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.Comment: 34 pages, 29 figures; V2 minor changes incorporating referees'
comments; to appear in "Proceedings of the 38th International Symposium on
Computational Geometry (SoCG 2022)
