For each closed –manifold and natural number , we define a simplicial complex , the –tunnel complex, whose vertices are knots of tunnel number at most . These complexes have a strong relation to disk complexes of handlebodies. We show that the complex is connected for the –sphere or a lens space. Using this complex, we define an invariant, the –tunnel complexity, for tunnel number knots. These invariants are shown to have a strong relation to toroidal bridge numbers and the hyperbolic structures.
"Tunnel complexes of $3$–manifolds." Algebr. Geom. Topol. 11 (1) 417 - 447, 2011. https://doi.org/10.2140/agt.2011.11.417