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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Cosmetic surgery and the link volume of hyperbolic $3$–manifolds

Yo’av Rieck and Yasushi Yamashita

Algebraic & Geometric Topology 16 (2016) 3445–3521
Abstract

We prove that for any V > 0 there exists a hyperbolic manifold MV such that Vol(MV ) < 2.03 and LinkVol(MV ) > V . This was conjectured by the authors in [Algebr. Geom. Topol. 13 (2013) 927–958, Conjecture 1.3].

The proof requires study of cosmetic surgery on links (equivalently, fillings of manifolds with boundary tori). There is no bound on the number of components of the link (or boundary components). For statements, see the second part of the introduction. Here are two examples of the results we obtain:

  1. Let K be a component of a link L in S3. Then “most” slopes on K cannot be completed to a cosmetic surgery on L, unless K becomes a component of a Hopf link.
  2. Let X be a manifold and ϵ > 0. Then all but finitely many hyperbolic manifolds obtained by filling X admit a geodesic shorter than ϵ. (Note that it is not true that there are only finitely many fillings fulfilling this condition.)
Keywords
link volume, hyperbolic volume, cosmetic surgery, Dehn surgery, 3–manifolds, hyperbolic manifolds, branched covering
Mathematical Subject Classification 2010
Primary: 57M12, 57M50
References
Publication
Received: 1 October 2015
Revised: 24 February 2016
Accepted: 27 March 2016
Published: 15 December 2016
Authors
Yo’av Rieck
Department of Mathematics
University of Arkansas
Fayetteville, AR 72701
United States
Yasushi Yamashita
Department of Information and Computer Sciences
Nara Women’s University Kitauoya
Nishimachi, Nara 630-0586
Japan