Additivity of Bridge Number of Knots

TL;DR

This paper offers a new proof for Schubert's results on how the bridge number of satellite and composite knots relates to their components, demonstrating additivity and inequalities in knot theory.

Contribution

It provides a novel proof of existing theorems on bridge numbers, clarifying the additivity properties for satellite and composite knots.

Findings

Bridge number of satellite knot ≥ product of index and companion's bridge number

Bridge number of composite knot is one less than sum of component bridge numbers

New proof simplifies understanding of bridge number additivity in knots

Abstract

We provide a new proof of the following results of H. Schubert: If K is a satellite knot with companion J and pattern L that lies in a solid torus T in which it has index k, then the bridge numbers satisfy the following: 1) The bridge number of K is greater than or equal to the product of k and the bridge number of J; 2) If K is a composite knot (this is the case k = 1), then the bridge number of K is one less than the sum of the bridge numbers of J and L.