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Guts, volume and skein modules of 3-manifolds

Volume 256 / 2022

Brandon Bavier, Efstratia Kalfagianni Fundamenta Mathematicae 256 (2022), 195-220 MSC: Primary 57K10; Secondary 57K14, 57K31, 57K32. DOI: 10.4064/fm996-1-2021 Published online: 7 June 2021

Abstract

We consider hyperbolic links that admit alternating projections on surfaces in compact, irreducible 3-manifolds. We show that, under some mild hypotheses, the volume of the complement of such a link is bounded below in terms of a Kauffman bracket function defined on link diagrams on the surface.

In the case that the 3-manifold is a thickened surface, this Kauffman bracket function leads to a Jones-type polynomial that is an isotopy invariant of links. We show that coefficients of this polynomial provide 2-sided linear bounds on the volume of hyperbolic alternating links in the thickened surface. As a corollary of the proof of this result, we deduce that the twist number of a reduced, twist-reduced, alternating link projection with checkerboard disk regions is an invariant of the link.

Authors

  • Brandon BavierDepartment of Mathematics
    Michigan State University
    East Lansing, MI 48824, U.S.A.
    e-mail
  • Efstratia KalfagianniDepartment of Mathematics
    Michigan State University
    East Lansing, MI 48824, U.S.A.
    e-mail

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