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The Gassner representation for string links. (English) Zbl 0989.57005
The authors investigate the extension of the Gassner representation to the group of concordance classes of string links. They present new perspectives on its definition by giving a simple homological one and a completely different probalistic one in terms of random walks on string links. This allows them to obtain general results. For example, they prove that the extended Gassner representation is unitary, that the pure braid group on \(n\)-components (\(n\geq 3\)) is not normal in the pure string link concordance group and that the entries in the generalized Gassner matrix are finite type invariants.
Furthermore, they use these definitions to establish a formula relating the Alexander polynomial of the closure of a string link and its generalized Gassner invariant, to give elementary arguments of the theorems of J.-Y. Le Dimet [Comment. Math. Helv. 67, No. 2, 306-315 (1992; Zbl 0759.57010)] and to obtain a useful and flexible generalization of a recent theorem of J. Levine [ibid. 74, No. 1, 27-52 (1999; Zbl 0918.57001)].

MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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