Holomorphic disks, link invariants and the multi-variable Alexander polynomial

Abstract

We define a Floer-homology invariant for links in S 3 , and study its properties.

Floer homology and singular knots

“…Definition of HFS(K) and HFS(K). Suppose now that K ⊂ S 3 is a singular knot and (Σ, α, β, w, z) is a balanced Heegaard diagram compatible with K. As usual (compare [8], [10]), we consider the (g + ℓ)-fold symmetric power Sym g+ℓ (Σ) of the genus-g surface Σ, equipped with the tori…”

“…A ′ is well-defined. Well-definedness of N ′ follows from standard properties of the Maslov index, see [10,Proposition 4.1]. Additivity of these quantities is straightforward.…”

“…Since the coefficients in our theories are chosen in F = Z/2Z, the arguments of [10] apply. Now the proof that ∂ 2 = 0 = ∂ 2 is an adaptation of [10,Lemma 4.3].…”

The sutured Floer homology polytope

“…This was extended to an invariant HFK of knots by Ozsváth and Szabó in [18], and independently by Rasmussen in [23]. Later, Ozsváth and Szabó [21] The proof of this striking result uses Gabai's theory of sutured manifolds [4; 7; 8], the Eliashberg-Thurston theory of confoliations [2], the contact invariant and cobordism maps in Heegaard Floer homology, symplectic semifillings and Lefschetz pencils.…”

Heegaard–Floer homology and string links

“…There is a relationship between the torsion of a string link and the multivariable Alexander polynomial of a simple link closure of the string link. Recently, P Ozsváth and Z Szabó have announced a version of Heegaard-Floer homology for links which enhances the multi-variable Alexander polynomial, [16], as the knot Floer homology enhanced the single-variable version. While there should be a relationship between the string link homology and the link homology, it should be noted that the string link homology appears to be a different beast; it does not share the symmetry under change of the components' orientation, for example.…”