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A trivial knot whose spanning disks have exponential size

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Jack SnoeyinkAuthors Info & Claims

SCG '90: Proceedings of the sixth annual symposium on Computational geometry

Pages 139 - 147

https://doi.org/10.1145/98524.98555

Published: 01 May 1990 Publication History

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Abstract

If a closed curve in space is a trivial knot (intuitively, one can untie it without cutting) then it is the boundary of some disk with no self-intersections. In this paper we investigate the minimum number of faces of a polyhedral spanning disk of a polygonal knot with n segments. We exhibit a knot whose minimal spanning disk has exp(cn) faces, for some positive constant c.

References

[1]

M. A. Armstrong. Basic Topology. McGraw- Hill, London, 1979.

Google Scholar

[2]

H. S. M. Coxeter and W. O. J. Moser. Ge'nerators and Relations for Discrete Groups. Springer Verlag, third edition, 1972.

Google Scholar

[3]

Richard H. Crowell and Ralph H. Fox. Introduction to Knot Theory. Blaisdell Publishing Co., New York, 1965.

Google Scholar

[4]

Mike Fellows. Problem Corner: 6. The complexity of knot triviality and the Arf invariant. In Graphs and Algorithms: Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference, volume 89 of Contemporary Mathematics~ pages 189-190. American Mathematical Society, 1989.

Google Scholar

[5]

Wolfgang Haken. Theorie der Normalfl~chen: Ein Isotopiekriterium fiir den Kreisknoten. Acta Mathematica, 105:245-375, 1961.

Crossref

Google Scholar

[6]

Dale Rolfsen. Knots and Links. Publish or Perish, Berkeley, 1976.

Google Scholar

[7]

C. P. Rourke and B. J. Sanderson. Introduction to Piecewise-Linear Topology. Springer Verlag, 1972.

Crossref

Google Scholar

[8]

Horst Schubert. Bestimmung der Primfaktorzerlegung yon Verkettungen. Mathematische Zeitschrifl, 76:116-148, 1961.

Crossref

Google Scholar

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Hass JLagarias JPippenger N(1997)The computational complexity of knot and link problemsProceedings 38th Annual Symposium on Foundations of Computer Science10.1109/SFCS.1997.646106(172-181)Online publication date: 1997
https://doi.org/10.1109/SFCS.1997.646106
Barequet GDickerson MEppstein DWhitesides S(1996)On triangulating three-dimensional polygonsProceedings of the twelfth annual symposium on Computational geometry10.1145/237218.237234(38-47)Online publication date: 1-May-1996
https://dl.acm.org/doi/10.1145/237218.237234
Welsh D(1993)The Complexity of KnotsQuo Vadis, Graph Theory? - A Source Book for Challenges and Directions10.1016/S0167-5060(08)70385-6(159-171)Online publication date: 1993
https://doi.org/10.1016/S0167-5060(08)70385-6

Index Terms

A trivial knot whose spanning disks have exponential size
1. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

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SCG '90: Proceedings of the sixth annual symposium on Computational geometry

May 1990

371 pages

ISBN:0897913620

DOI:10.1145/98524

Chairman:
Raimund Seidel

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

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Published: 01 May 1990

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SOCG: 6th Annual Conference on Computational Geometry

June 7 - 9, 1990

California, Berkley, USA

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Hass JLagarias JPippenger N(1997)The computational complexity of knot and link problemsProceedings 38th Annual Symposium on Foundations of Computer Science10.1109/SFCS.1997.646106(172-181)Online publication date: 1997
https://doi.org/10.1109/SFCS.1997.646106
Barequet GDickerson MEppstein DWhitesides S(1996)On triangulating three-dimensional polygonsProceedings of the twelfth annual symposium on Computational geometry10.1145/237218.237234(38-47)Online publication date: 1-May-1996
https://dl.acm.org/doi/10.1145/237218.237234
Welsh D(1993)The Complexity of KnotsQuo Vadis, Graph Theory? - A Source Book for Challenges and Directions10.1016/S0167-5060(08)70385-6(159-171)Online publication date: 1993
https://doi.org/10.1016/S0167-5060(08)70385-6

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Geometrically designed, variable knot regression splines

Multiple-knot and rational cubic beta-splines

Modifying a knot of B-spline curves

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Geometrically designed, variable knot regression splines

Multiple-knot and rational cubic beta-splines

Modifying a knot of B-spline curves

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