Bridge number and twist equivalence of Seifert surfaces

TL;DR

This paper introduces a new method using bridge spheres to distinguish twist equivalence classes of Seifert surfaces, establishing bounds on boundary link bridge numbers and demonstrating infinitely many classes for genus one surfaces.

Contribution

The authors develop a novel approach employing bridge spheres to classify twist equivalence of Seifert surfaces and prove the existence of infinitely many classes for certain surfaces.

Findings

Bound on boundary link bridge numbers depending only on the Seifert surface

Inequality relating link bridge number to genus and number of components

Existence of infinitely many twist classes of genus one Seifert surfaces

Abstract

Two Seifert surfaces of links in $S^3$ are said to be twist equivalent if one can be obtained from the other, up to isotopy, by repeatedly performing operations consisting of cutting along an embedded arc, applying a full twist near one copy of the arc, and re-gluing. By using bridge spheres for their boundary links, we provide a new method of distinguishing twist equivalence classes of Seifert surfaces of any given genus. Given a Seifert surface $F$ of a link $L$, we show that the bridge numbers of the boundary links of Seifert surfaces twist equivalent to $F$ are uniformly bounded above by a constant depending only on $F$. By computing this constant in one simple case and applying a result of Pfeuti, we deduce that $b(L)\leq 2g_c(L)+|L|$ for any link $L$ in $S^3$, where $g_c(L)$ denotes the canonical genus of $L$. Various consequences of this inequality are discussed. We then apply the main theorem to show that there are infinitely many distinct twist equivalence classes represented by free, genus one Seifert surfaces, answering a question of Pfeuti in the negative.