A knot is a circle piecewiselinearly embedded into the 3sphere. 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 3manifolds 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 2500 crossings. Other applications include finding pairs of knots with the same 0surgery, which relates to questions about slice knots and the smooth 4D Poincaré conjecture.
Computing a Link Diagram from its Exterior
A knot is a circle piecewiselinearly embedded into the 3sphere. 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 3manifolds 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 0surgery, which relates to questions about slice knots and the smooth 4D Poincaré conjecture.
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 Award ID(s):
 1811156
 NSFPAR ID:
 10413365
 Editor(s):
 Goaoc, Xavier; Kerber, Michael
 Date Published:
 Journal Name:
 Leibniz international proceedings in informatics
 Volume:
 224
 ISSN:
 18688969
 Page Range / eLocation ID:
 37:137:24
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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