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I've heard that there are different theories providing knot invariants in form of homologies. My understanding is that if you embed knot in a special way into a space, there is a special homology theory called Floer homology.

Question: what's the definition and properties of a Floer homology of a knot? How is it related to other knot homology theories?

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There is a really long Wikipedia article on Floer homology. Could you ask a more specific question? –  Qiaochu Yuan Oct 27 '09 at 22:02
    
Sorry, what is the article you're referring to? The one I found, en.wikipedia.org/wiki/Floer_homology, doesn't seem to answer my question about "what's ... Floer homology of a knot" –  Ilya Nikokoshev Oct 27 '09 at 22:13
    
I think this part of that wiki article answers your question: en.wikipedia.org/wiki/… (or maybe I don't understand your question..) –  David Zureick-Brown Oct 28 '09 at 0:27
    
The wiki article says that there is a knot invariant coming from a filtration on the Heegaard Floer homology of a 3-manifold, but nothing about what that filtration is or what properties the associated object has (other than categorifying the Alexander polynomial). My long answer below was an attempt to explain some of that. –  Steven Sivek Oct 28 '09 at 1:03
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Let's not have too too many of these "What is X?" questions. –  Scott Morrison Oct 28 '09 at 1:04

1 Answer 1

up vote 12 down vote accepted

I can say something about this for Heegaard Floer homology. Given a 3-manifold Y, you can take a Heegaard splitting, i.e. a decomposition of Y into two genus g handlebodies joined along their boundary. This can be represented by drawing g disjoint curves a1,...,ag and g disjoint curves b1,...,bg on a surface S of genus g; then you attach 1-handles along the ai and 2-handles along the bi, and fill in what's left of the boundary with 0-handles and 3-handles to get Y.

The products Ta=a1x...xag and Tb=b1x...xbg are Lagrangian tori in the symmetric product Symg(S), which has a complex structure induced from S, and applying typical constructions from Lagrangian Floer homology gives you a chain complex CF(Y) whose generators are points in the intersection of these tori and whose differential counts certain holomorphic disks in Symg(S). Miraculously, its homology HF(Y) turns out to be independent of every choice you made along the way. We can also pick a basepoint z in the surface S and identify a hypersurface {z}xSymg-1(S) in Symg(S), and we can count the number nz(u) of times these disks cross that hypersurface: if we only count disks where nz(u)=0, for example, we get the hat version of HF, and otherwise we get more complicated versions.

Given two points z and w on the surface S of any Heegaard splitting we can construct a knot in Y: draw one curve in S-{ai} and another in S-{bi} connecting z and w, and push these slightly into the corresponding handlebodies. In fact, for any knot K in Y there is a Heegaard splitting such that we can construct K in this fashion. But now this extra basepoint w gives a filtration on CF(Y); in the simplest form, if we only count holomorphic disks u with nz(u)=nw(u)=0 we get the invariant \hat{HFK}(Y,K), and otherwise we get other versions. The fact that this comes from a filtration also gives us a spectral sequence HFK(Y,K) => HF(Y).

This was constructed independently by Ozsvath-Szabo and Rasmussen, and it satisfies several interesting properties. Just to name a few:

  • for knots K in S3 it has a bigrading (a,m), and the Euler characteristic \sum (-1)^m HFKm(S3,K,a) is the Alexander polynomial of K;
  • there's a skein exact sequence relating HFK for K and various resolutions at a fixed crossing;
  • the filtered chain homotopy type of CFK tells you about the Heegaard Floer homology of various surgeries on K;
  • the highest a for which HFK*(S3,K,a) is nonzero is the Seifert genus of the knot;
  • If Y-K is irreducible and K is nullhomologous, then HFK(Y,K,g(K)) = Z if and only if K is fibered (proved by Ghiggini for genus 1 and Ni in general, and later by Juhasz as well).

For knots in S3 it is also known how to compute HFK(K) combinatorially: see papers by Manolescu-Ozsvath-Sarkar and Manolescu-Ozsvath-Szabo-Thurston.

The relation to other knot homology theories isn't all that well understood, but there are some results comparing it to Khovanov homology. For example, given a knot K in S3:

  • Just as Lee's spectral sequence for Khovanov homology gave a concordance invariant s(K), the spectral sequence from HFK(K) to HF(S3) gives a concordance invariant tau(K), and both of these provide lower bounds on the slice genus of K. (Hedden and Ording showed that these invariants are not equal.)
  • There's a spectral sequence from the Khovanov homology of the mirror of K to HF of the branched double cover of K.
  • For quasi-alternating knots, both Khovanov homology and HFK are determined entirely by the Jones and Alexander polynomials, respectively, as well as the signature; this can be proven using skein exact sequences for both (Manolescu-Ozsvath).

Anyway, that was long enough that I've probably made several mistakes above and still not been anywhere near rigorous. There's a nice overview that's now several years old (and thus probably missing some of the things I said above) on Zoltan Szabo's website, http://www.math.princeton.edu/~szabo/clay.pdf, if you want more details.

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