Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 11, Number 1 (2011), 417-447.
Tunnel complexes of $3$–manifolds
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.
Algebr. Geom. Topol., Volume 11, Number 1 (2011), 417-447.
Received: 25 April 2010
Revised: 18 September 2010
Accepted: 1 November 2010
First available in Project Euclid: 19 December 2017
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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