Computing the Matching Distance of 2-Parameter Persistence Modules from Critical Values

The exact computation of the matching distance for multi-parameter persistence modules is an active area of research in computational topology. Achieving an easily obtainable exact computation of this distance would permit multi-parameter persistent homology to be a viable option for data analysis. For this purpose, two approaches are currently available, limited to persistence with parameters from $\mathbb{R}^2$: authors of arXiv:1812.09085, arXiv:2111.10303 work in the discrete setting and apply the point-line duality; authors of arXiv:2210.16718, arXiv:2312.04201 work in the smooth setting while remaining in the primal plane. In this paper, we streamline the computation of the matching distance in the combinatorial setting while staying in the primal plane. In doing so, besides connecting results from the literature, we give explicit formulas for the switch points needed by all the available methods and we show that it is possible to avoid considering vertical and horizontal lines. For the latter, lines with slope 1 play an essential role.