Linking Numbers in Three-Manifolds

Abstract

Let M be a connected, closed, oriented three-manifold and K, L two rationally null-homologous oriented simple closed curves in M. We give an explicit algorithm for computing the linking number between K and L in terms of a presentation of M as an irregular dihedral three-fold cover of $$S^3$$ S 3 branched along a knot $$\alpha \subset S^3$$ α ⊂ S 3 . Since every closed, oriented three-manifold admits such a presentation, our results apply to all (well-defined) linking numbers in all three-manifolds. Furthermore, ribbon obstructions for a knot $$\alpha $$ α can be derived from dihedral covers of $$\alpha $$ α . The linking numbers we compute are necessary for evaluating one such obstruction. This work is a step toward testing potential counter-examples to the Slice-Ribbon Conjecture, among other applications.

Note on three-fold branched covers of S 4

“…It is possible to generalize the definition, allowing transversely immersed surfaces with any number of components, and with any number of singularities, to be put in bridge trisected position. We refer the reader to [3,4,20,21] for related discussions about singular branched coverings of bridge trisected surfaces. Definition 2.3.…”

“…In a diagram, each arc is colored by the image of its Wirtinger meridian. A Fox 3-coloring of a singular tri-plane diagram is defined in an analogous way, and it determines an irregular 3-fold cover of S 4 branched along a singular surface [3].…”

“…We use Fox 3-colored tri-plane diagrams to depict the branching sets and associated dihedral representations. This is sufficient to determine trisection diagrams of the corresponding irregular 3-fold covers [3]. to the given tri-plane diagram.…”

“…Hilden's proof leveraged the way in which Heegaard splittings of 3manifolds arise naturally as branched covers of bridge splittings of links. Trisections, which were introduced by Gay and Kirby as a 4-dimensional analog of Heegaard splittings [9], have a similar relationship with bridge trisections of knotted surfaces [24,25]; see also [3,20,21]. We extend Hilden's techniques to dimension four using trisections.…”

“…Families of 3-fold covers of S 4 by simply-connected manifolds where the branch sets are embedded with cone singularities are given in [2,3,4]. In these constructions, there is at most one singularity on each cover.…”