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In a paper of Casson and Gordon's "A loop theorem for duality spaces and fibred ribbon knots. Invent. Math. 74 (1983), no. 1, 119–137" they give a necessary criterion for a fibred knot to be a ribbon knot.

The criterion is that the monodromy of the bundle has to extend to an automorphism of some handlebody (technicality, the fibre has a circle boundary so one caps-off the surface with a disc to get a closed surface). The handlebody is not known from simply the knot in advance, as it comes from a generalized Dehn lemma that they prove.

My question:

  • Q: Given an outer automorphism of the fundamental group of a closed orientable surface, is there an efficient procedure to determine if the automorphism extends to some handlebody that has the surface as boundary?

By the regular 3-manifold Dehn lemma, an automorphism of the surface extends to some handlebody if and only if it preserves the class of curves that bound discs in the handlebody -- which can be phrased entirely in terms of the homomorphism $\pi_1 \Sigma \to \pi_1 H$ where $\Sigma$ is the surface is the boundary of the handlebody $\Sigma = \partial H$. So this is easy to check.

My question has the do with the case where you want to know if such a handlebody exists. Presumably there is a computable procedure for this, but it does not seem immediate to me.

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up vote 7 down vote accepted

This problem is solved in the paper "Algorithmic compression of surface automorphisms" by Casson and Long. They remark that their motivation is (partly) the paper of Casson and Gordon.

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Great! This is very helpful. I'll take a look at these papers tonight. – Ryan Budney Jun 13 '13 at 16:40

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