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https://doi.org/10.1007/s00454-021-00287-3

Cahn, Patricia; Kjuchukova, Alexandra (July 2021, Discrete & Computational Geometry)

Abstract LetMbe a connected, closed, oriented three-manifold andK, Ltwo rationally null-homologous oriented simple closed curves in M. We give an explicit algorithm for computing the linking number betweenKandLin terms of a presentation of Mas an irregular dihedral three-fold cover of $$S^3$$

S^{3}

branched along a knot$$\alpha \subset S^3$$

α \subset 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.

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