Tunnel complexes of 3–manifolds

Koda, Yuya¹

¹Mathematical Institute, Tohoku University, Sendai 980-8578, Japan

Algebraic and Geometric Topology, Tome 11 (2011) no. 1, pp. 417-447

Résumé

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.

Détail

DOI : 10.2140/agt.2011.11.417

Keywords: knot, unknotting tunnel, complex, toroidal bridge number

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