Towards formalization of the soliton counting technique for the Khovanov-Rozansky invariants in the deformed $\mathcal{R}$-matrix approach

Abstract

We consider recently developed Cohomological Field Theory soliton counting diagram technique for Khovanov and Khovanov-Rozansky invariants [1,2]. Although the expectation to obtain a new way for computing the invariants has not yet come true, we demonstrate that soliton counting technique can be totally formalized at an intermediate stage, at least in particular cases. We present the corresponding algorithm, based on the approach involving deformed $\mathcal{R}$-matrix and minimal positive division, developed previously in [3]. We start from a detailed review of the minimal positive division approach, comparing it with other methods, including the rigorous mathematical treatment [4]. Pieces of data obtained within our approach are presented in the Appendices.