Superbridge index of composite knots

TL;DR

This paper establishes an upper bound for the superbridge index of composite knots based on braid and bridge indices, revealing that the superbridge index can differ significantly from the sum of the indices of the summands.

Contribution

It introduces a new upper bound for the superbridge index of connected sums of knots, especially for torus knots, and highlights the potential for large differences compared to the sum of individual superbridge indices.

Findings

Upper bound of superbridge index in terms of braid index

Upper bound for torus knots using superbridge and bridge indices

Superbridge index difference can be arbitrarily large

Abstract

An upper bound of the superbridge index of the connected sum of two knots is given in terms of the braid index of the summands. Using this upper bound and minimal polygonal presentations, we give an upper bound in terms of the superbridge index and the bridge index of the summands when they are torus knots. In contrast to the fact that the difference between the sum of bridge indices of two knots and the bridge index of their connected sum is always one, the corresponding difference for the superbridge index can be arbitrarily large.