Bennequin's inequality and the positivity of the signature

Abstract

We use an algorithm for special diagrams to prove a Bennequin type inequality for the signature of an arbitrary link diagram, related to its Murasugi sum decomposition. We apply this inequality to show that the signature of a non-trivial positive 3-braid knot is greater than its genus, and that the signature of a positive braid link is minorated by an increasing function of its negated Euler characteristic. The latter property is conjectured to extend to positive links.