Braid Group Representations arising from the Yang Baxter Equation

The second Yang-Baxter homology for the Homflypt polynomial

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

Homology Theory for the Set-Theoretic Yang-Baxter Equation and Knot Invariants from Generalizations of Quandles

A homology theory is developed for set-theoretic Yang-Baxter equations, and knot invariants are constructed by generalized colorings by biquandles and Yang-Baxter cocycles.

Virtual Yang-Baxter cocycle invariants

We extend the Yang-Baxter cocycle invariants for virtual knots by augmenting Yang-Baxter 2-cocycles with cocycles from a cohomology theory associated to a virtual biquandle structure. These invariants coincide with the classical Yang-Baxter cocycle invariants for classical knots but provide extra information about virtual knots and links. In particular, they provide a method for detecting non-classicality of virtual knots and links.

Set-theoretic Yang-Baxter (co)homology theory of involutive non-degenerate solutions

W. Rump showed that there exists a one-to-one correspondence between involutive right non-degenerate solutions of the Yang-Baxter equation and Rump right quasigroups. J. S. Carter, M. Elhamdadi, and M. Saito, meanwhile, introduced a homology theory of set-theoretic solutions of the Yang-Baxter equation in order to define cocycle invariants of classical knots. In this paper, we introduce the normalized homology theory of an involutive right non-degenerate solution of the Yang-Baxter equation and prove that the set-theoretic Yang-Baxter homology of certain solutions can be split into the normali…

通用编织量子门

Universal Braiding Quantum Gates

本文展示了在任意维度下构造既酉又通用的辫子Yang-Baxter方程解的简单方法，并对广义Yang-Baxter方程的解进行了完整分类，证明了某些实例仅有标量倍单位矩阵解。

Set Theoretic Yang-Baxter Solutions via Fox Calculus

We construct solutions to the set-theoretic Yang-Baxter equation using braid group representations in free group automorphisms and their Fox differentials. The method resembles the extensions of groups and quandles.

Yang-Baxter operators arising from algebra structures and the Alexander polynomial of knots

In this note, we consider the problem of constructing knot invariants from Yang-Baxter operators associated to (unitary associative) algebra structures. We first compute the enhancements of these operators. Then, we conclude that Turaev's procedure to construct knot invariants, as modified by Murakami, invariably produces from those enhanced Yang-Baxter operators the Alexander polynomial.

Invariants of links from the generalized Yang-Baxter equation

Enhanced Yang-Baxter operators give rise to invariants of oriented links. We expand the enhancing method to generalized Yang-Baxter operators. At present two examples of generalized Yang-Baxter operators are known and recently three types of variations for one of these were discovered. We present the definition of enhanced generalized YB-operators and show that all known examples of generalized YB-operators can be enhanced to give corresponding invariants of oriented links. Most of these invariants are specializations of the polynomial invariant $P$. Invariants from generalized YB-operators ar…

Yang-Baxter deformations and rack cohomology

Every rack $Q$ provides a set-theoretic solution $c_Q$ of the Yang-Baxter equation. This article examines the deformation theory of $c_Q$ within the space of Yang-Baxter operators over a ring $\A$, a problem initiated by Freyd and Yetter in 1989. As our main result we classify deformations in the modular case, which had previously been left in suspense, and establish that every deformation of $c_Q$ is gauge-equivalent to a quasi-diagonal one. Stated informally, in a quasi-diagonal deformation only behaviourally equivalent elements interact. In the extreme case, where all elements of $Q$ are be…