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.