Fillings of genus-1 open books and 4-braids

TL;DR

This paper explores the diversity of Stein fillings and quasipositive factorizations in genus-1 open books and 4-braids, revealing infinite families with specific topological properties and minimal examples.

Contribution

It demonstrates the existence of contact 3-manifolds with genus-1 open books that have infinitely many Stein fillings with distinct homology, but only finitely many for genus-zero cases, and constructs 4-braids with infinitely many quasipositive factorizations.

Findings

Existence of contact 3-manifolds with infinitely many Stein fillings of support genus one.

Finiteness of homology possibilities for Stein fillings of genus-zero contact 3-manifolds.

Infinite family of knotted complex analytic annuli in the 4-ball from 4-strand braids.

Abstract

We show that there are contact 3-manifolds of support genus one which admit infinitely many Stein fillings, but do not admit arbitrarily large ones. These Stein fillings arise from genus-1 allowable Lefschetz fibrations with distinct homology groups, all filling a fixed minimal genus open book supporting the boundary contact 3-manifold. In contrast, we observe that there are only finitely many possibilities for the homology groups of Stein fillings of a given contact 3-manifold with support genus zero. We also show that there are 4-strand braids which admit infinitely many distinct Hurwitz classes of quasipositive factorizations, yielding in particular an infinite family of knotted complex analytic annuli in the 4-ball bounding the same transverse link up to transverse isotopy. These realize the smallest possible examples in terms of the number of boundary components a genus-1 mapping class and the number of strands a braid can have with infinitely many positive/quasipositive factorizations.