Non-minimal bridge positions of torus knots are stabilized

TL;DR

The paper proves that all non-minimal bridge decompositions of torus knots are stabilized and establishes the uniqueness of n-bridge decompositions for these knots, providing a characterization of torus knots in bridge positions.

Contribution

It demonstrates that non-minimal bridge decompositions of torus knots are stabilized and proves the uniqueness of n-bridge decompositions for any integer n.

Findings

Non-minimal bridge decompositions are stabilized

n-bridge decompositions are unique for torus knots

Characterization of torus knots via bridge positions

Abstract

We show that any non-minimal bridge decomposition of a torus knot is stabilized and that $n$-bridge decompositions of a torus knot are unique for any integer $n$. This implies that a knot in a bridge position is a torus knot if and only if there exists a torus containing the knot such that it intersects the bridge sphere in two essential loops.