# How much do homological knot invariants improve the classification problem of knots?

The mutation operation in knots appears to be detected by the Floer homological invariants. See the papers by Ozsvath, Szabo and by Baldwin, Gillam. In addition, the Khovanov homology turns out to be able to detect the unknot. See the papers by Elisenda Grigsby, Wehrli and by Kronheimer, Mrowka.

My question is the following.

How much do homological knot invariants improve the classification problem of knots? Is there something homological invariants cannot distinguish?

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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. –  Tim Perutz Sep 10 '12 at 4:23
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). –  Marco Golla Sep 10 '12 at 6:41
For (quasi-)alternating knots, $\widehat{HFK}$ is determined by the Alexander polynomial. –  Tim Perutz Sep 10 '12 at 14:32