  To any prime alternating link, we associate a collection of hyperbolic
right-angled ideal polyhedra by relating geometric, topological and
combinatorial methods to decompose the link complement. The sum of the
hyperbolic volumes of these polyhedra is a new geometric link invariant, which
we call the right-angled volume of the alternating link. We give an explicit
procedure to compute the right-angled volume from any alternating link diagram,
and prove that it is a new lower bound for the hyperbolic volume of the link.
