State cycles which represent the canonical class of Lee's homology of a knot (Q409582)

In the context of the Khovanov homology of a knot, E. S. Lee developed her chain complex of a diagram (Lee's complex) which led to the introduction of the Rasmussen invariant \(s(K)\) of a knot \(K\). The author goes through the definition of Lee's complex in detail to spot certain cycles as representing basis elements of the \(0\)-dimensional homology of Lee's complex (which carries the non-trivial part of that homology), in particular in Theorem 3.6 (for general diagrams) and Corollary 6.1 (for negative diagrams). This leads to a formula for the Rasmussen invariant of a knot with negative diagram involving one of the fundamental state cycles. Moreover, a new proof of the sharper slice-Bennequin inequality for the Rasmussen invariant is obtained. A final example gives a new calculation of the Rasmussen invariant of the pretzel knot \(P(3,-5,-7)\), an interesting result because this knot does not permit a homogeneous diagram.