Complexity of torus bundles over the circle with monodromy (2 1, 1 1)

Abstract

We find the exact values of complexity for an infinite series of 3-manifolds. Namely, by calculating hyperbolic volumes, we show that c(N_n)=2n, where $c$ is the complexity of a 3-manifold and N_n is the total space of the punctured torus bundle over S^1 with monodromy 2&1 1&1 ^n$. We also apply a recent result of Matveev and Pervova to show that c(M_n) \ge 2Cn with C\approx 0.598, where a compact manifold M_n is the total space of the torus bundle over S^1 with the same monodromy as N_n, and discuss an approach to the conjecture c(M_n)=2n+5 based on the equality c(N_n)=2n.