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2003 Algebraic linking numbers of knots in 3–manifolds
Rob Schneiderman
Algebr. Geom. Topol. 3(2): 921-968 (2003). DOI: 10.2140/agt.2003.3.921

Abstract

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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Rob Schneiderman. "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

Information

Received: 26 February 2003; Revised: 2 September 2003; Accepted: 5 September 2003; Published: 2003
First available in Project Euclid: 21 December 2017

zbMATH: 1039.57005
MathSciNet: MR2012959
Digital Object Identifier: 10.2140/agt.2003.3.921

Subjects:
Primary: 57M27
Secondary: 57M25, 57N10

Rights: Copyright © 2003 Mathematical Sciences Publishers

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