Tunnel complexes of $3$–manifolds

Abstract

For each closed $3$–manifold $M$ and natural number $t$, we define a simplicial complex ${\mathcal{T}}_t\left(M\right)$, the $t$–tunnel complex, whose vertices are knots of tunnel number at most $t$. These complexes have a strong relation to disk complexes of handlebodies. We show that the complex ${\mathcal{T}}_t\left(M\right)$ 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.