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Formalising Knot Theory in Isabelle/HOL

  • T. V. H. PrathameshEmail author
Conference paper
  • 618 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9236)

Abstract

This paper describes a formalization of some topics in knot theory. The formalization was carried out in the interactive proof assistant, Isabelle. The concepts that were formalized include definitions of tangles, links, framed links and various forms of equivalences between them. The formalization is based on a formulation of links in terms of tangles. We further construct and prove the invariance of the Bracket polynomial. Bracket polynomial is an invariant of framed links closely linked to the Jones polynomial. This is perhaps the first attempt to formalize any aspect of knot theory in an interactive proof assistant.

Keywords

Formalization of mathematics Knot theory Kauffman bracket Bracket polynomial 

Notes

Acknowledgments

I would like to thank Siddhartha Gadgil for conceptualising and supervising the project, apart from his many other invaluable suggestions.

References

  1. 1.
    Sawin, S.: Links, quantum groups and TQFTs. Bull. Amer. Math. Soc. (N.S.) 33, 413–445 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Sternagel, C., Thiemann, R.: Executable Matrix Operations on Matrices of Arbitrary Dimensions, Archive of Formal Proofs (2010). http://afp.sf.net/entries/Matrix.shtml
  3. 3.
    Kauffman, L.H.: On Knots. Princeton University Press, Princeton (1987)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia

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