Algorithms for Computing Invariants of Trisected Branched Covers

TL;DR

This paper introduces diagrammatic algorithms to compute invariants of 4-manifolds presented as branched covers, enabling automated calculations of complex topological properties from tri-plane diagrams.

Contribution

It provides new algorithms for computing group trisections, homology, and intersection forms of 4-manifolds from tri-plane diagrams, including applications to various branched covers.

Findings

Algorithms successfully applied to dihedral and cyclic covers of spun knots.

Automated computation of homotopy-ribbon obstructions for p-colorable knots.

Demonstrated effectiveness on examples like the Stevedore disk double.

Abstract

We give diagrammatic algorithms for computing the group trisection, homology groups, and intersection form of a closed, orientable, smooth 4-manifold, presented as a branched cover of a bridge-trisected surface in $\mathbb{S}^{4}$. The algorithm takes as input a tri-plane diagram, labelled with permutations according to the Wirtinger relations. We apply our algorithm to several examples, including dihedral and cyclic covers of spun knots, cyclic covers of Suciu's ribbon knots with the trefoil knot group, and an infinite family of irregular covers of the Stevedore disk double. As an application, we give a fully automated algorithm for computing Kjuchukova's homotopy-ribbon obstruction for a $p$-colorable knot, given an extension of that coloring over a ribbon surface in the 4-ball.