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A rational noncommutative invariant of boundary links. (English) Zbl 1075.57004
The major results of this long paper are the construction of an invariant \(Z^{\text{rat}}\) for \(\mu\)-component boundary links with values in the ring of rational functions in \(\mu\) free (noncommuting) variables (the Cohn localization of the free group ring \(\mathbb{Z}[F(\mu)]\) with respect to the augmentation) and the identification of this invariant with the Kontsevich integral for such functions. Such boundary links may be obtained by surgery on a framed link in the complement of a \(\mu\)-component trivial link. Moreover the linking matrix of the framed surgery link may be assumed to be invertible (over \(\mathbb{Z}\)). Kirby moves on the surgery link leave invariant the canonical \(F\)-structure determined by the trivial link, and so the construction of an invariant in terms of surgery data involves checking that it is well behaved with respect to Kirby moves and also with respect to change of \(F\)-structure (i.e., composition with automorphisms of \(F(\mu)\) inducing the identity on abelianization). The second result is verified by showing that \(Z^{\text{rat}}\) is the universal finite type invariant of boundary links with respect to the null move. The authors indicate several applications, including a realization theorem for \(Z^{\text{rat}}\), which may be considered as a generalization of Levine’s surgical realization for Alexander polynomials of knots, the correction of a subtle error of Freedman, and a formula for the Casson-Walker invariant of a cyclic branched cover of a knot.

MSC:
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
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