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].

A Bar-Natan homotopy type

“…The following proof is a flow category level analogue of the argument given by Wehrli in [Weh08, Remark 5.4]. A more detailed argument can be found in [Lew09], [Tur20] and [San20].…”

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“…In the following, we omit the subscript (h, t) from A, CKh and Kh when there is no confusion. Lee's theory has an amazing property that, for any link diagram D, its Q-Lee homology has dimension 2 |D| , and its generators, called the canonical generators, are constructed explicitly from D. In [San20] we extended this construction to the generalized version of Khovanov homology. Consider the following condition for (R, h, t):…”