Movatterモバイル変換

Holomorphic disks and knot invariants.(English)Zbl 1062.57019

Adv. Math.186, No. 1, 58-116 (2004).

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 \(S^3\).

Reviewer: Claus Ernst (Bowling Green)

Cited in39 Reviews

Cited in371 Documents

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

Keywords:

Heegaard Floer homology;3-manifold;Alexander-Conway polynomial

Citations:

Zbl 1052.57012;Zbl 1056.57020;Zbl 1073.57009;Zbl 1081.57013

Cite Review PDF

Full Text:DOI arXiv

References:

[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 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, (Hofer, H.; Taubes, C. H.; Weinstein, A.; Zehnder, E., The Floer Memorial. The Floer Memorial, Progress in Mathematics, Vol. 133 (1995), Birkhäuser: Birkhäuser Basel), 195-256 ·Zbl 0996.57516
[5]	Eliashberg, Y. M.; Thurston, W. P., Confoliations, University Lecture Series, Vol. 13 (1998), American Mathematical Society: 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.; 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: 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; 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.; 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.; 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.; 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.; 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.; 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.; 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.; 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

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.

Movatterモバイル変換

Examples

Fields

Operators