Geometry & Topology

Heegaard Floer homology and alternating knots

Peter Ozsváth and Zoltán Szabó

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Abstract

In an earlier paper, we introduced a knot invariant for a null-homologous knot K in an oriented three-manifold Y, which is closely related to the Heegaard Floer homology of Y. In this paper we investigate some properties of these knot homology groups for knots in the three-sphere. We give a combinatorial description for the generators of the chain complex and their gradings. With the help of this description, we determine the knot homology for alternating knots, showing that in this special case, it depends only on the signature and the Alexander polynomial of the knot (generalizing a result of Rasmussen for two-bridge knots). Applications include new restrictions on the Alexander polynomial of alternating knots.

Article information

Source
Geom. Topol., Volume 7, Number 1 (2003), 225-254.

Dates
Received: 1 November 2002
Revised: 19 March 2003
Accepted: 20 March 2003
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513883097

Digital Object Identifier
doi:10.2140/gt.2003.7.225

Mathematical Reviews number (MathSciNet)
MR1988285

Zentralblatt MATH identifier
1083.57013

Subjects
Primary: 57R58: Floer homology
Secondary: 57M27: Invariants of knots and 3-manifolds 53D40: Floer homology and cohomology, symplectic aspects 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

Keywords
alternating knots Kauffman states Floer homology

Citation

Ozsváth, Peter; Szabó, Zoltán. Heegaard Floer homology and alternating knots. Geom. Topol. 7 (2003), no. 1, 225--254. doi:10.2140/gt.2003.7.225. https://projecteuclid.org/euclid.gt/1513883097


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References

  • J W Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30 (1928) 275–306
  • D Bar-Natan, On Khovanov's categorification of the Jones polynomial, Algebraic and Geometric Topology 2 (2002) 337–370
  • G Burde, H Zieschang, Knots, de Gruyter Studies in Mathematics 5, Walter de Gruyter & Co. (1985)
  • R Crowell, Genus of alternating link types, Ann. of Math. (2) 69 (1959) 258–275
  • R H Crowell, Nonalternating links, Illinois J. Math. 3 (1959) 101–120
  • S K Donaldson, An application of gauge theory to four-dimensional topology, J. Differential Geom. 18 (1983) 279–315
  • Y Eliashberg, Classification of contact structures on ${\bf R}^3$, Internat. Math. Res. Notices 3 (1993) 87–91
  • N D Elkies, A characterization of the ${Z}\sp n$ lattice, Math. Res. Lett. 2 (1995) 321–326
  • R H Fox, Some problems in knot theory, from: “Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961)”, Prentice-Hall, Englewood Cliffs, N.J. (1962) 168–176
  • K A Frøyshov, The Seiberg-Witten equations and four-manifolds with boundary, Math. Res. Lett 3 (1996) 373–390
  • K A Frøyshov, An inequality for the h-invariant in instanton Floer theory (2001).
  • K A Frøyshov, Equivariant aspects of Yang-Mills Floer theory, Topology 41 (2002) 525–552
  • S Garoufalidis, A conjecture on Khovanov's invariants (2001), preprint
  • E Giroux (2001), lectures at Oberwohlfach
  • C McA Gordon, R A Litherland, On the signature of a link, Invent. Math. 47 (1978) 53–69
  • L H Kauffman, Formal knot theory, Mathematical Notes 30, Princeton University Press (1983)
  • L H Kauffman, On knots, Annals of Mathematics Studies 115, Princeton University Press (1987)
  • M Khovanov, A categorification of the Jones polynomial (1999), Duke Math. J. 101 (2000) 359–426
  • E S Lee, The support of the Khovanov's invariants for alternating knots (2002).
  • W B Raymond Lickorish, An introduction to knot theory, volume 175 of Graduate Texts in Mathematics, Springer-Verlag (1997)
  • J Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966) 358–426
  • K Murasugi, On the genus of the alternating knot. I, II, J. Math. Soc. Japan 10 (1958) 94–105, 235–248
  • K Murasugi, On the Alexander polynomial of alternating algebraic knots, J. Austral. Math. Soc. Ser. A 39 (1985) 317–333
  • P S Ozsváth, Z Szabó, Holomorphic disks and three-manifold invariants: properties and applications., to appear in Annals of Math.
  • P S Ozsváth, Z Szabó, Holomorphic disks and topological invariants for closed three-manifolds (2001)., to appear in Annals of Math.
  • P S Ozsváth, Z Szabó, Heegaard Floer homologies and contact structures (2002).
  • P S Ozsváth, Z Szabó, Holomorphic disks and knot invariants (2002).
  • P S Ozsváth, Z Szabó, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Advances in Mathematics 173 (2003) 179–261
  • P S Ozsváth, Z Szabó, Knot Floer homology, genus bounds, and mutation (2003).
  • J Rasmussen, Floer homologies of surgeries on two-bridge knots (2002), Algebr. Geom. Topol. 2 (2002) 757–789
  • W P Thurston, H E Winkelnkemper, On the existence of contact forms, Proc. Amer. Math. Soc. 52 (1975) 345–347
  • V Turaev, Torsion invariants of Spin$^c$-Structures on $3$-manifolds, Math. Research Letters 4 (1997) 679–695