Algebraic & Geometric Topology 11 (2011) 417–447

DOI: 10.2140/agt.2011.11.417

Abstract

For each closed 3–manifold M and natural number t, we define a simplicial complex T_t(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 T_t(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

Primary: 57M25

Secondary: 57M15, 57M27

References

Publication

Received: 25 April 2010
Revised: 18 September 2010
Accepted: 1 November 2010
Published: 25 January 2011

Authors