Tunnel complexes of 3-manifolds

For each closed 3-manifold M and natural number t, we define a simplicial complex T_t(M), the t-tunnel complex, whose vertices are knots of tunnel number at most t. These complexes have a strong relation to disk complexes of handle bodies. We show that the complex T_t(M) is connected for M the 3-sphere or a lens space. Using this complex, we define an invariant, the t-tunnel complexity, for tunnel number t knots. These invariants are shown to have a strong relation to toroidal bridge numbers and the hyperbolic structures.