A sharp signature bound for positive four-braids

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.

12 pages, 6 figures, comments welcome! Accepted for publication in Q. J. Math

References in corpus (2)

• On the topological 4-genus of torus knots
• The signature of positive braids is linearly bounded by their first Betti number

Cited by in corpus (4)

• Signature and concordance of positive knots
• Taut foliations, braid positivity, and unknot detection
• On the signature of a positive braid
• Divide knots of maximal genus defect