Mathematical Research Letters

Volume 18 (2011)

Number 6

Volume bounds for generalized twisted torus links

Pages: 1097 – 1120



Abhijit Champanerkar

David Futer

Ilya Kofman

Walter Neumann

Jessica S. Purcell


Twisted torus knots and links are given by twisting adjacent strands of a torus link. They are geometrically simple and contain many examples of the smallest volume hyperbolic knots. Many are also Lorenz links.

We study the geometry of twisted torus links and related generalizations. We determine upper bounds on their hyperbolic volumes that depend only on the number of strands being twisted. We exhibit a family of twisted torus knots for which this upper bound is sharp, and another family with volumes approaching infinity. Consequently, we show there exist twisted torus knots with arbitrarily large braid index and yet bounded volume.

Published 20 February 2012