Volume, twist number, and the Jones polynomial of hyperbolic knots

I will describe a few results that relate combinatorial data about a knot or link projection to geometric and topological information about the link complement. For a large family of knots and links, one single piece of diagrammatic data (the twist number) coarsely determines the volume of the link and several coefficients of its Jones polynomial, with explicit constants. The volume estimates rely on a new result about the change in volume under Dehn surgery. This is joint work with Effie Kalfagianni and Jessica Purcell.