Link diagrams vs. hyperbolic volume of the complement: the alternating case

6 November 2013
Antonio de Capua
A large class of links in $S^3$ has the property that the complement admits a complete hyperbolic metric of finite volume. But is this volume understandable from the link itself, or maybe from some nice diagram of it? Marc Lackenby in the early 2000s gave a positive answer for a class of diagrams, the alternating ones. The proof of this result involves an analysis of the JSJ decomposition of the link complement: in particular of how does it appear on the link diagram. I will tell you an outline of this proof, forgetting its most technical aspects and explaining the underlying ideas in an accessible way.