Narrow Results

We determine all 2- and 3-cocycles for Laver tables, an infinite sequence of finite structures obeying the left-self-distributivity law; in particular, we describe simple explicit bases. This provides a number of new positive braid invariants and paves the way for further potential topological applications. An important tool for constructing a combinatorially meaningful basis of 2-cocycles is the right-divisibility relation on Laver tables, which turns out to be a partial ordering.

The braids of $B_{\infty }$ can be equipped with a self-distributive operation ⊳ enjoying a number of deep properties. This text is a survey of known properties and open questions involving this structure, its quotients, and its extensions.

Self-distributive (SD) structures form an important class of solutions to the Yang–Baxter equation (YBE), which underlie spectacular knot-theoretic applications of self-distributivity (SD). It is less known that one can go the other way around, and construct an SD structure out of any left non-degenerate (LND) set-theoretic YBE solution. This structure captures important properties of the solution: invertibility, involutivity, biquandle-ness, the associated braid group actions. Surprisingly, the tools used to study these associated SD structures also apply to the cohomology of LND solutions, which generalizes SD cohomology. Namely, they yield an explicit isomorphism between two cohomology theories for these solutions, which until recently were studied independently. The whole story is full of open problems. One of them is the relation between the cohomologies of a YBE solution and its associated SD structure. These and related questions are covered in the present survey.

This paper recalls my first meeting with Patrick DeHornoy and its consequences.

A braided Frobenius algebra is a Frobenius algebra with a Yang–Baxter operator that commutes with the operations, that are related to diagrams of compact surfaces with boundary expressed as ribbon graphs. A heap is a ternary operation exemplified by a group with the operation $(x,y,z)\mapsto x{y}^{-1}z$, that is ternary self-distributive. Hopf algebras can be endowed with the algebra version of the heap operation. Using this, we construct braided Frobenius algebras from a class of certain Hopf algebras that admit integrals and cointegrals. For these Hopf algebras we show that the heap operation induces a Yang–Baxter operator on the tensor product, which satisfies the required compatibility conditions. Diagrammatic methods are employed for proving commutativity between Yang–Baxter operators and Frobenius operations.

We present a symbolic computations for developing cohomology theories of algebraic systems. The method is applied to coalgebra self-distributive maps to recover low dimensional differential maps.

We survey two of the many aspects of the standard braid order, namely its set theoretical roots, and the known connections with knot theory, including results by Netsvetaev, Malyutin, and Ito, and very recent work in progress by Fromentin and Gebhardt.