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, and 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
Poincare conjecture. This is joint work with Cameron Rudd and Malik Obeidin. Based on: arXiv:2112.03251.
