link-homotopy, J. Knot Theory Ramifications22(6) (2013) 1350026, doi:10.1142/S0218216513500260, MR3070837].

We investigate the relationship between the quandle and biquandle coloring invariant and obtain an enhancement of the quandle and biquandle coloring invariants using biquandle structures.

We also continue the study of biquandle homomorphisms into a medial biquandle begun in [Hom quandles, J. Knot Theory Ramifications23(2) (2014)], finding biquandle analogs of results therein. We describe the biquandle structure of the Hom-biquandle, and consider the relationship between the Hom-quandle and Hom-biquandle.

Biracks and biquandles, which are useful for studying the knot theory, are special families of solutions of the set-theoretic Yang–Baxter equation. A homology theory for the set-theoretic Yang–Baxter equation was developed by Carter et al. in order to construct knot invariants. In this paper, we construct a normalized (co)homology theory of a set-theoretic solution of the Yang–Baxter equation. We obtain some concrete examples of nontrivial $n$-cocycles for Alexander biquandles. For a biquandle $X,$ its geometric realization $BX$ is discussed, which has the potential to build invariants of links and knotted surfaces. In particular, we demonstrate that the second homotopy group of $BX$ is finitely generated if the biquandle $X$ is finite.

The Biquandle Bracket is a generalization of the Jones Polynomial. In this paper, we outline a Khovanov Homology-style construction which generalizes Khovanov Homology and attempts to categorify the Biquandle Bracket. The Biquandle Bracket is not always recoverable from our construction, so this is not a true categorification. However, this deficiency leads to a new invariant: a canonical biquandle 2-cocycle associated to a biquandle bracket.

We prove that parities on virtual knots come from invariant 1-cycles on the arcs of knot diagrams. In turn, the invariant cycles are determined by quasi-indices on the crossings of the diagrams. The found connection between the parities and the (quasi)-indices allows one to define a new series of parities on virtual knots.

A traditional method of finding topological invariants of a knot in 3-dimensional euclidean space is to colour the arcs of a planar diagram and use this to extract invariants of the knot such as the fundamental group. This method can be extended in many ways to other kinds of codimension 2 embeddings. Moreover the new algebraic objects which arise quite naturally from these generalisations such as quandles, racks, biquandles etc are of interest in their own right. In this paper we shall give a brief overview of the methods and the new algebraic structures.