Asymptotic Rasmussen Invariant

We use simple properties of the Rasmussen invariant of knots to study its asymptotic behaviour on the orbits of a smooth volume preserving vector field on a compact domain in the 3-space. A comparison with the asymptotic signature allows us to prove that asymptotic knots are non-alternating, in general. Further we show that the Rasmussen invariant defines a quasi-morphism on the braid groups and derive estimates for the stable commutator and torsion lengths of alternating braids.Comment: 12 pages, 5 figure

Positive braids of maximal signature

We characterise positive braid links with positive Seifert form via a finite number of forbidden minors. From this we deduce a one-to-one correspondence between prime positive braid links with positive Seifert form and simply laced Dynkin diagrams, as well as a simple classification of alternating positive braid knots.Comment: 7 pages, 7 figure

Scissor equivalence for torus links

This article is about a natural distance function induced by smooth cobordisms between links. We show that the cobordism distance of torus links is determined by the profiles of their signature functions, up to a constant factor.Comment: 15 pages, 5 figures, Theorem 1 adde