#### 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
##### Abstract

We view closed orientable $3$–manifolds as covers of ${S}^{3}$ branched over hyperbolic links. To a cover $M\stackrel{p}{\to }{S}^{3}$, of degree $p$ and branched over a hyperbolic link $L\subset {S}^{3}$, we assign the complexity $pVol\left({S}^{3}\setminus L\right)$. We define an invariant of $3$–manifolds, called the link volume and denoted by $LinkVol\left(M\right)$, that assigns to a 3-manifold $M$ the infimum of the complexities of all possible covers $M\to {S}^{3}$, 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 ${S}^{3}$.

We study the basic properties of the link volume and related invariants, in particular observing that for any hyperbolic manifold $M$, $Vol\left(M\right)$ is less than $LinkVol\left(M\right)$. 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.

##### Keywords
$3$–manifolds, hyperbolic volume, branched covers, knots and links
##### Mathematical Subject Classification 2010
Primary: 57M12, 57M50
Secondary: 57M27