What is Floer homology of a knot?

Question

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?

Answer 1

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,...,a_g and g disjoint curves b₁,...,b_g on a surface S of genus g; then you attach 1-handles along the a_i and 2-handles along the b_i, and fill in what's left of the boundary with 0-handles and 3-handles to get Y.

The products T_a=a₁x...xa_g and T_b=b₁x...xb_g are Lagrangian tori in the symmetric product Sym^g(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 Sym^g(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}xSym^g-1(S) in Sym^g(S), and we can count the number n_z(u) of times these disks cross that hypersurface: if we only count disks where n_z(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-{a_i} and another in S-{b_i} 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 n_z(u)=n_w(u)=0 we get the invariant $\widehat{HFK}(Y,K)$, 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 $\sum_m (-1)^m HFK_m(S^3,K,a)$ 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.