  The topological insulating phase results from inversion of the band gap due
to spin-orbit coupling at an odd number of time-reversal symmetric points. In
Bi$_2$Se$_3$, this inversion occurs at the $\Gamma$ point. For bulk
Bi$_2$Se$_3$, we have analyzed the effect of arbitrary strain on the $\Gamma$
point band gap using Density Functional Theory. By computing the band structure
both with and without spin-orbit interactions, we consider the effects of
strain on the gap via Coulombic interaction and spin-orbit interaction
separately. While compressive strain acts to decrease the Coulombic gap, it
also increases the strength of the spin-orbit interaction, increasing the
inverted gap. Comparison with Bi$_2$Te$_3$ supports the conclusion that effects
on both Coulombic and spin-orbit interactions are critical to understanding the
behavior of topological insulators under strain, and we propose that the
topological insulating phase can be effectively manipulated by inducing strain
through chemical substitution.
