Volume bounds for generalized twisted torus links

TL;DR

This paper investigates the hyperbolic volume bounds of twisted torus links, establishing upper bounds based on strand number, and demonstrates families with unbounded volume and arbitrarily large braid index.

Contribution

It provides new upper bounds on volumes of twisted torus links depending only on the number of strands, and constructs examples with large braid index but bounded volume.

Findings

Upper bounds on hyperbolic volumes depending only on strand count

Existence of twisted torus knots with arbitrarily large volume

Families with volume approaching infinity

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.