Volume 16, issue 6 (2016)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 2, 645–1281
Issue 1, 1–643

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Author Index
To Appear
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

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.)
link volume, hyperbolic volume, cosmetic surgery, Dehn surgery, 3–manifolds, hyperbolic manifolds, branched covering
Mathematical Subject Classification 2010
Primary: 57M12, 57M50
Received: 1 October 2015
Revised: 24 February 2016
Accepted: 27 March 2016
Published: 15 December 2016
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