Algebraic & Geometric Topology

Tunnel complexes of $3$–manifolds

Yuya Koda

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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

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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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57M15: Relations with graph theory [See also 05Cxx] 57M27: Invariants of knots and 3-manifolds

knot unknotting tunnel complex toroidal bridge number


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

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