Tunnel complexes of 3–manifolds

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.

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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

Volume 11, issue 1 (2011)

Yuya Koda

Abstract