The least spanning area of a knot and the optimal bounding chain problem

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Two fundamental objects in knot theory are the minimal genus surface and the least area surface bounded by a knot in a 3-dimensional manifold. When the knot is embedded in a general 3-manifold, the problems of finding these surfaces were shown to be NP-complete and NPhard respectively. However, there is evidence that the special case when the ambient manifold is ℝ3, or more generally when the second homology is trivial, should be considerably more tractable. Indeed, we show here that a natural discrete version of the least area surface can be found in polynomial time. The precise setting is that the knot is a 1-dimensional subcomplex of a triangulation of the ambient 3-manifold. The main tool we use is a linear programming formulation of the Optimal Bounding Chain Problem (OBCP), where one is required to find the smallest norm chain with a given boundary. While the decision variant of OBCP is NP-complete in general, we give conditions under which it can be solved in polynomial time. We then show that the least area surface can be constructed from the optimal bounding chain using a standard desingularization argument from 3-dimensional topology. We also prove that the related Optimal Homologous Chain Problem is NP-complete for homology with integer coefficients, complementing the corresponding result of Chen and Freedman for mod 2 homology.

Original languageEnglish (US)
Title of host publicationProceedings of the 27th Annual Symposium on Computational Geometry, SCG'11
Pages135-144
Number of pages10
DOIs
StatePublished - 2011
Event27th Annual ACM Symposium on Computational Geometry, SCG'11 - Paris, France
Duration: Jun 13 2011Jun 15 2011

Publication series

NameProceedings of the Annual Symposium on Computational Geometry

Other

Other27th Annual ACM Symposium on Computational Geometry, SCG'11
CountryFrance
CityParis
Period6/13/116/15/11

Keywords

  • Homology
  • Knot genus
  • Linear programming
  • Minimal surface
  • NP-complete
  • Topology

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Computational Mathematics

Fingerprint Dive into the research topics of 'The least spanning area of a knot and the optimal bounding chain problem'. Together they form a unique fingerprint.

Cite this