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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.
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