We show that for an immersed two-sided minimal surface in R^3,
there is a lower bound on the index depending on the genus and number of
ends. Using this, we show the nonexistence of an embedded minimal surface
in R^3 of index 2, as conjectured by Choe. Moreover, we show that the
index of an immersed two-sided minimal surface with embedded ends is
bounded from above and below by a linear function of the total curvature
of the surface. (This is joint work with Otis Chodosh)