Computing a Link Diagram From Its Exterior

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 2500 crossings. Other applications include finding pairs of knots with the same 0-surgery, which relates to questions about slice knots and the smooth 4D Poincaré conjecture.

[1]  K. Baker,et al.  Census L-space knots are braid positive, except for one that is not , 2022, 2203.12013.

[2]  James T. Allison,et al.  Systematic Enumeration and Identification of Unique Spatial Topologies of 3D Systems Using Spatial Graph Representations , 2021, Volume 3A: 47th Design Automation Conference (DAC).

[3]  Benjamin A. Burton,et al.  Hard Diagrams of the Unknot , 2021, Experimental Mathematics.

[4]  Ciprian Manolescu,et al.  From zero surgeries to candidates for exotic definite 4‐manifolds , 2021, Journal of the London Mathematical Society.

[5]  I. Dynnikov,et al.  Multiflypes of rectangular diagrams of links , 2020, 2009.02247.

[6]  Benjamin A. Burton.  The Next 350 Million Knots , 2020, SoCG.

[7]  Francesco Lin,et al.  Monopole Floer Homology, Eigenform Multiplicities, and the Seifert–Weber Dodecahedral Space , 2020, 2003.11165.

[8]  E. Flapan,et al.  Topological descriptions of protein folding , 2019, Proceedings of the National Academy of Sciences.

[9]  M. Baker,et al.  All Known Principal Congruence Links , 2019, 1902.04426.

[10]  N. Dunfield.  A census of exceptional Dehn fillings , 2018, Characters in Low-Dimensional Topology.

[11]  Jonathan Spreer,et al.  3-Manifold triangulations with small treewidth , 2018, SoCG.

[12]  D. Schütz.  A FAST ALGORITHM FOR CALCULATING S-INVARIANTS , 2018, Glasgow Mathematical Journal.

[13]  Martin Tancer,et al.  The Unbearable Hardness of Unknotting , 2018, SoCG.

[14]  Lisa Piccirillo.  The Conway knot is not slice , 2018, Annals of Mathematics.

[15]  M. Baker,et al.  All principal congruence link groups , 2018, Journal of Algebra.

[16]  P. Ozsváth,et al.  Bordered knot algebras with matchings , 2017, Quantum Topology.

[17]  Henry Segerman.  Connectivity of triangulations without degree one edges under 2-3 and 3-2 moves , 2016, 1605.09099.

[18]  Raphael Zentner,et al.  Integer homology 3-spheres admit irreducible representations in SL(2,C) , 2016, Duke Mathematical Journal.

[19]  J. Greene.  Alternating links and definite surfaces , 2015, 1511.06329.

[20]  J. Howie.  A characterisation of alternating knot exteriors , 2015, 1511.04945.

[21]  Greg Kuperberg,et al.  Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization , 2015, Pacific Journal of Mathematics.

[22]  Benjamin A. Burton.  The cusped hyperbolic census is complete , 2014, ArXiv.

[23]  Abhijit Champanerkar,et al.  The 500 simplest hyperbolic knots , 2013, 1307.4439.

[24]  J. Rubinstein,et al.  Inflations of ideal triangulations , 2013, 1302.6921.

[25]  Benjamin A. Burton.  Computational topology with Regina: Algorithms, heuristics and implementations , 2012, ArXiv.

[26]  Colin Adams,et al.  Triple Crossing Number of Knots and Links , 2012, 1207.7332.

[27]  Benjamin A. Burton.  The pachner graph and the simplification of 3-sphere triangulations , 2010, SoCG '11.

[28]  Benjamin A. Burton,et al.  The Weber-Seifert dodecahedral space is non-Haken , 2009, 0909.4625.

[29]  Robert E. Gompf,et al.  Man and machine thinking about the smooth 4-dimensional Poincaré conjecture , 2009, 0906.5177.

[30]  Sergei V Ivanov,et al.  The computational complexity of basic decision problems in 3-dimensional topology , 2008 .

[31]  D. Bar-Natan.  FAST KHOVANOV HOMOLOGY COMPUTATIONS , 2006, math/0606318.

[32]  J. Hyam Rubinstein,et al.  Layered-triangulations of 3-manifolds , 2006, math/0603601.

[33]  Saul Schleimer,et al.  SPHERE RECOGNITION LIES IN NP , 2004, math/0407047.

[34]  Stephan Tillmann.  Normal surfaces in topologically finite 3-manifolds , 2004, math/0406271.

[35]  D. Rolfsen.  Knots and Links , 2003 .

[36]  J. Weeks.  Computation of Hyperbolic Structures in Knot Theory , 2003, math/0309407.

[37]  A. Mijatović.  Simplifying triangulations of S3 , 2003 .

[38]  I. Dynnikov.  Arc-presentations of links. Monotonic simplification , 2002, math/0208153.

[39]  J. Rubinstein,et al.  0-Efficient Triangulations of 3-Manifolds , 2002, math/0207158.

[40]  Johann A. Makowsky,et al.  Colored Tutte polynomials and Kaufman brackets for graphs of bounded tree width , 2001, SODA '01.

[41]  A. Mijatović.  Simplifying triangulations of S3 , 2000, math/0008107.

[42]  C. Adams.  Isometric cusps in hyperbolic 3-manifolds. , 1999 .

[43]  E. Sedgwick,et al.  Decision problems in the space of Dehn fillings , 1998, math/9811031.

[44]  J. Weeks,et al.  The first 1,701,936 knots , 1998 .

[45]  M. Thistlethwaite,et al.  The rate of growth of the number of prime alternating links and tangles , 1998 .

[46]  Jeffrey C. Lagarias,et al.  The computational complexity of knot and link problems , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[47]  Ronald C. Read,et al.  The knot book: An elementary introduction to the mathematical theory of knots , 1997, Complex..

[48]  J. Simon.  ENERGY FUNCTIONS FOR POLYGONAL KNOTS , 1994 .

[49]  J. Weeks.  Convex hulls and isometries of cusped hyperbolic 3-manifolds , 1993 .

[50]  Udo Pachner,et al.  P.L. Homeomorphic Manifolds are Equivalent by Elementary 5hellingst , 1991, Eur. J. Comb..

[51]  D. Welsh,et al.  On the computational complexity of the Jones and Tutte polynomials , 1990, Mathematical Proceedings of the Cambridge Philosophical Society.

[52]  John Luecke,et al.  Knots are determined by their complements , 1989 .

[53]  W. Haken.  Theorie der Normalflächen , 1961 .

[54]  H. Seifert,et al.  Die beiden Dodekaederräume , 1933 .

[55]  Sergei Matveev,et al.  Algorithmic Topology and Classification of 3-Manifolds , 2003 .

[56]  Stefan Schirra,et al.  Robustness and Precision Issues in Geometric Computation , 2000, Handbook of Computational Geometry.

[57]  Erica Flapan,et al.  When Topology Meets Chemistry: A Topological Look at Molecular Chirality , 2000 .

[58]  Jeffrey R. Weeks,et al.  Symmetries, Isometries and Length Spectra of Closed Hyperbolic Three-Manifolds , 1994, Exp. Math..

[59]  K. Murasugi.  On the braid index of alternating links , 1991 .

[60]  Jun O'Hara,et al.  Energy of a knot , 1991 .

[61]  Riccardo Piergallini,et al.  Standard moves for standard polyhedra and spines , 1988 .

[62]  Shinji Fukuhara,et al.  ENERGY OF A KNOT , 1988 .

[63]  Robert Myers.  Simple knots in compact, orientable 3-manifolds , 1982 .

[64]  W. Thurston.  The geometry and topology of 3-manifolds , 1979 .