A description of Rasmussen’s invariant from the divisibility of Lee’s canonical class

Abstract

We give a description of Rasmussen’s [Formula: see text]-invariant from the divisibility of Lee’s canonical class. More precisely, given any link diagram [Formula: see text], for any choice of an integral domain [Formula: see text] and a non-zero, non-invertible element [Formula: see text], we define the [Formula: see text]-divisibility [Formula: see text] of Lee’s canonical class of [Formula: see text], and prove that a combination of [Formula: see text] and some elementary properties of [Formula: see text] yields a link invariant [Formula: see text]. Each [Formula: see text] possesses properties similar to [Formula: see text], which in particular reproves the Milnor conjecture. If we restrict to knots and take [Formula: see text], then our invariant coincides with [Formula: see text].