Khovanov homology is an unknot-detector
Kronheimer, P. B.1 ; Mrowka, T. S.2
1 Harvard University Cambridge, MA, 02138 USA
2 Massachusetts Institute of Technology Cambridge, MA, 02139 USA
Publications Mathématiques de l'IHÉS, Tome 113 (2011), pp. 97-208.

We prove that a knot is the unknot if and only if its reduced Khovanov cohomology has rank 1. The proof has two steps. We show first that there is a spectral sequence beginning with the reduced Khovanov cohomology and abutting to a knot homology defined using singular instantons. We then show that the latter homology is isomorphic to the instanton Floer homology of the sutured knot complement: an invariant that is already known to detect the unknot.

DOI : 10.1007/s10240-010-0030-y
Kronheimer, P. B. 1 ; Mrowka, T. S. 2

1 Harvard University Cambridge, MA, 02138 USA
2 Massachusetts Institute of Technology Cambridge, MA, 02139 USA 
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Kronheimer, P. B.; Mrowka, T. S. Khovanov homology is an unknot-detector. Publications Mathématiques de l'IHÉS, Tome 113 (2011), pp. 97-208. doi : 10.1007/s10240-010-0030-y. http://archive.numdam.org/articles/10.1007/s10240-010-0030-y/

[1.] Akbulut, S.; Mrowka, T.; Ruan, Y. Torsion classes and a universal constraint on Donaldson invariants for odd manifolds, Trans. Am. Math. Soc., Volume 347 (1995), pp. 63-76 | DOI | MR | Zbl

[2.] J. A. Baldwin, On the spectral sequence from Khovanov homology to Heegaard Floer homology, prepint (2008). | Zbl

[3.] J. Bloom, A link surgery spectral sequence in monopole Floer homology, prepint (2009). | MR | Zbl

[4.] Braam, P. J.; Donaldson, S. K. Floer’s work on instanton homology, knots and surgery, The Floer Memorial Volume (Progr. Math., 133), Birkhäuser, Basel (1995), pp. 195-256 | MR | Zbl

[5.] Dold, A.; Whitney, H. Classification of oriented sphere bundles over a 4-complex, Ann. Math. (2), Volume 69 (1959), pp. 667-677 | DOI | MR | Zbl

[6.] Donaldson, S. K. The orientation of Yang-Mills moduli spaces and 4-manifold topology, J. Differ. Geom., Volume 26 (1987), pp. 397-428 | MR | Zbl

[7.] Donaldson, S. K.; Kronheimer, P. B. The geometry of four-manifolds, Oxford University Press, New York, 1990 | MR | Zbl

[8.] Floer, A. An instanton-invariant for 3-manifolds, Commun. Math. Phys., Volume 118 (1988), pp. 215-240 | DOI | MR | Zbl

[9.] Floer, A. Instanton homology, surgery, and knots, Geometry of Low-Dimensional Manifolds, 1 (London Math. Soc. Lecture Note Ser., 150), Cambridge Univ. Press, Cambridge (1990), pp. 97-114 | MR | Zbl

[10.] Grigsby, J. E.; Wehrli, S. On the colored Jones polynomial, sutured Floer homology, and knot Floer homology, Adv. Math., Volume 223 (2010), pp. 2114-2165 | DOI | MR | Zbl

[11.] Hedden, M. Khovanov homology of the 2-cable detects the unknot, Math. Res. Lett., Volume 16 (2009), pp. 991-994 | MR | Zbl

[12.] Hedden, M.; Watson, L. Does Khovanov homology detect the unknot?, Am. J. Math. (2010) | DOI | MR | Zbl

[13.] Juhász, A. Holomorphic discs and sutured manifolds, Algebr. Geom. Topol., Volume 6 (2006), pp. 1429-1457 (electronic) | DOI | MR | Zbl

[14.] Juhász, A. Floer homology and surface decompositions, Geom. Topol., Volume 12 (2008), pp. 299-350 | DOI | MR | Zbl

[15.] Kawasaki, T. The index of elliptic operators over V-manifolds, Nagoya Math. J., Volume 84 (1981), pp. 135-157 | MR | Zbl

[16.] Khovanov, M. A categorification of the Jones polynomial, Duke Math. J., Volume 101 (2000), pp. 359-426 | DOI | MR | Zbl

[17.] Kronheimer, P. B. An obstruction to removing intersection points in immersed surfaces, Topology, Volume 36 (1997), pp. 931-962 | DOI | MR | Zbl

[18.] Kronheimer, P. B.; Mrowka, T. S. Gauge theory for embedded surfaces. I, Topology, Volume 32 (1993), pp. 773-826 | DOI | MR | Zbl

[19.] Kronheimer, P. B.; Mrowka, T. S. Gauge theory for embedded surfaces. II, Topology, Volume 34 (1995), pp. 37-97 | DOI | MR | Zbl

[20.] Kronheimer, P. B.; Mrowka, T. S. Monopoles and Three-Manifolds, New Mathematical Monographs, Cambridge University Press, Cambridge, 2007 | DOI | MR | Zbl

[21.] P. B. Kronheimer and T. S. Mrowka, Knot homology groups from instantons, preprint (2008). | Zbl

[22.] Kronheimer, P. B.; Mrowka, T. S. Instanton Floer homology and the Alexander polynomial, Algebr. Geom. Topol., Volume 10 (2010), pp. 1715-1738 | DOI | MR | Zbl

[23.] Kronheimer, P. B.; Mrowka, T. S. Knots, sutures, and excision, J. Differ. Geom., Volume 84 (2010), pp. 301-364 | MR | Zbl

[24.] Kronheimer, P.; Mrowka, T.; Ozsváth, P.; Szabó, Z. Monopoles and lens space surgeries, Ann. Math. (2), Volume 165 (2007), pp. 457-546 | DOI | MR | Zbl

[25.] Lee, E. S. An endomorphism of the Khovanov invariant, Adv. Math., Volume 197 (2005), pp. 554-586 | DOI | MR | Zbl

[26.] Manolescu, C. An unoriented skein exact triangle for knot Floer homology, Math. Res. Lett., Volume 14 (2007), pp. 839-852 | MR | Zbl

[27.] Manolescu, C.; Ozsváth, P. On the Khovanov and knot Floer homologies of quasi-alternating links, Gökova Geometry/Topology Conference (GGT) (2008), pp. 60-81 | MR | Zbl

[28.] Ozsváth, P.; Szabó, Z. Holomorphic disks and knot invariants, Adv. Math., Volume 186 (2004), pp. 58-116 | DOI | MR | Zbl

[29.] Ozsváth, P.; Szabó, Z. On the Heegaard Floer homology of branched double-covers, Adv. Math., Volume 194 (2005), pp. 1-33 | DOI | MR | Zbl

[30.] Rasmussen, J. Khovanov homology and the slice genus, Invent. Math. (2010) | DOI | MR | Zbl

[31.] J. Rasmussen, Floer homology and knot complements, PhD thesis, Harvard University, 2003. | MR

[32.] Rasmussen, J. Knot polynomials and knot homologies, Geometry and Topology of Manifolds (Fields Inst. Commun., 47), Am. Math. Soc., Providence (2005), pp. 261-280 | MR | Zbl

[33.] Taubes, C. H. Casson’s invariant and gauge theory, J. Differ. Geom., Volume 31 (1990), pp. 547-599 | MR | Zbl

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