Complex quantum coupling phenomena of halide perovskites are examined through ab-initio calculations and exact diagonalization of model Hamiltonians to formulate a set of fundamental guiding rules to engineer the bandgap through strain. The bandgap tuning in halides is crucial for photovoltaic applications and for establishing non-trivial electronic states. Using CsSnI$_3$ as the prototype material, we show that in the cubic phase, the bandgap reduces irrespective of the nature of strain. However, for the tetragonal phase, it reduces with tensile strain and increases with compressive strain, while the reverse is the case for the orthorhombic phase. The reduction can give rise to negative bandgap in the cubic and tetragonal phases leading to normal to topological insulator phase transition. Also, these halides tend to form a stability plateau in a space spanned by strain and octahedral rotation. In this plateau, with negligible cost to the total energy, the bandgap can be varied in a range of 1eV. Furthermore, we present a descriptor model for the perovskite to simulate their bandgap with strain and rotation. Analysis of band topology through model Hamiltonians led to the conceptualization of topological influencers that provide a quantitative measure of the contribution of each chemical bonding towards establishing a normal or topological insulator phase. On the technical aspect, we show that a four orbital based basis set (Sn-${s,p}$ for CsSnI$_3$) is sufficient to construct the model Hamiltonian which can explain the electronic structure of each polymorph of halide perovskites.
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