Timeline for How much do homological knot invariants improve the classification problem of knots?
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7 events
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| Sep 12, 2012 at 20:19 | comment | added | Tim Perutz | OK, so I would look for a pair of thin (e.g. alternating) knots which have the same values for Ozsvath-Szabo's invariant $\tau\in \mathbb{Z}$ and for Jen Hom's invariant $\varepsilon\in \{-1,0,1\}$ (e.g. they could be concordant). I now don't know any invariant that distinguishes their $CFK^\infty$'s, and I suspect that there isn't one. | |
| Sep 12, 2012 at 12:41 | comment | added | Daniel Moskovich | By HFH I meant the full theory, not just the hat version. | |
| Sep 11, 2012 at 0:06 | comment | added | Lee Mosher | Apropos "at least conjecturally", there is of course the Gordon-Luecke theorem MR0965210. | |
| Sep 10, 2012 at 14:32 | comment | added | Tim Perutz | For (quasi-)alternating knots, $\widehat{HFK}$ is determined by the Alexander polynomial. | |
| Sep 10, 2012 at 6:41 | comment | added | Marco Golla | gt.postech.ac.kr/~jccha/2009kmsams/tanaka.pdf gives some interesting examples (although I couldn't find a paper to back the slides up). In particular, they say that $\widehat{HFK}(8_{20})$ is isomorphic to $\widehat{HFK}(3_1\#3_1)$ (and this is easy to check). | |
| Sep 10, 2012 at 4:23 | comment | added | Tim Perutz | Don't think you're right that Heegaard Floer theory gives complete knot invariants. For $\widehat{HFK}$ at least, I believe that there are counterexamples in the literature; possibly also for $CFK^\infty$. For Khovanov homology, Liam Watson gave counterexamples. | |
| Sep 10, 2012 at 1:09 | history | answered | Daniel Moskovich | CC BY-SA 3.0 |