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].

Khovanov homology is functorial up to sign with respect to link cobordisms. The sign indeterminacy has been fixed by several authors, by extending the original theory both conceptually and algebraically. In this paper, we propose an alternative approach: we stay in the classical setup and fix the functoriality by simply adjusting the signs of the morphisms associated to the Reidemeister moves and the Morse moves.

A spatial refinement of Bar-Natan homology is given, that is, for any link diagram [Formula: see text] we construct a CW-spectrum [Formula: see text] whose reduced cellular cochain complex gives the Bar-Natan complex of [Formula: see text]. The stable homotopy type of [Formula: see text] is a link invariant and is described as the wedge sum of the “canonical sphere spectra”. We conjecture that the quantum filtration of Bar-Natan homology also lifts to the spatial level, and that it leads us to a cohomotopical refinement of the [Formula: see text]-invariant.

We give an algorithm to compute the reduced HOMFLY homology for knots. We determine the homologies for 695 prime knots, including all prime knots with up to 10 crossings, and all prime knots with 11 crossings and braid length up to 13.

Following Lipshitz-Sarkar's construction of Khovanov homotopy type, we construct for any link diagram L a CW spectrum XBN (L) whose reduced cellular cochain complex gives the Bar-Natan complex of L. We prove that XBN (L) is stably homotopy equivalent to the wedge sum of its canonical cells.