Legendrian Large Cables And New Phenomenon For Non-Uniformly Thick Knots

TL;DR

This paper introduces the concept of Legendrian large cables in knot theory, revealing new phenomena in non-uniformly thick knots, including virtually overtwisted contact structures and a family of ribbon knots with these properties.

Contribution

It defines Legendrian large cables, demonstrates their implications for non-uniform thickness, and constructs an infinite family of ribbon knots exhibiting these phenomena.

Findings

Existence of non-uniformly thick knots with Legendrian large cables

Presence of solid tori with virtually overtwisted contact structures

Construction of an infinite family of ribbon knots with these properties

Abstract

We define the notion of a knot type having Legendrian large cables and show that having this property implies that the knot type is not uniformly thick. Moreover, there are solid tori in this knot type that do not thicken to a solid torus with integer sloped boundary torus, and that exhibit new phenomena; specifically, they have virtually overtwisted contact structures. We then show that there exists an infinite family of ribbon knots that have Legendrian large cables. These knots fail to be uniformly thick in several ways not previously seen. We also give a general construction of ribbon knots, and show when they give similar such examples.