Abstract
We compute the Heegaard–Floer homology of \(S^{3}_{1}(K)\) (the (+1) surgery on the torus knot T p,q ) in terms of the semigroup generated by p and q, and we find a compact formula (involving Dedekind sums) for the corresponding Ozsváth–Szabó d-invariant. We relate the result to known knot invariants of T p,q as the genus and the Levine–Tristram signatures. Furthermore, we emphasize the striking resemblance between Heegaard–Floer homologies of (+1) and (−1) surgeries on torus knots. This relation is best seen at the level of τ functions.
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References
S. Akbulut and J. McCarthy, Casson’s Invariant for Oriented Homology 3-Spheres. An Exposition, Mathematical Notes 36, Princeton University Press (Princeton, 1990).
M. Borodzik and A. Némethi, Hodge-type structures as link invariants, preprint, arxiv:1005.2084, to appear in Ann. Inst. Fourier.
S. M. Gusein-Zade, F. Delgado and A. Campillo, On the monodromy of a plane curve singularity and the Poincaré series of the ring of functions on the curve, Functional Analysis and its Applications, 33 (1999), 56–57.
P. Kronheimer and T. Mrowka, Gauge theory for embedded surfaces I, Topology, 32 (1993), 773–826.
C. Lescop, Global Surgery Formula for the Casson–Walker Invariants, Annals of Mathematical Studies 140, Princeton University Press (1996).
R. A. Litherland, Signatures of iterated torus knots, in: Topology of Low-Dimensional Manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977). Lecture Notes in Math. 722, Springer (Berlin, 1979), pp. 71–84.
L. Moser, Elementary surgery along torus knots, Pacific J. Math., 38 (1971), 737–745.
C. Manolescu and B. Owens, A concordance invariant from the Floer homology of double branched covers, Int. Res. Math. Not. IMRN (2007), 20.
A. Némethi, Dedekind sums and the signature of f(x,y)+z N, Selecta Mathematica, New series, 4 (1998), 361–376.
A. Némethi, On the Ozsváth–Szabó invariants of negative definite plumbed manifolds, Geom. Topol., 9 (2005), 991–1042.
A. Némethi, On the Heegaard–Floer homology of \(S^{3}_{-p/q}(K)\), arxiv:math/0410570, published as part of [13].
A. Némethi, On the Heegaard–Floer homology of \(S^{3}_{-d}(K)\) and unicuspidal rational plane curves, in: Geometry and Topology of Manifolds, Fields Inst. Commun. 47, Amer. Math. Soc. (Providence, RI, 2005), pp. 219–234.
A. Némethi, Graded roots and singularities, in: Singularities in Geometry and Topology, World Sci. Publ. (Hackensack, NJ, 2007), pp. 394–463.
P. Ozsváth and Z. Szabó, Absolutely graded Floer homologies and intersection forms for four manifolds with boundary, Adv. Math., 173 (2003), 179–261.
P. Ozsváth and Z. Szabó, Heegaard–Floer homology and alternating knots, Geom. Topol., 7 (2003), 225–254.
P. S. Ozsváth and Z. Szabó, On the Floer homology of plumbed three-manifolds, Geom. Topol., 7 (2003), 185–224.
Th. Peters, A concordance invariant from the Floer homology of +/−1 surgeries, preprint, arxiv:1003.3038.
H. Rademacher and E. Grosswald, Dedekind Sums, The Carus Mathematical Monographs (1972).
J. Steenbrink, Mixed Hodge Structures on the Vanishing Cohomology, Nordic Summer School/NAVF, Symposium in Mathematics (Oslo, 1976).
C. T. C. Wall, Singular Points of Plane Curves, London Mathematical Society Student Texts 63, Cambridge University Press (Cambridge, 2004).
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Borodzik, M., Némethi, A. Heegaard–Floer homologies of (+1) surgeries on torus knots. Acta Math Hung 139, 303–319 (2013). https://doi.org/10.1007/s10474-012-0280-x
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DOI: https://doi.org/10.1007/s10474-012-0280-x
Key words and phrases
- torus knot
- Brieskorn sphere
- plumbed manifold
- Heegaard–Floer homology
- d-invariant
- surgery 3-manifold
- Levine–Tristram signature
- semigroup of algebraic knots
- graded root