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The Braid Approach to the HOMFLYPT Skein Module of the Lens Spaces L(p, 1)

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Algebraic Modeling of Topological and Computational Structures and Applications (AlModTopCom 2015)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 219))

Abstract

In this paper we present recent results toward the computation of the HOMFLYPT skein module of the lens spaces L(p, 1), \(\mathcal {S}\left( L(p,1) \right) \), via braids. Our starting point is the knot theory of the solid torus ST and the Lambropoulou invariant, X, for knots and links in ST, the universal analogue of the HOMFLYPT polynomial in ST. The relation between \(\mathcal {S}\left( L(p,1) \right) \) and \(\mathcal {S}(\mathrm{ST})\) is established in Diamantis et al. (J Knot Theory Ramif, 25:13, 2016, [5]) and it is shown that in order to compute \(\mathcal {S}\left( L(p,1) \right) \), it suffices to solve an infinite system of equations obtained by performing all possible braid band moves on elements in the basis of \(\mathcal {S}(\mathrm{ST})\), \(\varLambda \), presented in Diamantis and Lambropoulou (J Pure Appl Algebra, 220(2):577–605, 2016, [4]). The solution of this infinite system of equations is very technical and is the subject of a sequel work (Diamantis and Lambropoulou, The HOMFLYPT skein module of the lens spaces L(p, 1) via braids, in preparation, [2]).

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Authors and Affiliations

  1. International College Beijing, China Agricultural University, No.17 Qinghua East Road, Haidian District, Beijing, 100083, People’s Republic of China

    Ioannis Diamantis

  2. Departament of Applied Mathematics, National Technical University of Athens, Zografou Campus, 15780, Athens, Greece

    Sofia Lambropoulou

Authors
  1. Ioannis Diamantis
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  2. Sofia Lambropoulou
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Correspondence to Sofia Lambropoulou .

Editor information

Editors and Affiliations

  1. Department of Applied Mathematics, National Technical University of Athens, Athens, Greece

    Sofia Lambropoulou

  2. School of Chemical Engineering, National Technical University of Athens, Athens, Greece

    Doros Theodorou

  3. Department of Applied Mathematics, National Technical University of Athens, Athens, Greece

    Petros Stefaneas

  4. Department of Mathematics, University of Illinois at Chicago, Chicago, Illinois, USA

    Louis H. Kauffman

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Diamantis, I., Lambropoulou, S. (2017). The Braid Approach to the HOMFLYPT Skein Module of the Lens Spaces L(p, 1). In: Lambropoulou, S., Theodorou, D., Stefaneas, P., Kauffman, L. (eds) Algebraic Modeling of Topological and Computational Structures and Applications. AlModTopCom 2015. Springer Proceedings in Mathematics & Statistics, vol 219. Springer, Cham. https://doi.org/10.1007/978-3-319-68103-0_7

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