Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alexander, J.W.: Topological invariants of knots and links. Trans. Am. Math. Soc. 30(2), 275–306 (1928)
Baader, S.: Fibered knot with periodic homological monodromy. Math Overflow Response. http://mathoverflow.net/questions/128513
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)
Baker, K.L., Hoffman, N.R.: The Poincaré homology sphere, lens space surgeries, and some knots with tunnel number two. Preprint arXiv:1504.06682
Baker, K.L., Luecke, J.: Asymmetric L-space knots (in preparation)
Baker, K.L., Moore, A.H.: Montesinos knots, Hopf plumbings, and L-space surgeries. Preprint arXiv:1404.7585
Baldwin, J.A., Grigsby, J.E.: Categorified invariants and the braid group. Proc. Amer. Math. Soc. 143(7), 2801–2814 (2015)
Baldwin, J.A., Levine, A.S.: A combinatorial spanning tree model for knot Floer homology. Adv. Math. 231(3–4), 1886–1939 (2012)
Bankwitz, C.: Über die Torsionszahlen der alternierenden Knoten. Math. Ann. 103(1), 145–161 (1930)
Bar-Natan, D.: Khovanov’s homology for tangles and cobordisms. Geom. Topol. 9, 1443–1499 (2005)
Berge, J.: Some knots with surgeries yielding lens spaces. Unpublished manuscript
Boyer, S., Zhang, X.: Finite Dehn surgery on knots. J. Am. Math. Soc. 9(4), 1005–1050 (1996)
Burde, G., Zieschang, H.: Knots, volume 5 of de Gruyter Studies in Mathematics, 2nd edn. Walter de Gruyter & Co., Berlin (2003)
Clarkson, C.: Three-manifold mutations detected by Heegaard Floer homology. Algebr. Geom. Topol. 17(1), 1–16 (2017)
Cromwell, P.R.: Homogeneous links. J. Lond. Math. Soc. (2) 39(3), 535–552 (1989)
Farb, B., Margalit, D.: A Primer on Mapping Class Groups. Princeton Mathematical Series, vol. 49. Princeton University Press, Princeton, NJ (2012)
Frøyshov, K.A.: The Seiberg–Witten equations and four-manifolds with boundary. Math. Res. Lett. 3(3), 373–390 (1996)
Frøyshov, K.A.: Equivariant aspects of Yang–Mills Floer theory. Topology 41(3), 525–552 (2002)
Frøyshov, K.A.: An inequality for the \(h\)-invariant in instanton Floer theory. Topology 43(2), 407–432 (2004)
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)
Gabai, D.: Foliations and the topology of 3-manifolds II. J. Differ. Geom. 26, 461–478 (1987)
Ghiggini, P.: Knot Floer homology detects genus-one fibred knots. Am. J. Math. 130(5), 1151–1169 (2008)
Greene, J.E., Watson, L.: Turaev torsion, definite 4-manifolds, and quasi-alternating knots. Bull. Lond. Math. Soc. 45(5), 962–972 (2013)
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.)
Hedden, M.: On knot Floer homology and cabling. Algebr. Geom. Topol. 5, 1197–1222 (2005). (electronic)
Hedden, M.: Knot Floer homology of Whitehead doubles. Geom. Topol. 11, 2277–2338 (2007)
Hedden, M.: On knot Floer homology and cabling. II. Int. Math. Res. Not. IMRN 12, 2248–2274 (2009)
Hedden, M.: Notions of positivity and the Ozsváth–Szabó concordance invariant. J. Knot Theory Ramif. 19(5), 617–629 (2010)
Hedden, M.: On Floer homology and the Berge conjecture on knots admitting lens space surgeries. Trans. Am. Math. Soc. 363(2), 949–968 (2011)
Hedden, M., Ni, Y.: Khovanov module and the detection of unlinks. Geom. Topol. 17(5), 3027–3076 (2013)
Hom, J.: Personal correspondence
Hom, J.: A note on cabling and \(L\)-space surgeries. Algebr. Geom. Topol. 11(1), 219–223 (2011)
Hom, J.: Bordered Heegaard Floer homology and the tau-invariant of cable knots. J. Topol. 7(2), 287–326 (2014)
Hom, J.: The knot Floer complex and the smooth concordance group. Comment. Math. Helv. 89(3), 537–570 (2014)
Hom, J.: An infinite rank summand of topologically slice knots. Geom. Topol. 19(2), 1063–1110 (2015)
Honda, K., Kazez, W.H., Matić, G.: Right-veering diffeomorphisms of compact surfaces with boundary. Invent. Math. 169(2), 427–449 (2007)
Hughes, M.C.: A note on Khovanov–Rozansky \(sl_2\)-homology and ordinary Khovanov homology. J. Knot Theory Ramifications 23(12), 1450057 (2014)
Juhász, A.: Floer homology and surface decompositions. Geom. Topol. 12(1), 299–350 (2008)
Kanenobu, T.: Infinitely many knots with the same polynomial invariant. Proc. Am. Math. Soc. 97(1), 158–162 (1986)
Khovanov, M.: A categorification of the Jones polynomial. Duke Math. J. 101(3), 359–426 (2000)
Khovanov, M.: Patterns in knot cohomology. I. Exp. Math. 12(3), 365–374 (2003)
Khovanov, M., Rozansky, L.: Matrix factorizations and link homology. Fund. Math. 199(1), 1–91 (2008)
Khovanov, M., Rozansky, L.: Matrix factorizations and link homology. II. Geom. Topol. 12(3), 1387–1425 (2008)
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)
Kronheimer, P.B., Mrowka, T.S., Ozsváth, P.S., Szabó, Z.: Monopoles and lens space surgeries. Ann. Math. 165(2), 457–546 (2007)
Kronheimer, P.B., Mrowka, T.S.: Khovanov homology is an unknot-detector. Publ. Math. Inst. Hautes Étud. Sci. 113, 97–208 (2011)
Levine, A.S.: Slicing mixed Bing–Whitehead doubles. J. Topol. 5(3), 713–726 (2012)
Li, E., Ni, Y.: Half-integral finite surgeries on knots in \(S^3\). Ann. Fac. Sci. Toulouse Math. (6) 24(5), 1157–1178 (2015)
Lidman, T., Moore, A.H.: Pretzel knots with L-space surgeries. Michigan Math. J. 65(1), 105–130 (2016)
Lipshitz, R., Sarkar, S.: A refinement of Rasmussen’s \(S\)-invariant. Duke Math. J. 163(5), 923–952 (2014)
Lobb, A.: The Kanenobu knots and Khovanov–Rozansky homology. Proc. Am. Math. Soc. 142(4), 1447–1455 (2014)
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)
Manolescu, C., Ozsváth, P., Sarkar, S.: A combinatorial description of knot Floer homology. Ann. Math. (2) 169(2), 633–660 (2009)
Manolescu, C., Ozsváth, P., Szabó, Z., Thurston, D.: On combinatorial link Floer homology. Geom. Topol. 11, 2339–2412 (2007)
Moore, A.H.:: Behavior of knot Floer homology under conway and genus two mutation. Ph.D. thesis, University of Texas at Austin (2013)
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)
Mosher, L.: Mapping class groups are automatic. Ann. Math. (2) 142(2), 303–384 (1995)
Ni, Y.: Sutured Heegaard diagrams for knots. Algebr. Geom. Topol. 6, 513–537 (2006)
Ni, Y.: Knot Floer homology detects fibred knots. Invent. Math. 170(3), 577–608 (2007)
Ni, Y.: Heegaard Floer homology and fibred 3-manifolds. Am. J. Math. 131(4), 1047–1063 (2009)
Ni, Y.: Link Floer homology detects the Thurston norm. Geom. Topol. 13(5), 2991–3019 (2009)
Ni, Y.: Homological actions on sutured Floer homology. Math. Res. Lett. 21(5), 1177–1197 (2014)
Ozsváth, P., Szabó, Z.: On the skein exact squence for knot Floer homology. Preprint arXiv:0707.1165
Ozsváth, P., Szabó, Z.: The Dehn surgery characterization of the trefoil and the figure eight knot. Preprint arXiv:math/0604079
Ozsváth, P., Szabó, Z.: Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary. Adv. Math. 173(2), 179–261 (2003)
Ozsváth, P., Szabó, Z.: Heegaard Floer homology and alternating knots. Geom. Topol. 7, 225–254 (2003)
Ozsváth, P., Szabó, Z.: Holomorphic disks and knot invariants. Adv. Math. 186(1), 58–116 (2004)
Ozsváth, P., Szabó, Z.: Heegaard Floer homology and contact structures. Duke Math. J. 129(1), 39–61 (2005)
Ozsváth, P., Szabó, Z.: Knot Floer homology and integer surgeries. Algebr. Geom. Topol. 8(1), 101–153 (2008)
Ozsváth, P., Stipsicz, A., Szabó, Z.: Floer homology and singular knots. J. Topol. 2(2), 380–404 (2009)
Ozsváth, P., Szabó, Z.: On knot Floer homology and lens space surgeries. Topology 44(6), 1281–1300 (2005)
Ozsváth, P., Szabó, Z., Thurston, D.: Legendrian knots, transverse knots and combinatorial Floer homology. Geom. Topol. 12(2), 941–980 (2008)
Ozsváth, P.S., Szabó, Z.: Knot Floer homology and the four-ball genus. Geom. Topol. 7, 615–639 (2003)
Ozsváth, P.S., Szabó, Z.: Holomorphic disks and genus bounds. Geom. Topol. 8, 311–334 (2004)
Ozsváth, P.S., Szabó, Z.: Knot Floer homology and rational surgeries. Algebr. Geom. Topol. 11(1), 1–68 (2011)
Petkova, I.: Cables of thin knots and bordered Heegaard Floer homology. Quantum Topol. 4(4), 377–409 (2013)
Plamenevskaya, O.: A combinatorial description of the Heegaard Floer contact invariant. Algebr. Geom. Topol. 7, 1201–1209 (2007)
Rasmussen, J.: Lens space surgeries and L-space homology spheres. Preprint arXiv:0710.2531
Rasmussen, J.: Floer homology and knot complements. Ph.D. thesis, Harvard University (2003)
Rasmussen, J.: Lens space surgeries and a conjecture of Goda and Teragaito. Geom. Topol. 8, 1013–1031 (2004)
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)
Rasmussen, J.: Khovanov homology and the slice genus. Invent. Math. 182(2), 419–447 (2010)
Rasmussen, J.: Some differentials on Khovanov–Rozansky homology. Geom. Topol. 19(6), 3031–3104 (2015)
Rolfsen, D.: Knots and Links. Mathematics Lecture Series, No. 7. Publish or Perish Inc., Berkeley, CA (1976)
Sarkar, S., Wang, J.: An algorithm for computing some Heegaard Floer homologies. Ann. Math. (2) 171(2), 1213–1236 (2010)
Scharlemann, M.: Smooth spheres in \({ R}^4\) with four critical points are standard. Invent. Math. 79(1), 125–141 (1985)
Scharlemann, M., Thompson, A.: Link genus and the Conway moves. Comment. Math. Helv. 64(4), 527–535 (1989)
Seed, C.: Knotkit. https://github.com/cseed/knotkit
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)
Stoimenow, A.: Realizing Alexander polynomials by hyperbolic links. Expo. Math. 28(2), 133–178 (2010)
Thurston, W., Winkelnkemper, H.: On the existence of contact forms. Proc. Am. Math. Soc. 52, 345–347 (1975)
Thurston, D., Lipshitz, R., Ozsváth, P.: Bordered Heegaard Floer homology. Mem. Am. Math. Soc. (to appear). arXiv:0810.0687
Torisu, I.: Convex contact structures and fibered links in 3-manifolds. Int. Math. Res. Not. 9, 441–454 (2000)
Turner, P.R.: Calculating Bar-Natan’s characteristic two Khovanov homology. J. Knot Theory Ramif. 15(10), 1335–1356 (2006)
Watson, L.: Knots with identical Khovanov homology. Algebr. Geom. Topol. 7, 1389–1407 (2007)
Watson, L.: New proofs of certain finite filling results via Khovanov homology. Quantum Topol. 4(4), 353–376 (2013)
Watson, L.: Khovanov homology and the symmetry group of a knot. Adv. Math. 313, 915–946 (2017)
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author was partially supported by NSF Grants DMS-0706979 DMS-0906258, CAREER DMS-1150872, and an A.P. Sloan Research Fellowship.
The second author was partially supported by a Marie Curie Career Integration Grant (HFFUNDGRP).
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Hedden, M., Watson, L. On the geography and botany of knot Floer homology. Sel. Math. New Ser. 24, 997–1037 (2018). https://doi.org/10.1007/s00029-017-0351-5
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00029-017-0351-5