Positive knots have negative signature

TL;DR

This paper proves that all nontrivial positive links, including Lorentz knots, have negative signature, confirming a longstanding conjecture and clarifying the relationship between positivity and signature in knot theory.

Contribution

It establishes the general result that nontrivial positive links have negative signature, resolving a folklore conjecture in knot theory.

Findings

Nontrivial Lorentz knots have negative signature.

Positive braids have positive signature.

Nontrivial positive links have negative signature.

Abstract

It was asked by J.Birman, Williams, and L.Rudolph whether nontrivial Lorentz knots have always positive signature. Lorentz knots are examples of positive braids (in our convention they have all crossings negative so they are negative links). It was shown by L.Rudolph that positive braids have positive signature (if they represent nontrivial links). K.Murasugi has shown that nontrivial, alternating, positive links have negative signature. Here we prove in general the old folklore conjecture that nontrivial positive links have negative signature.