The mapping cone formula in Heegaard Floer homology and Dehn surgery on knots in S3

Gainullin, Fyodor¹

¹Department of Mathematics, Imperial College London, South Kensington Campus, London, SW7 2AZ, United Kingdom

Algebraic and Geometric Topology, Tome 17 (2017) no. 4, pp. 1917-1951

Résumé

We write down an explicit formula for the + version of the Heegaard Floer homology (as an absolutely graded vector space over an arbitrary field) of the results of Dehn surgery on a knot K in S3 in terms of homological data derived from CFK∞(K). This allows us to prove some results about Dehn surgery on knots in S3. In particular, we show that for a fixed manifold there are only finitely many alternating knots that can produce it by surgery. This is an improvement on a recent result by Lackenby and Purcell. We also derive a lower bound on the genus of knots depending on the manifold they give by surgery. Some new restrictions on Seifert fibred surgery are also presented.

Détail

DOI : 10.2140/agt.2017.17.1917

Classification : 57M27, 57M25
Keywords: Heegaard Floer homology, Dehn surgery

Bibliographie

[1] R H Crowell, Nonalternating links, Illinois J. Math. 3 (1959) 101

[2] J E Greene, L-space surgeries, genus bounds, and the cabling conjecture, J. Differential Geom. 100 (2015) 491 | DOI

[3] S Jabuka, Heegaard Floer genus bounds for Dehn surgeries on knots, J. Topol. 7 (2014) 523 | DOI

[4] S Jabuka, Heegaard Floer groups of Dehn surgeries, J. Lond. Math. Soc. 92 (2015) 499 | DOI

[5] Ç Karakurt, HFNem2 (2013)

[6] M Lackenby, Dehn surgery on knots in 3–manifolds, J. Amer. Math. Soc. 10 (1997) 835 | DOI

[7] M Lackenby, J S Purcell, Cusp volumes of alternating knots, Geom. Topol. 20 (2016) 2053 | DOI

[8] D Mccoy, Non-integer surgery and branched double covers of alternating knots, J. Lond. Math. Soc. 92 (2015) 311 | DOI

[9] A H Moore, L Starkston, Genus-two mutant knots with the same dimension in knot Floer and Khovanov homologies, Algebr. Geom. Topol. 15 (2015) 43 | DOI

[10] K Murasugi, On the Alexander polynomial of the alternating knot, Osaka Math. J. 10 (1958) 181

[11] Y Ni, Z Wu, Cosmetic surgeries on knots in S3, J. Reine Angew. Math. 706 (2015) 1 | DOI

[12] Y Ni, X Zhang, Characterizing slopes for torus knots, Algebr. Geom. Topol. 14 (2014) 1249 | DOI

[13] J K Osoinach Jr., Manifolds obtained by surgery on an infinite number of knots in S3, Topology 45 (2006) 725 | DOI

[14] P Ozsváth, Z Szabó, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Adv. Math. 173 (2003) 179 | DOI

[15] P Ozsváth, Z Szabó, On the Floer homology of plumbed three-manifolds, Geom. Topol. 7 (2003) 185 | DOI

[16] P Ozsváth, Z Szabó, Holomorphic disks and genus bounds, Geom. Topol. 8 (2004) 311 | DOI

[17] P Ozsváth, Z Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004) 58 | DOI

[18] P Ozsváth, Z Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. 159 (2004) 1027 | DOI

[19] P Ozsváth, Z Szabó, Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. 159 (2004) 1159 | DOI

[20] P Ozsváth, Z Szabó, On Heegaard Floer homology and Seifert fibered surgeries, from: "Proceedings of the Casson Fest" (editors C Gordon, Y Rieck), Geom. Topol. Monogr. 7, Geom. Topol. Publ. (2004) 181 | DOI

[21] P Ozsváth, Z Szabó, On knot Floer homology and lens space surgeries, Topology 44 (2005) 1281 | DOI

[22] P S Ozsváth, Z Szabó, Knot Floer homology and integer surgeries, Algebr. Geom. Topol. 8 (2008) 101 | DOI

[23] P S Ozsváth, Z Szabó, Knot Floer homology and rational surgeries, Algebr. Geom. Topol. 11 (2011) 1 | DOI

[24] J A Rasmussen, Floer homology and knot complements, PhD thesis, Harvard (2003)

[25] J Rasmussen, Lens space surgeries and a conjecture of Goda and Teragaito, Geom. Topol. 8 (2004) 1013 | DOI

[26] D Rolfsen, Knots and links, 7, Publish or Perish (1990)

[27] M Teragaito, A Seifert fibered manifold with infinitely many knot-surgery descriptions, Int. Math. Res. Not. 2007 (2007) | DOI

[28] J Wang, Cosmetic surgeries on genus one knots, Algebr. Geom. Topol. 6 (2006) 1491 | DOI

[29] Z Wu, On mapping cones of Seifert fibered surgeries, J. Topol. 5 (2012) 366 | DOI