Reduced Khovanov-Rozansky homology

We introduce a variant of reduced Khovanov-Rozansky homology defined in terms of the variables alpha_1, …, alpha_{n-1} that correspond to the regions between the strands. We rigorously reestablish several folklore results in terms of our framework, including the equivalence between Khovanov’s and Rasmussen’s approaches. Furthermore, we show that Rasmussen’s chain complex is a free resolution of the Rouquier chain complex.