Classifying divisor topologies for string phenomenology

In this article we present a pheno-inspired classification for the divisor topologies of the favorable Calabi Yau (CY) threefolds with $1 \leq h^{1,1}(CY) \leq 5$ arising from the four-dimensional reflexive polytopes of the Kreuzer-Skarke database. Based on some empirical observations we conjecture that the topologies of the so-called coordinate divisors can be classified into two categories: (i). $\chi_{_h}(D) \geq 1$ with Hodge numbers given by $\{h^{0,0} = 1, \, h^{1,0} = 0, \, h^{2,0} = \chi_{_h}(D) -1, \, h^{1,1} = \chi(D) - 2 \chi_{_h}(D) \}$ and (ii). $\chi_{_h}(D) \leq 1$ with Hodge numbers given by $\{h^{0,0} = 1, \, h^{1,0} = 1 - \chi_{_h}(D), \, h^{2,0} = 0, \, h^{1,1} = \chi(D) + 2 - 4 \chi_{_h}(D)\}$, where $\chi_{_h}(D)$ denotes the Arithmetic genus while $\chi(D)$ denotes the Euler characteristic of the divisor $D$. We present the Hodge numbers of around 140000 coordinate divisors corresponding to all the CY threefolds with $1 \leq h^{1,1}(CY) \leq 5$ which corresponds to a total of nearly 16000 distinct CY geometries. Subsequently we argue that our conjecture can help in ``bypassing" the need of cohomCalg for computing Hodge numbers of coordinate divisors, and hence can be significantly useful for studying the divisor topologies of CY threefolds with higher $h^{1,1}$ for which cohomCalg gets too slow and sometimes even breaks as well. We also demonstrate how these scanning results can be directly used for phenomenological model building, e.g. in estimating the $D3$-brane tadpole charge (under reflection involutions) which is a central ingredient for constructing explicit global models due to several different reasons/interests such as the de-Sitter uplifting through anti-$D3$ brane and (flat) flux vacua searches.