Narrow Results

It is shown that except in three cases conjugacy classes of classical Weyl groups $W(B_n)$ and $W(D_n)$ are of type $D$. This proves that Nichols algebras of irreducible Yetter–Drinfeld modules over the classical Weyl groups ${𝕎}_n$ (i.e. $H_n⋊{𝕊}_n$) are infinite dimensional, except the class of type $(2,3),(1^2,3)$ in ${𝕊}_5$, and $(1^{n-2},2)$ in ${𝕊}_n$ for $n>5$.

We introduce (co)homology theory for multiple group racks and construct cocycle invariants of compact oriented surfaces in the 3-sphere using their 2-cocycles, where a multiple group rack is a rack consisting of a disjoint union of groups. We provide a method to find new rack cocycles and multiple group rack cocycles from existing rack cocycles and quandle cocycles. Evaluating cocycle invariants, we determine several symmetry types, caused by chirality and invertibility, of compact oriented surfaces in the 3-sphere.

In this paper, we show that conjugacy classes of classical Weyl groups $W(B_n)$ and $W(D_n)$ are of type D. Consequently, we obtain that Nichols algebras of irreducible Yetter–Drinfeld modules over the classical Weyl groups ${𝕎}_n$ ($n\ge 5$) are infinite dimensional.

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 paper presents the first complete calculation of the cohomology of any nontrivial quandle, establishing that this cohomology exhibits a very rich and interesting algebraic structure. Rack and quandle cohomology have been applied in recent years to attack a number of problems in the theory of knots and their generalizations like virtual knots and higher-dimensional knots. An example of this is estimating the minimal number of triple points of surface knots [E. Hatakenaka, An estimate of the triple point numbers of surface knots by quandle cocycle invariants, Topology Appl139(1–3) (2004) 129–144.]. The theoretical importance of rack cohomology is exemplified by a theorem [R. Fenn, C. Rourke and B. Sanderson, James bundles and applications, Proc. London Math. Soc. (3)89(1) (2004) 217–240] identifying the homotopy groups of a rack space with a group of bordism classes of high-dimensional knots. There are also relations with other fields, like the study of solutions of the Yang–Baxter equations.

Given a skew left brace $B$, a method is given to construct all the non-degenerate set-theoretic solutions $(X,r)$ of the Yang–Baxter equation such that the associated permutation group ${𝒢}(X,r)$ is isomorphic, as a skew left brace, to $B$. This method depends entirely on the brace structure of $B$. We then adapt this method to show how to construct solutions with additional properties, like square-free, involutive or irretractable solutions. Using this result, it is even possible to recover racks from their permutation group.

A quandle is a set equipped with a binary operation satisfying three quandle axioms. It also can be expressed as a sequence of permutations of the underlying set satisfying certain conditions. In this paper, we will calculate the second rack homology group of the disjoint union of two finite quandles and the second and third rack homology groups of certain type of finite quandles.

In 2006, Nelson and Wong introduced that every finite quandle can be decomposed into a disjoint union of some quandles and for given $n$ quandles satisfying certain conditions, there is a quandle which is a disjoint union of the $n$ quandles. In 2008, Ehrman, Gurpinar, Thibault and Yetter also introduced similar results about the decomposition of quandles. In this paper, we observe an operation table $Q$ which consists of four sub-operation tables, where diagonal tables are quandle operations. We study which conditions make $Q$ a quandle operation table.

In this paper, we adjust the Yang–Baxter operators constructed by Jones for the HOMFLYPT polynomial. Then we compute the second homology for this family of Yang–Baxter operators. It has the potential to yield 2-cocycle invariants for links.

We introduce a bi-Legendrian rack and show that a bi-Legendrian rack coloring number is an invariant of Legendrian knots. We prove that bi-Legendrian rack coloring numbers can distinguish all Legendrian unknots with the same Thurston–Bennequin number. We also consider pairs of Legendrian knots which cannot be distinguished by bi-Legendrian rack coloring numbers.

In this paper, we introduce and investigate a natural family of metrics on connected components of a rack. The metrics are closely related to certain bi-invariant metrics on the group of inner automorphisms of the rack. We also introduce a bounded cohomology of racks and quandles, relate them to the above metrics and prove a vanishing result for racks and quandles with amenable group of inner automorphisms.

We obtain an R-matrix or matrix representation of the Artin braid group acting in a canonical way on the vector space of every (super)-Lie algebra or braided-Lie algebra. The same result applies for every (super)-Hopf algebra or braided-Hopf algebra. We recover some known representations such as those associated to racks. We also obtain new representations such as a non-trivial one on the ring k[x] of polynomials in one variable, regarded as a braided-line. Representations of the extended Artin braid group for braids in the complement of S¹ are also obtained by the same method.

We prove that the rack and quandle spaces of links in 3-manifolds, considered only as topological spaces (disregarding their cubical structure), are closely related to certain subspaces of the loop spaces on the 3-manifold, which we call the vertical and the straight loop space of the link. Using these models we prove that the homotopy type of the non-augmented rack and quandle spaces of a link L in a 3-manifold M depends essentially only on the homotopy type of the pair (M,M -L).

Generalized knot groups G_n(K) were introduced independently by Kelly [5] and Wada [10]. We prove that G₂(K) determines the unoriented knot type and sketch a proof of the same for G_n(K) for n > 2.

We show that Nichols algebras of most simple Yetter–Drinfeld modules over the projective symplectic linear group over a finite field, corresponding to unipotent orbits, have infinite dimension. We give a criterion to deal with unipotent classes of general finite simple groups of Lie type and apply it to regular classes in Chevalley and Steinberg groups.

We show that except in several cases conjugacy classes of classical Weyl groups $W(B_n)$ and $W(D_n)$ are of type $D$. We prove that except in three cases Nichols algebras of irreducible Yetter–Drinfeld (YD) modules over the classical Weyl groups are infinite dimensional.

This short survey contains some recent developments of the algebraic theory of racks and quandles. We report on some elements of representation theory of quandles and ring theoretic approach to quandles.

A rack is a non-empty set with a binary operation such that right multiplication is an automorphism. New racks closely related to the braid groups and the mapping class groups will be introduced using "cords" on surfaces, and their structure will be investigated. Emphasis will be laid on the racks of cords on the plane and the sphere. In particular, their presentations and the center of the associated group of the rack of cords on the sphere will be given.

Knot and link diagrams are used to represent nonstandard sets, and to represent the formalism of combinatory logic (lambda calculus). These diagrammatics create a two-way street between the topology of knots and links in three dimensional space and key considerations in the foundations of mathematics.

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.