Metaplectic link invariants
The authors construct a large collection of link invariants, one for each of a certain set of class functions on the ascending union of the braid groups \(B_ n\). Setting their objetive as finding a complex valued function w on the union of the braid groups which is constant on oriented links, the authors invoke Markov's theorem and observe that such a function must satisfy \[ w(\alpha \gamma \alpha^{-1})=w(\gamma),\quad \alpha,\gamma \in B_{n+1} \] \[ w(\gamma \sigma^{\pm 1}_{n+1})=w(\gamma),\quad \sigma_{n+1} \text{ a standard generator and }\gamma \in B_{n+1}. \] This leads them to consider class functions \(\tau\) on \(B_{n+1}\) by virtue of the first equation, and then, to force the second equation to restrict such \(\tau\) to those satisfying \[ \tau (\sigma_ i^{- 1})=\tau (\sigma i)\quad and\quad \tau (\gamma \sigma^{\pm 1}_{n+1})=\tau (\gamma)\tau (\sigma_{n+1}). \] By setting \(w(\gamma)=\tau (\gamma)/\tau (\sigma_ n)^ n\) for \(\gamma \in B_{n+1}\) the invariant function w is obtained. The Burau representation and results of \textit{C. Squier} [Proc. Am. Math. Soc. 90, 199-202 (1984; Zbl 0542.20022)] are applied to construct the representations \(\tau\). Specialization leads to connections with the classical Alexander module and the Jones polynomial.
scientific article; zbMATH DE number 4111602