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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 \(S^3\).

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57R58 Floer homology
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[1] Akbulut, S; McCarthy, J, Casson’s invariant for oriented homology 3-spheres—an exposition, Annals of mathematics studies, Vol. 36, (1990), Princeton University Press Princeton, NJ · Zbl 0695.57011
[2] Bar-Natan, D, On Khovanov’s categorification of the Jones polynomial, Algebraic geom. topol., 2, 337-370, (2002) · Zbl 0998.57016
[3] Bertram, A; Thaddeus, M, On the quantum cohomology of a symmetric product of an algebraic curve, Duke math. J., 108, 2, 329-362, (2001) · Zbl 1050.14052
[4] Braam, P; Donaldson, S.K, Floer’s work on instanton homology, knots, and surgery, (), 195-256 · Zbl 0996.57516
[5] Eliashberg, Y.M; Thurston, W.P, Confoliations, University lecture series, Vol. 13, (1998), American Mathematical Society Providence RI · Zbl 0893.53001
[6] Fintushel, R; Stern, R.J, Knots, links, and 4-manifolds, Invent. math., 134, 2, 363-400, (1998) · Zbl 0914.57015
[7] A. Floer, Instanton homology, surgery, and knots, in: Geometry of Low-Dimensional Manifolds, 1 (Durham, 1989), London Mathematical Society, Lecture Note Series, Vol. 150, 1990, Cambridge University Press, Cambridge, pp. 97-114. · Zbl 0788.57008
[8] Gabai, D, Foliations and the topology of 3-manifolds, J. differential geom., 18, 3, 445-503, (1983) · Zbl 0533.57013
[9] Gompf, R.E; Stipsicz, A.I, 4-manifolds and kirby calculus, Graduate studies in mathematics, Vol. 20, (1999), American Mathematical Society Privdence, RI · Zbl 0933.57020
[10] Hoste, J, Sewn-up r-link exteriors, Pacific J. math., 112, 2, 347-382, (1984) · Zbl 0539.57004
[11] M. Hutchings, M. Sullivan, The periodic Floer homology of a Dehn twist, 2002. http://math.berkeley.edu/hutching/pub/index.htm. · Zbl 1089.57021
[12] Khovanov, M, A categorification of the Jones polynomial, Duke math. J., 101, 359-426, (2000) · Zbl 0960.57005
[13] Kronheimer, P.B; Mrowka, T.S, Scalar curvature and the Thurston norm, Math. res. lett., 4, 6, 931-937, (1997) · Zbl 0892.57011
[14] MacDonald, I.G, Symmetric products of an algebraic curve, Topology, 1, 319-343, (1962) · Zbl 0121.38003
[15] Meng, G; Taubes, C.H, SW=milnor torsion, Math. res. lett., 3, 661-674, (1996) · Zbl 0870.57018
[16] Morgan, J.W; Szabó, Z; Taubes, C.H, A product formula for Seiberg-Witten invariants and the generalized thom conjecture, J. differential geom., 44, 706-788, (1996) · Zbl 0974.53063
[17] V. Muñoz, B.-L. Wang, Seiberg-Witten-Floer homology of a surface times a circle, math.DG/9905050.
[18] P.S. Ozsváth, Z. Szabó, Holomorphic disks and three-manifold invariants: properties and applications, math.SG/0105202, Ann. of Math., to appear.
[19] P.S. Ozsváth, Z. Szabó, Holomorphic triangles and invariants for smooth four-manifolds, math.SG/0110169.
[20] P.S. Ozsváth, Z. Szabó, Holomorphic disks and topological invariants for closed three-manifolds, math.SG/0101206, Ann. of Math., 2001, to appear.
[21] P.S. Ozsváth, Z. Szabó, Heegaard Floer homologies and contact structures, math.SG/0210127, 2002.
[22] Ozsváth, P.S; Szabó, Z, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Adv. math., 173, 2, 179-261, (2003) · Zbl 1025.57016
[23] Ozsváth, P.S; Szabó, Z, Heegaard Floer homology and alternating knots, Geom. topol., 7, 225-254, (2003) · Zbl 1130.57303
[24] P.S. Ozsváth, Z. Szabó, Knot Floer homology, genus bounds, and mutation, math.GT/0303225, 2003.
[25] Rasmussen, J.A, Floer homologies of surgeries on two-bridge knots, Algebr. geom. topol., 2, 757-789, (2002) · Zbl 1013.57020
[26] J. Rasmussen, Floer homology and knot complements, Ph.D. Thesis, Harvard University, 2003.
[27] Seidel, P, The symplectic Floer homology of a Dehn twist, Math. res. lett., 3, 6, 829-834, (1996) · Zbl 0876.57022
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