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Kauffman bracket skein module of the connected sum of handlebodies: a counterexample

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Abstract

In this paper we provide a counterexample to a 22-year-old theorem about the structure of the Kauffman bracket skein module of the connected sum of two handlebodies. We achieve this by analysing handle slidings on compressing discs in a handlebody. We find more relations than previously predicted for the Kauffman bracket skein module of the connected sum of handlebodies, when one of them is not a solid torus. Additionally, we speculate on the structure of the Kauffman bracket skein module of the connected sum of two solid tori.

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Notes

  1. We use Kauffman’s diagrammatic visualisation [6] of the Temperley–Lieb algebra \(TL_k\), where \(Id_k\) denotes the identity element and \(e_i\) denotes caps connecting the marked points \(x_i\) with \(x_{i+1}\) and \(x_{2k-i}\) with \(x_{2k-i+1}\).

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Acknowledgements

The second author was partially supported by Simons Collaboration Grant-637794 and the CCAS Enhanced Travel award. The authors would like to thank Charles Frohman due to whom they decided to provide a proof for Theorem 1.2 and ended up disproving it. We would also like to thank Thang T. Q. Lê for helpful comments. Added in Proof: Conjecture 5.1 has been proved by Thang T. Q. Lê and the authors of this paper.

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Authors and Affiliations

  1. Department of Mathematics, The George Washington University, Washington, DC, USA

    Rhea Palak Bakshi & Józef H. Przytycki

  2. Department of Mathematics, University of Gdańsk, Gdańsk, Poland

    Józef H. Przytycki

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  1. Rhea Palak Bakshi
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  2. Józef H. Przytycki
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Correspondence to Rhea Palak Bakshi.

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Bakshi, R.P., Przytycki, J.H. Kauffman bracket skein module of the connected sum of handlebodies: a counterexample. manuscripta math. 167, 809–820 (2022). https://doi.org/10.1007/s00229-021-01288-5

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