Bridge number and tangle products

TL;DR

This paper investigates the complexity of punctured spheres in link complements, introduces tangle products as a generalization of link operations, and establishes a lower bound on the bridge number of tangle products based on their factors.

Contribution

It defines tangle products and proves a lower bound on their bridge number, extending understanding of link complexity and operations.

Findings

Essential punctured spheres have bounded complexity in certain link complements.

Tangle product operation generalizes connected sum and Conway product.

Bridge number of tangle product is at least the sum of the factors' bridge numbers, up to a constant.

Abstract

We show that essential punctured spheres in the complement of links with distance three bridge spheres have bounded complexity. We define the operation of tangle product, a generalization of both connected sum and Conway product. Finally, we use the bounded complexity of essential punctured spheres to show that the bridge number of a tangle product is at least the sum of the bridge numbers of the two factor links up to a constant error.