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Metaplectic link invariants. (English) Zbl 0678.57007
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 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.
Reviewer: L.P.Neuwirth

##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010) 20F36 Braid groups; Artin groups
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