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 sphericalknots, 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
–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.
