Volume bounds for generalized twisted torus links

Abstract

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.

Revised version to appear in Mathematical Research Letters. 21 pages, 14 figures

References in corpus (3)

• A new twist on Lorenz links
• Symmetric links and Conway sums: volume and Jones polynomial
• Cusp areas of Farey manifolds and applications to knot theory

Cited by in corpus (10)

• Cosmetic crossings of twisted knots
• Unexpected Essential Surfaces among Exteriors of Twisted Torus Knots
• Fully Augmented Links in the Thickened Torus
• Alternating links on surfaces and volume bounds
• Simplicial volume of links from link diagrams
• Twist positivity, L-space knots, and concordance
• Modular links: Bunch algorithm and upper volume bounds
• Intercusp Geodesics and Cusp Shapes of Fully Augmented Links
• Bridge spectra of twisted torus knots
• Independence of volume and genus $g$ bridge numbers