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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 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.

Keywords
knot, unknotting tunnel, complex, toroidal bridge number
Mathematical Subject Classification 2000
Primary: 57M25
Secondary: 57M15, 57M27
References
Publication
Received: 25 April 2010
Revised: 18 September 2010
Accepted: 1 November 2010
Published: 25 January 2011
Authors
Yuya Koda
Mathematical Institute
Tohoku University
Sendai 980-8578
Japan
http://www.math.tohoku.ac.jp/~koda/index_e.html