Link Floer homology and the Thurston norm

Ozsváth, Peter; Szabó, Zoltán

doi:10.1090/S0894-0347-08-00586-9

Link Floer homology and the Thurston norm
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by Peter Ozsváth and Zoltán Szabó

J. Amer. Math. Soc. 21 (2008), 671-709

DOI: https://doi.org/10.1090/S0894-0347-08-00586-9

Published electronically: January 22, 2008

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Abstract:

We show that link Floer homology detects the Thurston norm of a link complement. As an application, we show that the Thurston polytope of an alternating link is dual to the Newton polytope of its multi-variable Alexander polynomial. To illustrate these techniques, we also compute the Thurston polytopes of several specific link complements.

References

Kunio Murasugi, On the genus of the alternating knot. I, II, J. Math. Soc. Japan 10 (1958), 94–105, 235–248. MR 99664, DOI 10.2969/jmsj/01010094
S. K. Donaldson, Lefschetz pencils on symplectic manifolds, J. Differential Geom. 53 (1999), no. 2, 205–236. MR 1802722, DOI 10.4310/jdg/1214425535
David Eisenbud and Walter Neumann, Three-dimensional link theory and invariants of plane curve singularities, Annals of Mathematics Studies, vol. 110, Princeton University Press, Princeton, NJ, 1985. MR 817982
John B. Etnyre, On symplectic fillings, Algebr. Geom. Topol. 4 (2004), 73–80. MR 2023278, DOI 10.2140/agt.2004.4.73
Yakov M. Eliashberg and William P. Thurston, Confoliations, University Lecture Series, vol. 13, American Mathematical Society, Providence, RI, 1998. MR 1483314, DOI 10.1090/ulect/013
John B. Etnyre, On symplectic fillings, Algebr. Geom. Topol. 4 (2004), 73–80. MR 2023278, DOI 10.2140/agt.2004.4.73
Andreas Floer, Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988), no. 3, 513–547. MR 965228
Andreas Floer, The unregularized gradient flow of the symplectic action, Comm. Pure Appl. Math. 41 (1988), no. 6, 775–813. MR 948771, DOI 10.1002/cpa.3160410603
David Gabai, Foliations and the topology of $3$-manifolds, J. Differential Geom. 18 (1983), no. 3, 445–503. MR 723813
David Gabai, Foliations and the topology of $3$-manifolds. II, J. Differential Geom. 26 (1987), no. 3, 461–478. MR 910017
Matthew Hedden, On knot Floer homology and cabling, Algebr. Geom. Topol. 5 (2005), 1197–1222. MR 2171808, DOI 10.2140/agt.2005.5.1197
P. B. Kronheimer and T. S. Mrowka, Scalar curvature and the Thurston norm, Math. Res. Lett. 4 (1997), no. 6, 931–937. MR 1492131, DOI 10.4310/MRL.1997.v4.n6.a12
P. Kronheimer, T. Mrowka, P. Ozsváth, and Z. Szabó, Monopoles and lens space surgeries, Ann. of Math. (2) 165 (2007), no. 2, 457–546. MR 2299739, DOI 10.4007/annals.2007.165.457
Curtis T. McMullen, The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology, Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 2, 153–171 (English, with English and French summaries). MR 1914929, DOI 10.1016/S0012-9593(02)01086-8
Kunio Murasugi, On the Alexander polynomial of alternating algebraic knots, J. Austral. Math. Soc. Ser. A 39 (1985), no. 3, 317–333. MR 802722, DOI 10.1017/S1446788700026094
Yi Ni, A note on knot Floer homology of links, Geom. Topol. 10 (2006), 695–713. MR 2240902, DOI 10.2140/gt.2006.10.695
Yi Ni, Sutured Heegaard diagrams for knots, Algebr. Geom. Topol. 6 (2006), 513–537. MR 2220687, DOI 10.2140/agt.2006.6.513
Peter Ozsváth and Zoltán Szabó, Heegaard Floer homology and alternating knots, Geom. Topol. 7 (2003), 225–254. MR 1988285, DOI 10.2140/gt.2003.7.225
Peter Ozsváth and Zoltán Szabó, Heegaard diagrams and holomorphic disks, Different faces of geometry, Int. Math. Ser. (N. Y.), vol. 3, Kluwer/Plenum, New York, 2004, pp. 301–348. MR 2102999, DOI 10.1007/0-306-48658-X_{7}
Peter Ozsváth and Zoltán Szabó, Holomorphic disks and genus bounds, Geom. Topol. 8 (2004), 311–334. MR 2023281, DOI 10.2140/gt.2004.8.311
Peter Ozsváth and Zoltán Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004), no. 1, 58–116. MR 2065507, DOI 10.1016/j.aim.2003.05.001
Peter Ozsváth and Zoltán Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. (2) 159 (2004), no. 3, 1027–1158. MR 2113019, DOI 10.4007/annals.2004.159.1027
Peter Ozsváth and Zoltán Szabó, Heegaard Floer homology and contact structures, Duke Math. J. 129 (2005), no. 1, 39–61. MR 2153455, DOI 10.1215/S0012-7094-04-12912-4

Floer homology and knot complements

Dale Rolfsen, Knots and links, Mathematics Lecture Series, vol. 7, Publish or Perish, Inc., Houston, TX, 1990. Corrected reprint of the 1976 original. MR 1277811
William P. Thurston, A norm for the homology of $3$-manifolds, Mem. Amer. Math. Soc. 59 (1986), no. 339, i–vi and 99–130. MR 823443
Vladimir Turaev, Torsions of 3-manifolds, Invariants of knots and 3-manifolds (Kyoto, 2001) Geom. Topol. Monogr., vol. 4, Geom. Topol. Publ., Coventry, 2002, pp. 295–302. MR 2002617, DOI 10.2140/gtm.2002.4.295