Rectangle condition and its applications

TL;DR

This paper introduces a rectangle condition for n-bridge decompositions of knots, showing it guarantees a minimum Hempel distance and helps construct specific alternating 3-bridge knots.

Contribution

It defines a new rectangle condition for n-bridge decompositions and demonstrates its implications for Hempel distance and knot construction.

Findings

Rectangle condition guarantees Hempel distance ≥ 2

Constructs a family of alternating 3-bridge knots

Uses modified train track argument for knot analysis

Abstract

In this paper, we define the rectangle condition on the bridge sphere for a $n$-bridge decomposition of a knot whose definition is analogous to the definition of the rectangle condition for Heegaard splittings of $3$-manifolds. We show that the satisfaction of the rectangle condition for a $n$-bridge decomposition can guarantee that the Hempel distance for the $n$-bridge decomposition is greater than or equal to $2$. In particular, we give an interesting family of alternating 3-bridge knots by using the rectangle condition and a modified train track argument.