Bridge Spectra of Twisted Torus Knots

Abstract

We compute the genus zero bridge numbers and give lower bounds on the genus one bridge numbers for a large class of sufficiently generic hyperbolic twisted torus knots. As a result, the bridge spectra of these knots have two gaps which can be chosen to be arbitrarily large, providing the first known examples of hyperbolic knots exhibiting this property. In addition, we show that there are Berge and Dean knots with arbitrarily large genus one bridge numbers, and as a result, we give solutions to problems of Eudave-Muñoz concerning tunnel number one knots.

Independence of volume and genus 𝑔 bridge numbers

“…Note that the case g = 0 follows from results on twisted torus knots in [5] and [6]. We give a concrete proof of this result, in which we build a sequence of knots and show they satisfy the requirements of the theorem.…”

“…In [4], the authors define a catching surface for the pair (R , K ). In the present work, we use a more specific definition from [5] which will suffice for our purposes. Lemma 3.4.…”

On the Knot Floer Homology of Twisted Torus Knots

“…Note that m is the number of intersection points in the genus one Heegaard diagram of K. It is assumed, fixingα a connected component of π −1 (α), that we need n connected components of π −1 (β) to obtain a complete list of all the m intersection points between α and β downstairs. (b) A portion of the Heegaard diagram for the (3,4) torus knot with two positive full twists on two adjacent strands, lifted to C. Note that the base points specified in the picture depicted above are the only relevant base points needed to compute CF K − . .…”

“…A(3,4) torus knot with two positive full twists on two adjacent strands. The one-bridge is indicated by τ .…”

Additional cases of positive twisted torus knots

“…Moriah and Sedgwick provide an infinite class of hyperbolic twisted torus knots with a unique minimal genus Heegaard splitting [10]. Bowman, Taylor and Zupan compute bridge numbers for a family of hyperbolic twisted torus knots and show that each knot in this family has two arbitrarily large gaps in its bridge spectrum [1]. Doleshal [4] provides a class of twisted torus knots that are fibered by proving the following theorem.…”