Holomorphic disks and knot invariants
This article is one of a series of articles by the authors [Ann. Math. (2) 159, No.~3, 1027--1158 (2004; Zbl 1073.57009); Ann. Math. (2) 159, No.3, 1159-1245 (2004; Zbl 1081.57013) and Topology Appl. 141, No.1--3, 59--85 (2004; Zbl 1052.57012)] in which the authors develop a new invariant for a 3-manifold \(M\), namely the Heegaard-Floer homology groups \(\widehat {HF}(M)\) and \(HF^{\pm}(M)\). In the article under review the authors define closely related Floer-homologies for a null-homologous knot \(K\) in an oriented 3-manifold \(M\), denoted by \(\widehat {HFK}(M,K)\). These invariants are graded Abelian groups with a \(\mathbb{Z}\) or a \(1/2 +\mathbb{Z}\) grading. In the case of classical knot and links in \(S^3\) the invariant is related to the Alexander-Conway polynomial \(\Delta_K(t)\) of the knot \(K\) as follows: \[ \sum_j \chi ( \widehat {HFK} (S^3,K,j,\mathbb{Q})) t^j = (t^{-1/2}-t^{1/2})^{n-1} \Delta_K(t), \] where \(n\) denotes the number of components in the link \(K\), \(\chi\) is the Euler characteristic and \(i\) is the filtration index. These new invariants are stronger than the Alexander-Conway polynomial since they do not vanish for split links where \(\Delta_K(t)=0\). These invariants also satisfy a skein exact sequence. Let \(K_0\), \(K_-\) and \(K_+\) be three knots/links which have projections that differ only at a single crossing (as usual in skein theory), then there is a long exact sequence between \(\widehat {HFK}\) for the three knots/links \(K_0\), \(K_-\) and \(K_+\). In the article the authors conjecture that Floer-homologies \(\widehat {HFK} (S^3,K)\) determine the genus of a knot. The authors prove this conjecture in Geom. Topol. 8, 311--334 (2004; Zbl 1056.57020)]. The article also contains some sample calculations for two bridge knots, a calculation for the 3-bridge knot \(9_{42}\), and calculations of \(HF^+\) for certain simple 3-manifolds obtained by surgeries on some special knots in \(S^3\).
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