Given a hyperbolic 3–manifold with torus boundary, we bound the change in volume under a Dehn filling where all slopes have length at least 2π. This result is applied to give explicit diagrammatic bounds on the volumes of many knots and links, as well as their Dehn fillings and branched covers. Finally, we use this result to bound the volumes of knots in terms of the coefficients of their Jones polynomials.
"Dehn filling, volume, and the Jones polynomial." J. Differential Geom. 78 (3) 429 - 464, March 2008. https://doi.org/10.4310/jdg/1207834551