Twisted conjugation braidings and link invariants

Abstract

This work is about link invariants arising from enhanced Yang-Baxter operators. For each enhanced Yang-Baxter operator $\mathcal{R}=(R, D, \lambda, \beta)$ and any braid $Br(n)$ Turaev defined a link invariant $T_{\mathcal{R}}(\xi)= \lambda^{-\omega(\xi)} \beta^{-n} trace (b_{R}(\xi) \circ D^{\otimes n}),$ where $\omega: Br(n) \rightarrow \mathbb Z$ is a homomorphism and $b_{R}$ is the representation of the Artin braid group $Br(n)$ arising from the solution of the Yang-Baxter equation $R.$ Therefore, we first introduce new solutions of the Yang-Baxter equation $B^{\varphi}:V^{\otimes 2} \rightarrow V^{\otimes 2},$ $B^{\varphi}(a \otimes b)= ab \varphi(a)^{-1} \otimes \varphi(a),$ for $V=\mathbb K[G],$ $\varphi \in Aut(G),$ where $G$ is any group. We call these solutions twisted conjugation braidings.<br /> Then we give sufficient and necessary conditions for a map $D$ to decide whether the quadruple $(B^{\varphi}, D, \lambda, \beta)$ is an EYB-operator. Moreover, we prove that the twisted conjugation braidings $B^{\varphi}$ can be enhanced using character theory. These enhancements are called character enhancements. It turns out that for every character enhancement $D$ of the twisted conjugation brading $B^{\varphi}$ the link invariant is constantly 1, i.e., $T_{\mathcal{B}}(\xi)=1$ for all $\xi \in Br(n).$ In general, we prove that the link invariant for all $\xi \in Br(n)$ and for every enhancement $D$ of the twisted conjugation braiding $B^{\varphi}$ is a map $T_{\mathcal{B}}(\xi)=\beta^{-n} trace (b_{\mathcal{B}^{\varphi}}) \circ D^{\otimes n}$ .<br /> Our main result is the following theorem.<br /> Let $\gamma$ be a fixed invertible element of $\mathbb K$ and let $D$ denote a linear map. Asumme that $D \otimes D$ commutes with the twisted conjugation braiding $B^{\varphi}$. Then <br /> 1. $Sp_2((B^{\varphi})^{\pm 1} \circ (D \otimes D))= \gamma D \; \Longrightarrow \; D^2= \gamma D$<br /> 2. $Sp_2(B^{\varphi} \circ (D \otimes D))= \gamma D \; \Longleftrightarrow \; Sp_2 ( (B^{\varphi})^{-1} \circ (D \otimes D))= \gamma D$<br /> In the last part of this work, we prove that for finite groups $G$ the twisted conjugation braiding $B^{\varphi}$ satisfies $(B^{\varphi})^l(a \otimes b)= a \otimes b,$ with $l=2 \cdot lcm (ord(a), ord(b).$ From this follows that the link invariant is $T_{\mathcal{B}}(\xi)=\left( \frac{m_1}{\beta}\right)^n$, for braids $\xi$ in $Br(n)$, with $\xi= \sigma_{\sigma_{i_1}}^{\epsilon_1} \dots \sigma_{i_l}^{\epsilon_l}$, and with $\epsilon_1, \dots, \epsilon_l \equiv \; 0 \; \text {mod} \; l,$ where $m_1= trace (D)$. We call such braids mod-l braids. Furthermore, it follows that the link invariant is $T_{\mathcal{B}}=\left( \frac{m_1}{\beta} \right)^{n-1}$ for braids $\xi \in Br(n)$ such that $\xi=\sigma_i^{\epsilon}$, with $\epsilon \equiv \; 0 \text{mod} \; l$. We call these braids single-power braids. Moreover, we wrote a program in JAVA programming language which computes the link invariants for the enhancement $D= \gamma I,$ ($\gamma \in \mathbb K^*$) for braids $\xi \in Br(p),$ (p prime) with $\xi=(\sigma_1 \sigma_2 \dots \sigma_{p-1})^q,$ and with $(p, q)=1$ for the cases $G=\Sigma_n$ and $G=\mathbb Z/ n \mathbb Z.$ In the cases were we have computed the link invariants $T_{\mathcal{B}}$ ``the polynomial is constant,'' i.e.,$T_{\mathcal{B}} \in \mathbb{K}$, since the only braindings we consider are permutations of the basis $\mathbb{K}[G]^{\otimes 2}$.