Signatures, Heegaard Floer correction terms and quasi–alternating links
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- by Paolo Lisca and Brendan Owens PDF
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Abstract:
Turaev showed that there is a well–defined map assigning to an oriented link $L$ in the three–sphere a Spin structure $\mathbf {t}_0$ on $\Sigma (L)$, the two–fold cover of $S^3$ branched along $L$. We prove, generalizing results of Manolescu–Owens and Donald–Owens, that for an oriented quasi–alternating link $L$ the signature of $L$ equals minus four times the Heegaard Floer correction term of $(\Sigma (L), \mathbf {t}_0)$.References
- Abhijit Champanerkar and Ilya Kofman, Twisting quasi-alternating links, Proc. Amer. Math. Soc. 137 (2009), no. 7, 2451–2458. MR 2495282, DOI 10.1090/S0002-9939-09-09876-1
- A. Champanerkar and P. Ording, A note on quasi-alternating Montesinos links, arXiv preprint 1205.5261.
- Andrew Donald and Brendan Owens, Concordance groups of links, Algebr. Geom. Topol. 12 (2012), no. 4, 2069–2093. MR 3020201, DOI 10.2140/agt.2012.12.2069
- Joshua Greene, Homologically thin, non-quasi-alternating links, Math. Res. Lett. 17 (2010), no. 1, 39–49. MR 2592726, DOI 10.4310/MRL.2010.v17.n1.a4
- Joshua Evan Greene and Liam Watson, Turaev torsion, definite 4-manifolds, and quasi-alternating knots, Bull. Lond. Math. Soc. 45 (2013), no. 5, 962–972. MR 3104988, DOI 10.1112/blms/bds096
- S. Jablan and R. Sazdanović, Quasi-alternating links and odd homology: computations and conjectures, arXiv preprint 0901.0075.
- Robion C. Kirby, The topology of $4$-manifolds, Lecture Notes in Mathematics, vol. 1374, Springer-Verlag, Berlin, 1989. MR 1001966, DOI 10.1007/BFb0089031
- W. B. Raymond Lickorish, An introduction to knot theory, Graduate Texts in Mathematics, vol. 175, Springer-Verlag, New York, 1997. MR 1472978, DOI 10.1007/978-1-4612-0691-0
- Paolo Lisca and András I. Stipsicz, Ozsváth-Szabó invariants and tight contact three-manifolds. II, J. Differential Geom. 75 (2007), no. 1, 109–141. MR 2282726
- C. Manolescu and B. Owens, A concordance invariant from the Floer homology of double branched covers, International Mathematics Research Notices 2007 (2007), no. 20, Art. ID rnm077.
- Ciprian Manolescu and Peter Ozsváth, On the Khovanov and knot Floer homologies of quasi-alternating links, Proceedings of Gökova Geometry-Topology Conference 2007, Gökova Geometry/Topology Conference (GGT), Gökova, 2008, pp. 60–81. MR 2509750
- David Mullins, The generalized Casson invariant for $2$-fold branched covers of $S^3$ and the Jones polynomial, Topology 32 (1993), no. 2, 419–438. MR 1217078, DOI 10.1016/0040-9383(93)90029-U
- Peter Ozsváth and Zoltán Szabó, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Adv. Math. 173 (2003), no. 2, 179–261. MR 1957829, DOI 10.1016/S0001-8708(02)00030-0
- Peter Ozsváth and Zoltán Szabó, On the Heegaard Floer homology of branched double-covers, Advances in Mathematics 194 (2005), no. 1, 1–33.
- Peter Ozsváth and Zoltán Szabó, Holomorphic triangles and invariants for smooth four-manifolds, Adv. Math. 202 (2006), no. 2, 326–400. MR 2222356, DOI 10.1016/j.aim.2005.03.014
- K. Qazaqzeh, N. Chbili, and B. Qublan, Characterization of quasi-alternating Montesinos links, arXiv preprint 1205.4650.
- K. Qazaqzeh, B. Qublan, and A. Jaradat, A new property of quasi-alternating links, arXiv preprint 1205.4291.
- R. Rustamov, Surgery formula for the renormalized Euler characteristic of Heegaard Floer homology, arXiv preprint math/0409294.
- Nikolai Saveliev, A surgery formula for the $\overline \mu$-invariant, Topology Appl. 106 (2000), no. 1, 91–102. MR 1769335, DOI 10.1016/S0166-8641(99)00075-9
- András I. Stipsicz, On the $\overline \mu$-invariant of rational surface singularities, Proc. Amer. Math. Soc. 136 (2008), no. 11, 3815–3823. MR 2425720, DOI 10.1090/S0002-9939-08-09439-2
- V. G. Turaev, Classification of oriented Montesinos links via spin structures, Topology and geometry—Rohlin Seminar, Lecture Notes in Math., vol. 1346, Springer, Berlin, 1988, pp. 271–289. MR 970080, DOI 10.1007/BFb0082779
- Liam Watson, A surgical perspective on quasi-alternating links, Low-dimensional and symplectic topology, Proc. Sympos. Pure Math., vol. 82, Amer. Math. Soc., Providence, RI, 2011, pp. 39–51. MR 2768652, DOI 10.1090/pspum/082/2768652
- Tamara Widmer, Quasi-alternating Montesinos links, J. Knot Theory Ramifications 18 (2009), no. 10, 1459–1469. MR 2583805, DOI 10.1142/S0218216509007518
Additional Information
- Paolo Lisca
- Affiliation: Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy
- Brendan Owens
- Affiliation: Department of Mathematics, University of Glasgow, University Gardens, Glasgow G12 8QW, United Kingdom
- Email: b.owens@maths.gla.ac.uk
- Received by editor(s): March 7, 2013
- Received by editor(s) in revised form: May 20, 2013
- Published electronically: October 17, 2014
- Additional Notes: The present work is part of the first author’s activities within CAST, a Research Network Program of the European Science Foundation, and the PRIN–MIUR research project 2010–2011 “Varietà reali e complesse: geometria, topologia e analisi armonica”.
The second author was supported in part by EPSRC grant EP/I033754/1. - Communicated by: Daniel Ruberman
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 143 (2015), 907-914
- MSC (2010): Primary 57M25, 57M27; Secondary 57Q60
- DOI: https://doi.org/10.1090/S0002-9939-2014-12265-9
- MathSciNet review: 3283677