## Holomorphic disks and knot invariants.(English)Zbl 1062.57019

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 $$S^3$$.

### MathOverflow Questions:

On the genus of thin knots and the degree of the Alexander polynomial

### MSC:

 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 57R58 Floer homology

### Citations:

Zbl 1052.57012; Zbl 1056.57020; Zbl 1073.57009; Zbl 1081.57013
Full Text:

### References:

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