Comments on: Slice-Ribbon Conjecture in danger!
https://ldtopology.wordpress.com/2010/12/02/slice-ribbon-conjecture-in-danger/
Recent Progress and Open ProblemsWed, 22 Dec 2010 16:28:54 +0000
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By: dmoskovich
https://ldtopology.wordpress.com/2010/12/02/slice-ribbon-conjecture-in-danger/#comment-1297
Wed, 22 Dec 2010 16:28:54 +0000http://ldtopology.wordpress.com/?p=1508#comment-1297Here’s a completely different ribbon obstruction which seems to have been unjustly ignored (mathscinet lists no citations):
Theorem H of V. Turaev, Multiplace generalizations of the Seifert form of a classical knot, Math. USSR, Sb. 44(3) (1983), 335–361.
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By: Stefan Friedl
https://ldtopology.wordpress.com/2010/12/02/slice-ribbon-conjecture-in-danger/#comment-1263
Tue, 07 Dec 2010 22:29:11 +0000http://ldtopology.wordpress.com/?p=1508#comment-1263A knot $K$ is called homotopy ribbon if it bounds a locally flat disk $D$ such that the map $\pi_1(S^3\setminus K)\to \pi_1(D^4\setminus D)$ is surjective.
If a knot is ribbon, then it is also homotopy ribbon. There are various obstructions to being homotopy ribbon, e.g. the ones Ian mentioned, but presumably such invariants are not helpful in this case.
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By: dmoskovich
https://ldtopology.wordpress.com/2010/12/02/slice-ribbon-conjecture-in-danger/#comment-1248
Sat, 04 Dec 2010 23:17:50 +0000http://ldtopology.wordpress.com/?p=1508#comment-1248Yeah… invariants association to metabelian reps should crack it if these knots are indeed not ribbon (and if they are indeed counterexamples to Generalized Property R, I see no reason why they would be). And if not, Cappell-Shaneson should extend also to polycyclic groups, just by iterating their construction, so one can also look deeper into the derived series of the knot group.
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By: Ian Agol
https://ldtopology.wordpress.com/2010/12/02/slice-ribbon-conjecture-in-danger/#comment-1241
Fri, 03 Dec 2010 21:04:37 +0000http://ldtopology.wordpress.com/?p=1508#comment-1241Looks like Friedl has some ribbon obstructions based on metabelian reps.: http://front.math.ucdavis.edu/0305.5402
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By: dmoskovich
https://ldtopology.wordpress.com/2010/12/02/slice-ribbon-conjecture-in-danger/#comment-1240
Fri, 03 Dec 2010 12:47:27 +0000http://ldtopology.wordpress.com/?p=1508#comment-1240:D
Thanks for the question! Of course, being a “coloured knots” person, I was thinking of Cappell and Shaneson’s formula for the Rohlin invariant of the irregular dihedral cover. The formula can be extended to any irregular covering space associated to a finite metabelian group (use intersection homology instead of homology in the construction, because the cobordism they use has singularities when the group is more general) (at least if the commutator subgroup is a cyclic group to some power, of order coprime to the order of the abelianization). I started doing this but never carried it through- anyway, there’s no reason for an obstruction associated to a dihedral cover not to be enough. It’s a ribbon obstruction, because it contains terms which depend on the choice of characteristic knot (which is information about the colouring), which will contribute zero if the knot is ribbon, by the way the cobordism is constructed (the detailed construction appeared a decade later, in Linking Numbers in Branched Covers).
Given a Seifert matrix for the knot, and another one for the characteristic knot, you can calculate the expression. You can find the characteristic knot because its homology class is the Alexander dual to the cohomology class represented by the colouring under the universal coefficient theorem. So I expect it can all be programmed into a computer without too much trouble (easier said than done, like everything in this world).
So there you have a zillion ribbon obstructions. One ribbon obstruction for each representation of the knot group onto a dihedral group whose commutator subgroup is cyclic of order p, where p ranges over the first zillion odd primes.
Maybe there are many other ribbon obstructions known in the literature- but these are the ones I was thinking of.
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By: Ian Agol
https://ldtopology.wordpress.com/2010/12/02/slice-ribbon-conjecture-in-danger/#comment-1237
Fri, 03 Dec 2010 06:41:32 +0000http://ldtopology.wordpress.com/?p=1508#comment-1237Could you give some references to the zillions of ribbon obstructions?
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