Distance and bridge position

Abstract

definition of the distance of a Heegaard surface generalizes to a notion of complexity for any knot that is in bridge position with respect to a Heegaard surface. Our main result is that the distance of a knot in bridge position is bounded above by twice the genus, plus the number of boundary components, of an essential surface in the knot complement. As a consequence knots constructed via sufficiently high powers of pseudoAnosov maps have minimal bridge presentations which are thin.

Distance of Heegaard splittings of knot complements

“…How surfaces in a manifold restrict the distance of a Heegaard splitting in various settings has been studied in several papers, including [Bachman and Schleimer 2005;Scharlemann and Tomova 2006a;Tomova 2007]. We will take advantage of some of these results.…”

Bridge number and tangle products

“…Now that there are no maxima of S between h −1 (a) and h −1 (b) we can similarly isotope all the maxima of K above h −1 (b) via an isotopy that fixes S and is supported on a neighborhood of the maxima of h K . Hence, we have achieved (1) in the definition of (K, S) bridge position while preserving the number of maxima of K and the number of saddles of S. By a symmetric argument, we can also achieve (2) in the definition of (K, S) bridge position while preserving the number of maxima of K and the number of saddles of S. After these isotopies, our choice of a and b guarantee that (3) in the definition of (K, S) bridge position is satisfied. Definition 2.4.…”

“…Bachman and Schleimer showed that twice the genus plus the number of boundary components of of an essential surface serves as an upper bound for the distance of any bridge sphere for a knot [1]. In other words, a high distance bridge sphere forces a high intrinsic complexity for any essential embedded surface in a knot complement.…”

A random link via bridge position is hyperbolic

“…Now we can state the following given by Bachman and Schleimer in [1]. On the other hand, in this paper, we consider not the arc and curve complex but the curve complex.…”

“…We will basically use the Bachman-Schleimer's criterion for a link to be hyperbolic obtained in [1]. To state the result in [1], we recall the following, which is a brief summery of [1,Section 2].…”