Volumes of knot complements

The complement of a knot or link is a 3-manifold which admits a geometric

structure. However, given a diagram of a knot or link, it seems to be a

difficult problem to determine geometric information about the link

complement. The volume is one piece of geometric information. For large

classes of knots and links with complement admitting a hyperbolic

structure, we show the volume of the link complement is bounded by the

number of twist regions of a diagram. We prove this result for a large

collection of knots and links using a theorem that estimates the change in

volume under Dehn filling. This is joint work with Effie Kalfagianni and

David Futer