  Relative self-linking and linking "numbers" for pairs of knots 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 cyclic (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 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 geometric equivalence relation is
shown to admit a group structure.
