Low Dimensional Topology

November 18, 2007

WYSIWYG Hyperbolic knots

Filed under: Heegaard splittings,Hyperbolic geometry,Knot theory,WYSIWYG topology — Jesse Johnson @ 12:55 pm

(Reprinted from my old ldt blog)

For my first blog entry I decided to write about a problem that combines a topic about which I know something (unknotting tunnels for knots) with a topic about which I should know more (hyperbolic geometry). Here’s the question: is every unknotting tunnel for a hyperbolic knot isotopic to a geodesic in the hyperbolic structure on the knot complement?

Colin Adams [1] showed that this is true for tunnel-number-one links (with two components) using the following trick: Given a one tunnel link in the three sphere, there is an automorphism that is orientation preserving on S^3, reverses the orientation of the knot, and preserves the unknotting tunnel pointwise. This automorphism is isotopic to a hyperbolic isometry, and it just so happens that any set that is preserved by a hyperbolic isometry is convex. Thus the isotopy that makes the automorphism into an isometry takes the unknotting tunnel to a geodesic.

It turns out the same trick works for two bridge knots. However, David Futer showed [2] that these are the only one-tunnel knots that admit such an inversion, so the problem remains open for higher bridge number knots. Incidentally, Cho and McCullough’s work [3] on the tree of unknotting tunnels gives a new proof of Futer’s result.

The problem falls under what Steve Kerckhoff calls WYSIWYG (what you see is what you get) topology. The idea is that even though the recently proved geometrization conjecture tells that all 3-manifolds are composed of geometric pieces, it doesn’t tell us how the combinatorial structure corresponds to the geometry. The goal of WYSIWYG topology is to see exactly how the combinatorial picture corresponds to the geometric picture. Here’s another closely related question: Given a minimal crossing diagram for a hyperbolic knot, draw an arc from the upper arc at a crossing straight down to the lower arc. Is this arc isotopic to a geodesic in the hyperbolic structure? Both this and the original question ask whether a certain combinatorially simple arc corresponds to a geometrically simple arc.

[1] C. Adams. Unknotting tunnels in hyperbolic 3-manifolds. Math. Ann. 302 (1995), 177–195.

[2] and [3] available on the ArXiv via the links above.

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