Volume 11, issue 1 (2011)

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Tunnel complexes of $3$–manifolds

Yuya Koda

Algebraic & Geometric Topology 11 (2011) 417–447
Abstract

For each closed $3$–manifold $M$ and natural number $t$, we define a simplicial complex ${\mathsc{T}}_{t}\left(M\right)$, 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 ${\mathsc{T}}_{t}\left(M\right)$ 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.

Keywords
knot, unknotting tunnel, complex, toroidal bridge number
Mathematical Subject Classification 2000
Primary: 57M25
Secondary: 57M15, 57M27