Distance two links

Abstract

In this paper, we characterize all links in S 3 with bridge number at least three that have a bridge sphere of distance two. We show that if a link L has a bridge sphere of distance at most two then it falls into at least one of three categories:• The exterior of L contains an essential meridional sphere.• L can be decomposed as a tangle product of a Montesinos tangle with an essential tangle in a way that respects the bridge surface and either the Montesinos tangle is rational or the essential tangle contains an incompressible, boundary-incompressible annulus. • L is obtained by banding from another link L that has a bridge sphere of the same Euler characteristic as the bridge sphere for L but of distance 0 or 1.This paper was written as part of an AIM SQuaREs project.

Alternating links have representativity 2

“…In particular, since the representativity of any (p, q) torus knot is r(T p,q ) = min{p, q} (more to the point and easier to check is r(T p,q ) ≥ min{p, q}), so that δ(T p,q ) → ∞ as p, q → ∞, Pardon was able to answer Gromov's question in the negative. Current work of Blair, Campisi, Taylor and Tomova [2] provides a lower bound for δ(K) in terms of distance and bridge numbers. Our main result implies that their lower bound improves Pardon's lower bound in the case of alternating knots with sufficiently large distance:…”

Knots and surfaces

“…The cabling conjecture was solved on satellite knots( [172]), strongly invertible knots ( [42]), alternating knots ( [120]), symmetric knots ( [82]), knots with bridge decomposing sphere of Hempel distance greater than or equal to 3 ([16]). In particular, since knots with Hempel distance of 2 are classified ( [17]), it would be one way for the solution to consider the cabling conjecture on this class. And if the manifold obtained by a Dehn surgery is a connected sum of lens spaces, then the cabling conjecture is true ( [65]).…”