Algebraic & Geometric Topology

Algebraic linking numbers of knots in 3–manifolds

Rob Schneiderman

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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.

Article information

Algebr. Geom. Topol., Volume 3, Number 2 (2003), 921-968.

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 57N10: Topology of general 3-manifolds [See also 57Mxx] 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

concordance invariant knots linking number 3–manifold


Schneiderman, Rob. Algebraic linking numbers of knots in 3–manifolds. Algebr. Geom. Topol. 3 (2003), no. 2, 921--968. doi:10.2140/agt.2003.3.921.

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