Computing a Link Diagram from its Exterior

TL;DR

This paper presents the first practical algorithm to reconstruct knot diagrams from the triangulated exterior of the knot, enabling analysis of complex links and applications in topology.

Contribution

It introduces a novel algorithm that converts triangulations of knot exteriors into planar diagrams, applicable to complex links and large datasets.

Findings

Successfully recovered diagrams for links with hundreds of crossings

Generated the first diagrams for 23 principal congruence arithmetic link exteriors

Identified knot pairs with identical 0-surgeries, impacting topology questions

Abstract

A knot is a circle piecewise-linearly embedded into the 3-sphere. The topology of a knot is intimately related to that of its exterior, which is the complement of an open regular neighborhood of the knot. Knots are typically encoded by planar diagrams, whereas their exteriors, which are compact 3-manifolds with torus boundary, are encoded by triangulations. Here, we give the first practical algorithm for finding a diagram of a knot given a triangulation of its exterior. Our method applies to links as well as knots, allows us to recover links with hundreds of crossings. We use it to find the first diagrams known for 23 principal congruence arithmetic link exteriors; the largest has over 2,500 crossings. Other applications include finding pairs of knots with the same 0-surgery, which relates to questions about slice knots and the smooth 4D Poincar\'e conjecture.