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 –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.
"Algebraic linking numbers of knots in 3–manifolds." Algebr. Geom. Topol. 3 (2) 921 - 968, 2003. https://doi.org/10.2140/agt.2003.3.921