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Formalising Knot Theory in Isabelle/HOL

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Interactive Theorem Proving (ITP 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9236))

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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.

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References

  1. Sawin, S.: Links, quantum groups and TQFTs. Bull. Amer. Math. Soc. (N.S.) 33, 413–445 (1996)

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  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. Kauffman, L.H.: On Knots. Princeton University Press, Princeton (1987)

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Acknowledgments

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

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Science, Bangalore, India

    T. V. H. Prathamesh

Authors
  1. T. V. H. Prathamesh
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Correspondence to T. V. H. Prathamesh .

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Editors and Affiliations

  1. Department of Informatics, King's College London, London, United Kingdom

    Christian Urban

  2. PLA University of Science and Technology, Nanjing, China

    Xingyuan Zhang

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© 2015 Springer International Publishing Switzerland

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Prathamesh, T.V.H. (2015). Formalising Knot Theory in Isabelle/HOL. In: Urban, C., Zhang, X. (eds) Interactive Theorem Proving. ITP 2015. Lecture Notes in Computer Science(), vol 9236. Springer, Cham. https://doi.org/10.1007/978-3-319-22102-1_29

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  • DOI: https://doi.org/10.1007/978-3-319-22102-1_29

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-22101-4

  • Online ISBN: 978-3-319-22102-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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