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

TL;DR

This paper formalizes a soliton counting technique for Khovanov-Rozansky invariants using a deformed R-matrix approach, providing an algorithm and detailed comparisons with other methods, though it does not yet produce new invariants.

Contribution

It develops a formalized algorithm for soliton counting in Khovanov-Rozansky invariants based on deformed R-matrices and minimal positive division, clarifying the method's theoretical foundation.

Findings

The soliton counting technique can be formalized at an intermediate stage.

An explicit algorithm based on deformed R-matrix and minimal positive division is presented.

Comparison with other methods and mathematical treatments is provided.

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.