##
**The volume of hyperbolic alternating link complements, with an appendix by Ian Agol and Dylan Thurston.**
*(English)*
Zbl 1041.57002

The most natural geometric (and topological) invariant for a hyperbolic link is its volume. In the paper under review, an estimate of the volume is given for an interesting class of hyperbolic links, namely those admitting a connected prime alternating diagram. These links were proved to be hyperbolic (unless their diagram is the standard one for the \((2,n)\)-torus link) by W. Menasco [Topology 23, 37–44 (1984; Zbl 0525.57003)]. The estimate
\[
v_3(t(D)-2)/2\leq \text{Vol}({\mathbb S}^3\setminus K)\leq16v_3(t(D)-1)
\]
is given in terms of the combinatorial data of the diagram, more precisely of the twist number \(t(D)\), i.e., the number of sequences of twists in the diagram. Here \(v_3\) denotes the volume of a regular hyperbolic ideal tetrahedron.

To compute the upper bound, the author observes that performing Dehn surgery decreases the volume of manifolds. Thus sequences of twists can be removed from the link up to adding new unknotted components which leads to a new hyperbolic “augmented” link with larger volume. An explicit ideal triangulation for the augmented link is then given. Note that to compute the upper bound, the link is not required to be alternating.

To compute the lower bound, a result of Agol, giving a lower bound on the volume of a hyperbolic manifold \(M\) admitting an incompressible and \(\partial\)-incompressible surface \(S\) in terms of the Euler characteristic of its guts, is exploited. Recall that the “guts” are, roughly speaking, the hyperbolic part of a characteristic decomposition for \(M\) cut open along \(S\).

In the appendix, I. Agol and D. Thurston give an improved upper bound, i.e., \(Vol({\mathbb S}^3\setminus K)\leq10v_3(t(D)-1)\), and show that this bound is asymptotically sharp in the sense that there exists a sequence of alternating links \(K_i\) with associated prime alternating diagrams \(D_i\) such that \(\text{Vol}({\mathbb S}^3\setminus K_i)/10v_3t(D_i)\to1\) as \(i\to\infty\).

To compute the upper bound, the author observes that performing Dehn surgery decreases the volume of manifolds. Thus sequences of twists can be removed from the link up to adding new unknotted components which leads to a new hyperbolic “augmented” link with larger volume. An explicit ideal triangulation for the augmented link is then given. Note that to compute the upper bound, the link is not required to be alternating.

To compute the lower bound, a result of Agol, giving a lower bound on the volume of a hyperbolic manifold \(M\) admitting an incompressible and \(\partial\)-incompressible surface \(S\) in terms of the Euler characteristic of its guts, is exploited. Recall that the “guts” are, roughly speaking, the hyperbolic part of a characteristic decomposition for \(M\) cut open along \(S\).

In the appendix, I. Agol and D. Thurston give an improved upper bound, i.e., \(Vol({\mathbb S}^3\setminus K)\leq10v_3(t(D)-1)\), and show that this bound is asymptotically sharp in the sense that there exists a sequence of alternating links \(K_i\) with associated prime alternating diagrams \(D_i\) such that \(\text{Vol}({\mathbb S}^3\setminus K_i)/10v_3t(D_i)\to1\) as \(i\to\infty\).

Reviewer: Luisa Paoluzzi (Dijon)

### MSC:

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

57M50 | General geometric structures on low-dimensional manifolds |