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Homotopy invariants of Gauss words

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Abstract

By defining combinatorial moves, we can define an equivalence relation on Gauss words called homotopy. In this paper we define a homotopy invariant of Gauss words. We use this to show that there exist Gauss words that are not homotopic to the empty Gauss word, disproving a conjecture by Turaev. In fact, we show that there are an infinite number of equivalence classes of Gauss words under homotopy.

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Authors and Affiliations

  1. Department of Mathematics, Tokyo Institute of Technology, Oh-okayama, Meguro, Tokyo, 152-8551, Japan

    Andrew Gibson

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  1. Andrew Gibson
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Correspondence to Andrew Gibson.

Additional information

This work was supported by a Scholarship from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

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Gibson, A. Homotopy invariants of Gauss words. Math. Ann. 349, 871–887 (2011). https://doi.org/10.1007/s00208-010-0536-0

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  • DOI: https://doi.org/10.1007/s00208-010-0536-0

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