Volume 13, issue 2 (2013)

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The link volume of $3$–manifolds

Yo’av Rieck and Yasushi Yamashita

Algebraic & Geometric Topology 13 (2013) 927–958

We view closed orientable 3–manifolds as covers of S3 branched over hyperbolic links. To a cover MpS3, of degree p and branched over a hyperbolic link L S3, we assign the complexity pVol(S3 L). We define an invariant of 3–manifolds, called the link volume and denoted by LinkVol(M), that assigns to a 3-manifold M the infimum of the complexities of all possible covers M S3, where the only constraint is that the branch set is a hyperbolic link. Thus the link volume measures how efficiently M can be represented as a cover of S3.

We study the basic properties of the link volume and related invariants, in particular observing that for any hyperbolic manifold M, Vol(M) is less than LinkVol(M). We prove a structure theorem that is similar to (and uses) the celebrated theorem of Jørgensen and Thurston. This leads us to conjecture that, generically, the link volume of a hyperbolic 3–manifold is much bigger than its volume.

Finally we prove that the link volumes of the manifolds obtained by Dehn filling a manifold with boundary tori are linearly bounded above in terms of the length of the continued fraction expansion of the filling curves.

$3$–manifolds, hyperbolic volume, branched covers, knots and links
Mathematical Subject Classification 2010
Primary: 57M12, 57M50
Secondary: 57M27
Received: 7 May 2012
Revised: 21 September 2012
Accepted: 26 October 2012
Published: 5 April 2013
Yo’av Rieck
Department of Mathematics
University of Arkansas
Fayetteville, AR 72701
Yasushi Yamashita
Department of Information and Computer Sciences
Nara Women’s University
Nara 630-8506