[Submitted on 6 Dec 2021 (v1), last revised 16 Mar 2023 (this version, v3)]
Download the PDF of the paper entitled Computing a Link Diagram from its Exterior by Nathan M. Dunfield and two other authors
overview: A knot is a circle that is piecewise linearly embedded in a three-sphere. A knot’s topology is closely related to its outer topology. This complements the open regular neighborhood of knots. A knot is usually encoded by a plane diagram, but its exterior, a compact 3-manifold with a torus boundary, is encoded by a triangulation. Here we present the first practical algorithm for finding a knot figure given a knot contour triangulation. Our method applies to links as well as knots, and can recover links at hundreds of intersections. We use this to find the first figure known for the 23 principal congruential arithmetic link exteriors. The largest has over 2,500 intersections. Other applications include finding pairs of knots with the same 0-surgery, related to questions about sliced knots and smooth his 4D Poincaré conjecture.
From: Nathan M. Dunfield [view email]
Mon, Dec 6, 2021 18:55:25 UTC (3,998 KB)
Saturday, March 26, 2022 21:50:06 UTC (4,271 KB)
Thursday, March 16, 2023 21:28:57 UTC (4,100 KB)