A sharp signature bound for positive four-braids

TL;DR

This paper establishes the optimal linear bound for the signature of positive four-braids based on the three-genus of their closures, improving previous bounds for positive braid links.

Contribution

It introduces the sharp linear bound for positive four-braids' signature, combining bounds for positive three-braids with Gordon and Litherland's approach.

Findings

Optimal linear signature bound for positive four-braids.

Improved bounds for positive braid links in terms of first Betti number.

Examples demonstrating sharpness of the bounds.

Abstract

We provide the optimal linear bound for the signature of positive four-braids in terms of the three-genus of their closures. As a consequence, we improve previously known linear bounds for the signature in terms of the first Betti number for all positive braid links. We obtain our results by combining bounds for positive three-braids with Gordon and Litherland's approach to signature via unoriented surfaces and their Goeritz forms. Examples of families of positive four-braids for which the bounds are sharp are provided.