Algebraic and Geometric Topology 3 (2003),
paper no. 31, pages 921-968.
Algebraic linking numbers of knots in 3-manifolds
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.
Concordance invariant, knots, linking number, 3--manifold
AMS subject classification.
Secondary: 57N10, 57M25.
Submitted: 26 February 2003.
(Revised: 17 July and 2 September 2003.)
Accepted: 5 September 2003.
Published: 2 October 2003.
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