Linking Numbers in Three-Manifolds

Patricia Cahn, Alexandra Kjuchukova

Published 2016-11-30Version 1

We give an explicit algorithm for computing linking numbers between curves in an irregular dihedral $p$-fold branched cover of $S^3$. This work extends a combinatorial algorithm by Perko which computes the linking number between the branch curves in the case $p=3$. Owing to the fact that every closed oriented three-manifold is a dihedral three-fold branched cover of $S^3$, the algorithm given here can be used to compute linking numbers in any three-manifold, provided that the manifold is presented as a dihedral cover of the sphere. The algorithm has been implemented in Python, and we include the code in an Appendix.