Algebraic linking numbers of knots in 3-manifolds. (English) Zbl 1039.57005

The reviewer cannot improve on the author’s abstract, which follows. Relative self-linking and linking “numbers” for pairs of oriented knots and 2-component links in oriented 3-manifolds are defined in terms of intersection invariants of immersed surfaces in 4-manifolds. The resulting concordance invariants generalize the usual homological notion of linking by taking into account the fundamental group of the ambient manifold and often map onto infinitely generated groups. The knot invariants generalize the type 1 invariants of Kirk and Livingston and when taken with respect to certain preferred knots, called spherical knots, relative self-linking numbers are characterized geometrically as the complete obstruction to the existence of a singular concordance which has all singularities paired by Whitney disks. This geometric equivalence relation, called W-equivalence, is also related to finite type 1-equivalence (in the sense of Habiro and Goussarov) via the work of Conant and Teichner and represents a “first order” improvement to an arbitrary singular concordance. For null-homotopic knots, a slightly weaker equivalence relation is shown to admit a group structure.


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