Links of intervals and the Gassner representation
Let \(F\) be the free group on \(n\) generators and \(G\) its algebraic closure (in the sense of \textit{J. P. Levine} [ibid. 64, 236-255 (1989; Zbl 0692.57009) and Invent. Math. 96, 571-592 (1989; Zbl 0692.57010)]). The Cohn localization and homological ideas are used to extend the free differential calculus to \(G\) and hence to define a representation of the group of automorphisms of \(G\) which induce the identity on the abelianization into \(GL(n,\Lambda_ S)\), where \(\Lambda_ S\) is the localization of the integral Laurent polynomial ring in \(n\) commuting variables with respect to the multiplicative system of elements with augmentation 1. This representation is then used to extend the Gassner representation of the pure braid group on \(n\) strings to the group of concordance classes of links of \(n\) intervals in the 3-disc.
Gassner representation of the pure braid group
scientific article; zbMATH DE number 93810