Selecta Mathematica

, Volume 24, Issue 2, pp 997–1037 | Cite as

On the geography and botany of knot Floer homology

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This paper explores two questions: (1) Which bigraded groups arise as the knot Floer homology of a knot in the three-sphere? (2) Given a knot, how many distinct knots share its Floer homology? Regarding the first, we show there exist bigraded groups satisfying all previously known constraints of knot Floer homology which do not arise as the invariant of a knot. This leads to a new constraint for knots admitting lens space surgeries, as well as a proof that the rank of knot Floer homology detects the trefoil knot. For the second, we show that any non-trivial band sum of two unknots gives rise to an infinite family of distinct knots with isomorphic knot Floer homology. We also prove that the fibered knot with identity monodromy is strongly detected by its knot Floer homology, implying that Floer homology solves the word problem for mapping class groups of surfaces with non-empty boundary. Finally, we survey some conjectures and questions and, based on the results described above, formulate some new ones.


  1. 1.
    Alexander, J.W.: Topological invariants of knots and links. Trans. Am. Math. Soc. 30(2), 275–306 (1928)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Baader, S.: Fibered knot with periodic homological monodromy. Math Overflow Response.
  3. 3.
    Baker, K.L., Elisenda Grigsby, J., Hedden, M.: Grid diagrams for lens spaces and combinatorial knot Floer homology. Int. Math. Res. Not. IMRN (10):Art. ID rnm024, 39 (2008)Google Scholar
  4. 4.
    Baker, K.L., Hoffman, N.R.: The Poincaré homology sphere, lens space surgeries, and some knots with tunnel number two. Preprint arXiv:1504.06682
  5. 5.
    Baker, K.L., Luecke, J.: Asymmetric L-space knots (in preparation)Google Scholar
  6. 6.
    Baker, K.L., Moore, A.H.: Montesinos knots, Hopf plumbings, and L-space surgeries. Preprint arXiv:1404.7585
  7. 7.
    Baldwin, J.A., Grigsby, J.E.: Categorified invariants and the braid group. Proc. Amer. Math. Soc. 143(7), 2801–2814 (2015)Google Scholar
  8. 8.
    Baldwin, J.A., Levine, A.S.: A combinatorial spanning tree model for knot Floer homology. Adv. Math. 231(3–4), 1886–1939 (2012)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bankwitz, C.: Über die Torsionszahlen der alternierenden Knoten. Math. Ann. 103(1), 145–161 (1930)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Bar-Natan, D.: Khovanov’s homology for tangles and cobordisms. Geom. Topol. 9, 1443–1499 (2005)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Berge, J.: Some knots with surgeries yielding lens spaces. Unpublished manuscriptGoogle Scholar
  12. 12.
    Boyer, S., Zhang, X.: Finite Dehn surgery on knots. J. Am. Math. Soc. 9(4), 1005–1050 (1996)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Burde, G., Zieschang, H.: Knots, volume 5 of de Gruyter Studies in Mathematics, 2nd edn. Walter de Gruyter & Co., Berlin (2003)Google Scholar
  14. 14.
    Clarkson, C.: Three-manifold mutations detected by Heegaard Floer homology. Algebr. Geom. Topol. 17(1), 1–16 (2017)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Cromwell, P.R.: Homogeneous links. J. Lond. Math. Soc. (2) 39(3), 535–552 (1989)Google Scholar
  16. 16.
    Farb, B., Margalit, D.: A Primer on Mapping Class Groups. Princeton Mathematical Series, vol. 49. Princeton University Press, Princeton, NJ (2012)Google Scholar
  17. 17.
    Frøyshov, K.A.: The Seiberg–Witten equations and four-manifolds with boundary. Math. Res. Lett. 3(3), 373–390 (1996)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Frøyshov, K.A.: Equivariant aspects of Yang–Mills Floer theory. Topology 41(3), 525–552 (2002)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Frøyshov, K.A.: An inequality for the \(h\)-invariant in instanton Floer theory. Topology 43(2), 407–432 (2004)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Gabai, D.: The Murasugi sum is a natural geometric operation. In: Low-Dimensional Topology (San Francisco. CA, 1981), Volume 20 of Contemporary Mathematics, pp. 131–143. American Mathematical Society, Providence, RI (1983)Google Scholar
  21. 21.
    Gabai, D.: Foliations and the topology of 3-manifolds II. J. Differ. Geom. 26, 461–478 (1987)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Ghiggini, P.: Knot Floer homology detects genus-one fibred knots. Am. J. Math. 130(5), 1151–1169 (2008)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Greene, J.E., Watson, L.: Turaev torsion, definite 4-manifolds, and quasi-alternating knots. Bull. Lond. Math. Soc. 45(5), 962–972 (2013)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Hancock, S., Hom, J., Newman, M.: On the knot Floer filtration of the concordance group. J. Knot Theory Ramif. 22(14), 1350084 (2013). (30 pp.)Google Scholar
  25. 25.
    Hedden, M.: On knot Floer homology and cabling. Algebr. Geom. Topol. 5, 1197–1222 (2005). (electronic)Google Scholar
  26. 26.
    Hedden, M.: Knot Floer homology of Whitehead doubles. Geom. Topol. 11, 2277–2338 (2007)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Hedden, M.: On knot Floer homology and cabling. II. Int. Math. Res. Not. IMRN 12, 2248–2274 (2009)MathSciNetMATHGoogle Scholar
  28. 28.
    Hedden, M.: Notions of positivity and the Ozsváth–Szabó concordance invariant. J. Knot Theory Ramif. 19(5), 617–629 (2010)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Hedden, M.: On Floer homology and the Berge conjecture on knots admitting lens space surgeries. Trans. Am. Math. Soc. 363(2), 949–968 (2011)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Hedden, M., Ni, Y.: Khovanov module and the detection of unlinks. Geom. Topol. 17(5), 3027–3076 (2013)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Hom, J.: Personal correspondenceGoogle Scholar
  32. 32.
    Hom, J.: A note on cabling and \(L\)-space surgeries. Algebr. Geom. Topol. 11(1), 219–223 (2011)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Hom, J.: Bordered Heegaard Floer homology and the tau-invariant of cable knots. J. Topol. 7(2), 287–326 (2014)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Hom, J.: The knot Floer complex and the smooth concordance group. Comment. Math. Helv. 89(3), 537–570 (2014)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Hom, J.: An infinite rank summand of topologically slice knots. Geom. Topol. 19(2), 1063–1110 (2015)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Honda, K., Kazez, W.H., Matić, G.: Right-veering diffeomorphisms of compact surfaces with boundary. Invent. Math. 169(2), 427–449 (2007)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Hughes, M.C.: A note on Khovanov–Rozansky \(sl_2\)-homology and ordinary Khovanov homology. J. Knot Theory Ramifications 23(12), 1450057 (2014)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Juhász, A.: Floer homology and surface decompositions. Geom. Topol. 12(1), 299–350 (2008)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Kanenobu, T.: Infinitely many knots with the same polynomial invariant. Proc. Am. Math. Soc. 97(1), 158–162 (1986)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Khovanov, M.: A categorification of the Jones polynomial. Duke Math. J. 101(3), 359–426 (2000)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Khovanov, M.: Patterns in knot cohomology. I. Exp. Math. 12(3), 365–374 (2003)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Khovanov, M., Rozansky, L.: Matrix factorizations and link homology. Fund. Math. 199(1), 1–91 (2008)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Khovanov, M., Rozansky, L.: Matrix factorizations and link homology. II. Geom. Topol. 12(3), 1387–1425 (2008)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Kirby, R. (ed.): Problems in low-dimensional topology. Geometric Topology (Athens, GA, 1993), Volume 2 of AMS/IP Studies in Advanced Mathematics, pp. 35–473. American Mathematical Society, Providence, RI (1997)Google Scholar
  45. 45.
    Kronheimer, P.B., Mrowka, T.S., Ozsváth, P.S., Szabó, Z.: Monopoles and lens space surgeries. Ann. Math. 165(2), 457–546 (2007)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Kronheimer, P.B., Mrowka, T.S.: Khovanov homology is an unknot-detector. Publ. Math. Inst. Hautes Étud. Sci. 113, 97–208 (2011)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Levine, A.S.: Slicing mixed Bing–Whitehead doubles. J. Topol. 5(3), 713–726 (2012)Google Scholar
  48. 48.
    Li, E., Ni, Y.: Half-integral finite surgeries on knots in \(S^3\). Ann. Fac. Sci. Toulouse Math. (6) 24(5), 1157–1178 (2015)Google Scholar
  49. 49.
    Lidman, T., Moore, A.H.: Pretzel knots with L-space surgeries. Michigan Math. J. 65(1), 105–130 (2016)Google Scholar
  50. 50.
    Lipshitz, R., Sarkar, S.: A refinement of Rasmussen’s \(S\)-invariant. Duke Math. J. 163(5), 923–952 (2014)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Lobb, A.: The Kanenobu knots and Khovanov–Rozansky homology. Proc. Am. Math. Soc. 142(4), 1447–1455 (2014)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Manolescu, C., Ozsváth, P.: On the Khovanov and knot Floer homologies of quasi-alternating links. In: Proceedings of Gökova Geometry-Topology Conference 2007, pp. 60–81. Gökova Geometry/Topology Conference (GGT), Gökova (2008)Google Scholar
  53. 53.
    Manolescu, C., Ozsváth, P., Sarkar, S.: A combinatorial description of knot Floer homology. Ann. Math. (2) 169(2), 633–660 (2009)Google Scholar
  54. 54.
    Manolescu, C., Ozsváth, P., Szabó, Z., Thurston, D.: On combinatorial link Floer homology. Geom. Topol. 11, 2339–2412 (2007)MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Moore, A.H.:: Behavior of knot Floer homology under conway and genus two mutation. Ph.D. thesis, University of Texas at Austin (2013)Google Scholar
  56. 56.
    Moore, A.H., Starkston, L.: Genus-two mutant knots with the same dimension in knot Floer and Khovanov homologies. Algebr. Geom. Topol. 15(1), 43–63 (2015)MathSciNetCrossRefMATHGoogle Scholar
  57. 57.
    Mosher, L.: Mapping class groups are automatic. Ann. Math. (2) 142(2), 303–384 (1995)Google Scholar
  58. 58.
    Ni, Y.: Sutured Heegaard diagrams for knots. Algebr. Geom. Topol. 6, 513–537 (2006)MathSciNetCrossRefMATHGoogle Scholar
  59. 59.
    Ni, Y.: Knot Floer homology detects fibred knots. Invent. Math. 170(3), 577–608 (2007)MathSciNetCrossRefMATHGoogle Scholar
  60. 60.
    Ni, Y.: Heegaard Floer homology and fibred 3-manifolds. Am. J. Math. 131(4), 1047–1063 (2009)MathSciNetCrossRefMATHGoogle Scholar
  61. 61.
    Ni, Y.: Link Floer homology detects the Thurston norm. Geom. Topol. 13(5), 2991–3019 (2009)MathSciNetCrossRefMATHGoogle Scholar
  62. 62.
    Ni, Y.: Homological actions on sutured Floer homology. Math. Res. Lett. 21(5), 1177–1197 (2014)MathSciNetCrossRefMATHGoogle Scholar
  63. 63.
    Ozsváth, P., Szabó, Z.: On the skein exact squence for knot Floer homology. Preprint arXiv:0707.1165
  64. 64.
    Ozsváth, P., Szabó, Z.: The Dehn surgery characterization of the trefoil and the figure eight knot. Preprint arXiv:math/0604079
  65. 65.
    Ozsváth, P., Szabó, Z.: Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary. Adv. Math. 173(2), 179–261 (2003)MathSciNetCrossRefMATHGoogle Scholar
  66. 66.
    Ozsváth, P., Szabó, Z.: Heegaard Floer homology and alternating knots. Geom. Topol. 7, 225–254 (2003)MathSciNetCrossRefMATHGoogle Scholar
  67. 67.
    Ozsváth, P., Szabó, Z.: Holomorphic disks and knot invariants. Adv. Math. 186(1), 58–116 (2004)MathSciNetCrossRefMATHGoogle Scholar
  68. 68.
    Ozsváth, P., Szabó, Z.: Heegaard Floer homology and contact structures. Duke Math. J. 129(1), 39–61 (2005)MathSciNetCrossRefMATHGoogle Scholar
  69. 69.
    Ozsváth, P., Szabó, Z.: Knot Floer homology and integer surgeries. Algebr. Geom. Topol. 8(1), 101–153 (2008)MathSciNetCrossRefMATHGoogle Scholar
  70. 70.
    Ozsváth, P., Stipsicz, A., Szabó, Z.: Floer homology and singular knots. J. Topol. 2(2), 380–404 (2009)MathSciNetCrossRefMATHGoogle Scholar
  71. 71.
    Ozsváth, P., Szabó, Z.: On knot Floer homology and lens space surgeries. Topology 44(6), 1281–1300 (2005)MathSciNetCrossRefMATHGoogle Scholar
  72. 72.
    Ozsváth, P., Szabó, Z., Thurston, D.: Legendrian knots, transverse knots and combinatorial Floer homology. Geom. Topol. 12(2), 941–980 (2008)MathSciNetCrossRefMATHGoogle Scholar
  73. 73.
    Ozsváth, P.S., Szabó, Z.: Knot Floer homology and the four-ball genus. Geom. Topol. 7, 615–639 (2003)MathSciNetCrossRefMATHGoogle Scholar
  74. 74.
    Ozsváth, P.S., Szabó, Z.: Holomorphic disks and genus bounds. Geom. Topol. 8, 311–334 (2004)MathSciNetCrossRefMATHGoogle Scholar
  75. 75.
    Ozsváth, P.S., Szabó, Z.: Knot Floer homology and rational surgeries. Algebr. Geom. Topol. 11(1), 1–68 (2011)MathSciNetCrossRefMATHGoogle Scholar
  76. 76.
    Petkova, I.: Cables of thin knots and bordered Heegaard Floer homology. Quantum Topol. 4(4), 377–409 (2013)MathSciNetCrossRefMATHGoogle Scholar
  77. 77.
    Plamenevskaya, O.: A combinatorial description of the Heegaard Floer contact invariant. Algebr. Geom. Topol. 7, 1201–1209 (2007)MathSciNetCrossRefMATHGoogle Scholar
  78. 78.
    Rasmussen, J.: Lens space surgeries and L-space homology spheres. Preprint arXiv:0710.2531
  79. 79.
    Rasmussen, J.: Floer homology and knot complements. Ph.D. thesis, Harvard University (2003)Google Scholar
  80. 80.
    Rasmussen, J.: Lens space surgeries and a conjecture of Goda and Teragaito. Geom. Topol. 8, 1013–1031 (2004)MathSciNetCrossRefMATHGoogle Scholar
  81. 81.
    Rasmussen, J.: Knot polynomials and knot homologies. In: Geometry and Topology of Manifolds. Volume 47 of Fields Institute Communication, pp. 261–280. American Mathematical Society, Providence, RI (2005)Google Scholar
  82. 82.
    Rasmussen, J.: Khovanov homology and the slice genus. Invent. Math. 182(2), 419–447 (2010)MathSciNetCrossRefMATHGoogle Scholar
  83. 83.
    Rasmussen, J.: Some differentials on Khovanov–Rozansky homology. Geom. Topol. 19(6), 3031–3104 (2015)MathSciNetCrossRefMATHGoogle Scholar
  84. 84.
    Rolfsen, D.: Knots and Links. Mathematics Lecture Series, No. 7. Publish or Perish Inc., Berkeley, CA (1976)Google Scholar
  85. 85.
    Sarkar, S., Wang, J.: An algorithm for computing some Heegaard Floer homologies. Ann. Math. (2) 171(2), 1213–1236 (2010)Google Scholar
  86. 86.
    Scharlemann, M.: Smooth spheres in \({ R}^4\) with four critical points are standard. Invent. Math. 79(1), 125–141 (1985)MathSciNetCrossRefMATHGoogle Scholar
  87. 87.
    Scharlemann, M., Thompson, A.: Link genus and the Conway moves. Comment. Math. Helv. 64(4), 527–535 (1989)MathSciNetCrossRefMATHGoogle Scholar
  88. 88.
  89. 89.
    Stoimenow, A.: On the crossing number of positive knots and braids and braid index criteria of Jones and Morton–Williams–Franks. Trans. Am. Math. Soc. 354(10), 3927–3954 (2002). (electronic)Google Scholar
  90. 90.
    Stoimenow, A.: Realizing Alexander polynomials by hyperbolic links. Expo. Math. 28(2), 133–178 (2010)MathSciNetCrossRefMATHGoogle Scholar
  91. 91.
    Thurston, W., Winkelnkemper, H.: On the existence of contact forms. Proc. Am. Math. Soc. 52, 345–347 (1975)MathSciNetCrossRefMATHGoogle Scholar
  92. 92.
    Thurston, D., Lipshitz, R., Ozsváth, P.: Bordered Heegaard Floer homology. Mem. Am. Math. Soc. (to appear). arXiv:0810.0687
  93. 93.
    Torisu, I.: Convex contact structures and fibered links in 3-manifolds. Int. Math. Res. Not. 9, 441–454 (2000)MathSciNetCrossRefMATHGoogle Scholar
  94. 94.
    Turner, P.R.: Calculating Bar-Natan’s characteristic two Khovanov homology. J. Knot Theory Ramif. 15(10), 1335–1356 (2006)MathSciNetCrossRefMATHGoogle Scholar
  95. 95.
    Watson, L.: Knots with identical Khovanov homology. Algebr. Geom. Topol. 7, 1389–1407 (2007)MathSciNetCrossRefMATHGoogle Scholar
  96. 96.
    Watson, L.: New proofs of certain finite filling results via Khovanov homology. Quantum Topol. 4(4), 353–376 (2013)MathSciNetCrossRefMATHGoogle Scholar
  97. 97.
    Watson, L.: Khovanov homology and the symmetry group of a knot. Adv. Math. 313, 915–946 (2017)MathSciNetCrossRefMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsMichigan State UniversityEast LansingUSA
  2. 2.School of Mathematics and StatisticsUniversity of GlasgowGlasgowUK
  3. 3.Département de MathématiquesUniversité de SherbrookeSherbrookeCanada

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