## Algebraic & Geometric Topology

### Tunnel complexes of $3$–manifolds

Yuya Koda

#### Abstract

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

#### Article information

Source
Algebr. Geom. Topol., Volume 11, Number 1 (2011), 417-447.

Dates
Revised: 18 September 2010
Accepted: 1 November 2010
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715189

Digital Object Identifier
doi:10.2140/agt.2011.11.417

Mathematical Reviews number (MathSciNet)
MR2783233

Zentralblatt MATH identifier
1216.57004

#### Citation

Koda, Yuya. Tunnel complexes of $3$–manifolds. Algebr. Geom. Topol. 11 (2011), no. 1, 417--447. doi:10.2140/agt.2011.11.417. https://projecteuclid.org/euclid.agt/1513715189

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