Knot Floer homology obstructs ribbon concordance

Abstract

We prove that the map on knot Floer homology induced by a ribbon concordance is injective. As a consequence, we prove that the Seifert genus is monotonic under ribbon concordance. Generalizing theorems of Gabai and Scharlemann, we also prove that the Seifert genus is super-additive under band connected sums of arbitrarily many knots. Our results give evidence for a conjecture of Gordon that ribbon concordance is a partial order on the set of knots.

  • [ChantraineNonCollarable] Go to document B. Chantraine, "Some non-collarable slices of Lagrangian surfaces," Bull. Lond. Math. Soc., vol. 44, iss. 5, pp. 981-987, 2012.
    @ARTICLE{ChantraineNonCollarable,
      author = {Chantraine, Baptiste},
      title = {Some non-collarable slices of {L}agrangian surfaces},
      journal = {Bull. Lond. Math. Soc.},
      fjournal = {Bulletin of the London Mathematical Society},
      volume = {44},
      year = {2012},
      number = {5},
      pages = {981--987},
      issn = {0024-6093},
      mrclass = {57R17 (57M50)},
      mrnumber = {2975156},
      mrreviewer = {Jonathan David Evans},
      doi = {10.1112/blms/bds026},
      url = {https://doi.org/10.1112/blms/bds026},
      zblnumber = {1251.57017},
      }
  • [CornwallNgSivek] Go to document C. Cornwell, L. Ng, and S. Sivek, "Obstructions to Lagrangian concordance," Algebr. Geom. Topol., vol. 16, iss. 2, pp. 797-824, 2016.
    @ARTICLE{CornwallNgSivek,
      author = {Cornwell, Christopher and Ng, Lenhard and Sivek, Steven},
      title = {Obstructions to {L}agrangian concordance},
      journal = {Algebr. Geom. Topol.},
      fjournal = {Algebraic \& Geometric Topology},
      volume = {16},
      year = {2016},
      number = {2},
      pages = {797--824},
      issn = {1472-2747},
      mrclass = {57M25 (53D12 53D42 57R17)},
      mrnumber = {3493408},
      mrreviewer = {Georgios Dimitroglou Rizell},
      doi = {10.2140/agt.2016.16.797},
      url = {https://doi.org/10.2140/agt.2016.16.797},
      zblnumber = {1346.57009},
      }
  • [Ekholm-Honda-Kalman] Go to document T. Ekholm, K. Honda, and T. Kálmán, "Legendrian knots and exact Lagrangian cobordisms," J. Eur. Math. Soc. (JEMS), vol. 18, iss. 11, pp. 2627-2689, 2016.
    @ARTICLE{Ekholm-Honda-Kalman,
      author = {Ekholm, Tobias and Honda, Ko and K\'{a}lm\'{a}n, Tam\'{a}s},
      title = {Legendrian knots and exact {L}agrangian cobordisms},
      journal = {J. Eur. Math. Soc. (JEMS)},
      fjournal = {Journal of the European Mathematical Society (JEMS)},
      volume = {18},
      year = {2016},
      number = {11},
      pages = {2627--2689},
      issn = {1435-9855},
      mrclass = {53D42 (53D10 53D40 57M27 57R90)},
      mrnumber = {3562353},
      mrreviewer = {Georgios Dimitroglou Rizell},
      doi = {10.4171/JEMS/650},
      url = {https://doi.org/10.4171/JEMS/650},
      zblnumber = {1357.57044},
      }
  • [GabaiBand] Go to document D. Gabai, "Genus is superadditive under band connected sum," Topology, vol. 26, iss. 2, pp. 209-210, 1987.
    @ARTICLE{GabaiBand,
      author = {Gabai, David},
      title = {Genus is superadditive under band connected sum},
      journal = {Topology},
      fjournal = {Topology. An International Journal of Mathematics},
      volume = {26},
      year = {1987},
      number = {2},
      pages = {209--210},
      issn = {0040-9383},
      mrclass = {57M25},
      mrnumber = {0895573},
      mrreviewer = {Martin Scharlemann},
      doi = {10.1016/0040-9383(87)90061-9},
      url = {https://doi.org/10.1016/0040-9383(87)90061-9},
      zblnumber = {0621.57004},
      }
  • [GhigginiFibered] Go to document P. Ghiggini, "Knot Floer homology detects genus-one fibred knots," Amer. J. Math., vol. 130, iss. 5, pp. 1151-1169, 2008.
    @ARTICLE{GhigginiFibered,
      author = {Ghiggini, Paolo},
      title = {Knot {F}loer homology detects genus-one fibred knots},
      journal = {Amer. J. Math.},
      fjournal = {American Journal of Mathematics},
      volume = {130},
      year = {2008},
      number = {5},
      pages = {1151--1169},
      issn = {0002-9327},
      mrclass = {57M25 (57R58)},
      mrnumber = {2450204},
      doi = {10.1353/ajm.0.0016},
      url = {https://doi.org/10.1353/ajm.0.0016},
      zblnumber = {1149.57019},
      }
  • [Gordon] Go to document . M. C. Gordon, "Ribbon concordance of knots in the $3$-sphere," Math. Ann., vol. 257, iss. 2, pp. 157-170, 1981.
    @ARTICLE{Gordon,
      author = {Gordon, C. {\relax McA}.},
      title = {Ribbon concordance of knots in the {$3$}-sphere},
      journal = {Math. Ann.},
      fjournal = {Mathematische Annalen},
      volume = {257},
      year = {1981},
      number = {2},
      pages = {157--170},
      issn = {0025-5831},
      mrclass = {57M25},
      mrnumber = {0634459},
      mrreviewer = {J. P. Levine},
      doi = {10.1007/BF01458281},
      url = {https://doi.org/10.1007/BF01458281},
      zblnumber = {0451.57001},
      }
  • [HeddenWatsonBotanyGeography] Go to document M. Hedden and L. Watson, "On the geography and botany of knot Floer homology," Selecta Math. (N.S.), vol. 24, iss. 2, pp. 997-1037, 2018.
    @ARTICLE{HeddenWatsonBotanyGeography,
      author = {Hedden, Matthew and Watson, Liam},
      title = {On the geography and botany of knot {F}loer homology},
      journal = {Selecta Math. (N.S.)},
      fjournal = {Selecta Mathematica. New Series},
      volume = {24},
      year = {2018},
      number = {2},
      pages = {997--1037},
      issn = {1022-1824},
      mrclass = {57M27 (20F36)},
      mrnumber = {3782416},
      doi = {10.1007/s00029-017-0351-5},
      url = {https://doi.org/10.1007/s00029-017-0351-5},
      zblnumber = {06862004},
      }
  • [HKMTQFT] K. Honda, W. Kazez, and G. Matić, Contact structures, sutured Floer homology and TQFT, 2008.
    @MISC{HKMTQFT,
      author = {Honda, K. and Kazez, W. and Matić, G.},
      title = {Contact structures, sutured {F}loer homology and {TQFT}},
      arxiv = {0807.2431},
      year = {2008},
      zblnumber = {},
      }
  • [JuhaszSurfaceDecomp] Go to document A. Juhász, "Floer homology and surface decompositions," Geom. Topol., vol. 12, iss. 1, pp. 299-350, 2008.
    @ARTICLE{JuhaszSurfaceDecomp,
      author = {Juh\'{a}sz, Andr\'{a}s},
      title = {Floer homology and surface decompositions},
      journal = {Geom. Topol.},
      fjournal = {Geometry \& Topology},
      volume = {12},
      year = {2008},
      number = {1},
      pages = {299--350},
      issn = {1465-3060},
      mrclass = {57M27 (57R58)},
      mrnumber = {2390347},
      mrreviewer = {Stanislav Jabuka},
      doi = {10.2140/gt.2008.12.299},
      url = {https://doi.org/10.2140/gt.2008.12.299},
      zblnumber = {1167.57005},
      }
  • [JCob] Go to document A. Juhász, "Cobordisms of sutured manifolds and the functoriality of link Floer homology," Adv. Math., vol. 299, pp. 940-1038, 2016.
    @ARTICLE{JCob,
      author = {Juh\'{a}sz, Andr\'{a}s},
      title = {Cobordisms of sutured manifolds and the functoriality of link {F}loer homology},
      journal = {Adv. Math.},
      fjournal = {Advances in Mathematics},
      volume = {299},
      year = {2016},
      pages = {940--1038},
      issn = {0001-8708},
      mrclass = {57M27 (57R58)},
      mrnumber = {3519484},
      doi = {10.1016/j.aim.2016.06.005},
      url = {https://doi.org/10.1016/j.aim.2016.06.005},
      zblnumber = {1358.57021},
      }
  • [JMConcordance] Go to document A. Juhász and M. Marengon, "Concordance maps in knot Floer homology," Geom. Topol., vol. 20, iss. 6, pp. 3623-3673, 2016.
    @ARTICLE{JMConcordance,
      author = {Juh\'{a}sz, Andr\'{a}s and Marengon, Marco},
      title = {Concordance maps in knot {F}loer homology},
      journal = {Geom. Topol.},
      fjournal = {Geometry \& Topology},
      volume = {20},
      year = {2016},
      number = {6},
      pages = {3623--3673},
      issn = {1465-3060},
      mrclass = {57M27 (57R58)},
      mrnumber = {3590358},
      mrreviewer = {Kenneth Lee Baker},
      doi = {10.2140/gt.2016.20.3623},
      url = {https://doi.org/10.2140/gt.2016.20.3623},
      zblnumber = {1364.57013},
      }
  • [JMComputeCobordismMaps] Go to document A. Juhász and M. Marengon, "Computing cobordism maps in link Floer homology and the reduced Khovanov TQFT," Selecta Math. (N.S.), vol. 24, iss. 2, pp. 1315-1390, 2018.
    @ARTICLE{JMComputeCobordismMaps,
      author = {Juh\'{a}sz, Andr\'{a}s and Marengon, Marco},
      title = {Computing cobordism maps in link {F}loer homology and the reduced {K}hovanov {TQFT}},
      journal = {Selecta Math. (N.S.)},
      fjournal = {Selecta Mathematica. New Series},
      volume = {24},
      year = {2018},
      number = {2},
      pages = {1315--1390},
      issn = {1022-1824},
      mrclass = {57M27 (57R58)},
      mrnumber = {3782423},
      doi = {10.1007/s00029-017-0368-9},
      url = {https://doi.org/10.1007/s00029-017-0368-9},
      zblnumber = {1397.57026},
      }
  • [JuhaszZemkeContactHandles] A. Juhász and I. Zemke, Contact handles, duality, and sutured Floer homology, 2018.
    @MISC{JuhaszZemkeContactHandles,
      author = {Juhász, András and Zemke, Ian},
      title = {Contact handles, duality, and sutured {F}loer homology},
      arxiv = {1803.04401},
      year = {2018},
      note = {{\em Geom. Topol.},
      to appear},
      zblnumber = {},
      }
  • [Kochloukova] Go to document D. H. Kochloukova, "Some Novikov rings that are von Neumann finite and knot-like groups," Comment. Math. Helv., vol. 81, iss. 4, pp. 931-943, 2006.
    @ARTICLE{Kochloukova,
      author = {Kochloukova, Dessislava H.},
      title = {Some {N}ovikov rings that are von {N}eumann finite and knot-like groups},
      journal = {Comment. Math. Helv.},
      fjournal = {Commentarii Mathematici Helvetici. A Journal of the Swiss Mathematical Society},
      volume = {81},
      year = {2006},
      number = {4},
      pages = {931--943},
      issn = {0010-2571},
      mrclass = {20C07 (20E34 46L10 57M25)},
      mrnumber = {2271229},
      mrreviewer = {J. S. Birman},
      doi = {10.4171/CMH/81},
      url = {https://doi.org/10.4171/CMH/81},
      zblnumber = {1166.20042},
      }
  • [MiyazakiBandSumRibbon] Go to document K. Miyazaki, "Band-sums are ribbon concordant to the connected sum," Proc. Amer. Math. Soc., vol. 126, iss. 11, pp. 3401-3406, 1998.
    @ARTICLE{MiyazakiBandSumRibbon,
      author = {Miyazaki, Katura},
      title = {Band-sums are ribbon concordant to the connected sum},
      journal = {Proc. Amer. Math. Soc.},
      fjournal = {Proceedings of the American Mathematical Society},
      volume = {126},
      year = {1998},
      number = {11},
      pages = {3401--3406},
      issn = {0002-9939},
      mrclass = {57M25 (57Q60)},
      mrnumber = {1451821},
      mrreviewer = {Daniel Silver},
      doi = {10.1090/S0002-9939-98-04352-4},
      url = {https://doi.org/10.1090/S0002-9939-98-04352-4},
      zblnumber = {0913.57002},
      }
  • [MiyazakiFiberedBand] Go to document K. Miyazaki, "A note on genera of band sums that are fibered," J. Knot Theory Ramifications, vol. 27, iss. 12, p. 1871002, 2018.
    @ARTICLE{MiyazakiFiberedBand,
      author = {Miyazaki, Katura},
      title = {A note on genera of band sums that are fibered},
      journal = {J. Knot Theory Ramifications},
      fjournal = {Journal of Knot Theory and its Ramifications},
      volume = {27},
      year = {2018},
      number = {12},
      pages = {1871002, 3},
      issn = {0218-2165},
      mrclass = {57M25 (57M27)},
      mrnumber = {3876350},
      mrreviewer = {Yuichi Yamada},
      doi = {10.1142/S0218216518710025},
      url = {https://doi.org/10.1142/S0218216518710025},
      zblnumber = {1402.57011},
      }
  • [NiFibered] Go to document Y. Ni, "Knot Floer homology detects fibred knots," Invent. Math., vol. 170, iss. 3, pp. 577-608, 2007.
    @ARTICLE{NiFibered,
      author = {Ni, Yi},
      title = {Knot {F}loer homology detects fibred knots},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {170},
      year = {2007},
      number = {3},
      pages = {577--608},
      issn = {0020-9910},
      mrclass = {57R58 (57M27 57R30)},
      mrnumber = {2357503},
      mrreviewer = {Stanislav Jabuka},
      doi = {10.1007/s00222-007-0075-9},
      url = {https://doi.org/10.1007/s00222-007-0075-9},
      zblnumber = {1138.57031},
      }
  • [OSgenusbounds] Go to document P. Ozsváth and Z. Szabó, "Holomorphic disks and genus bounds," Geom. Topol., vol. 8, pp. 311-334, 2004.
    @ARTICLE{OSgenusbounds,
      author = {Ozsv\'{a}th, Peter and Szabó,
      Zolt\'{a}n},
      title = {Holomorphic disks and genus bounds},
      journal = {Geom. Topol.},
      fjournal = {Geometry and Topology},
      volume = {8},
      year = {2004},
      pages = {311--334},
      issn = {1465-3060},
      mrclass = {57M27 (53D35 57N10 57R58)},
      mrnumber = {2023281},
      mrreviewer = {Jacob Andrew Rasmussen},
      doi = {10.2140/gt.2004.8.311},
      url = {https://doi.org/10.2140/gt.2004.8.311},
      zblnumber = {1056.57020},
      }
  • [OSKnots] Go to document P. Ozsváth and Z. Szabó, "Holomorphic disks and knot invariants," Adv. Math., vol. 186, iss. 1, pp. 58-116, 2004.
    @ARTICLE{OSKnots,
      author = {Ozsv\'{a}th, Peter and Szabó,
      Zolt\'{a}n},
      title = {Holomorphic disks and knot invariants},
      journal = {Adv. Math.},
      fjournal = {Advances in Mathematics},
      volume = {186},
      year = {2004},
      number = {1},
      pages = {58--116},
      issn = {0001-8708},
      mrclass = {57M27 (57R58)},
      mrnumber = {2065507},
      mrreviewer = {Stanislav Jabuka},
      doi = {10.1016/j.aim.2003.05.001},
      url = {https://doi.org/10.1016/j.aim.2003.05.001},
      zblnumber = {1062.57019},
      }
  • [OSDisks] Go to document P. Ozsváth and Z. Szabó, "Holomorphic disks and topological invariants for closed three-manifolds," Ann. of Math. (2), vol. 159, iss. 3, pp. 1027-1158, 2004.
    @ARTICLE{OSDisks,
      author = {Ozsv\'{a}th, Peter and Szabó,
      Zolt\'{a}n},
      title = {Holomorphic disks and topological invariants for closed three-manifolds},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {159},
      year = {2004},
      number = {3},
      pages = {1027--1158},
      issn = {0003-486X},
      mrclass = {57M27 (32Q65 57R58)},
      mrnumber = {2113019},
      mrreviewer = {Thomas E. Mark},
      doi = {10.4007/annals.2004.159.1027},
      url = {https://doi.org/10.4007/annals.2004.159.1027},
      zblnumber = {1073.57009},
      }
  • [OSLinks] Go to document P. Ozsváth and Z. Szabó, "Holomorphic disks, link invariants and the multi-variable Alexander polynomial," Algebr. Geom. Topol., vol. 8, iss. 2, pp. 615-692, 2008.
  • [RasmussenKnots] Go to document J. A. Rasmussen, Floer Homology and Knot Complements, ProQuest LLC, Ann Arbor, MI, 2003.
    @BOOK{RasmussenKnots,
      author = {Rasmussen, Jacob Andrew},
      title = {Floer Homology and Knot Complements},
      note = {Thesis (Ph.D.)--Harvard University},
      publisher = {ProQuest LLC, Ann Arbor, MI},
      year = {2003},
      pages = {126},
      isbn = {978-0496-39374-9},
      mrclass = {Thesis},
      mrnumber = {2704683},
      url = {http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3091665},
      zblnumber = {},
      }
  • [SarkarMovingBasepoints] Go to document S. Sarkar, "Moving basepoints and the induced automorphisms of link Floer homology," Algebr. Geom. Topol., vol. 15, iss. 5, pp. 2479-2515, 2015.
    @ARTICLE{SarkarMovingBasepoints,
      author = {Sarkar, Sucharit},
      title = {Moving basepoints and the induced automorphisms of link {F}loer homology},
      journal = {Algebr. Geom. Topol.},
      fjournal = {Algebraic \& Geometric Topology},
      volume = {15},
      year = {2015},
      number = {5},
      pages = {2479--2515},
      issn = {1472-2747},
      mrclass = {57R58 (57M25 57M27)},
      mrnumber = {3426686},
      mrreviewer = {Yi Ni},
      doi = {10.2140/agt.2015.15.2479},
      url = {https://doi.org/10.2140/agt.2015.15.2479},
      zblnumber = {1331.57015},
      }
  • [ScharlemannSphere] Go to document M. Scharlemann, "Smooth spheres in ${\bf R}^4$ with four critical points are standard," Invent. Math., vol. 79, iss. 1, pp. 125-141, 1985.
    @ARTICLE{ScharlemannSphere,
      author = {Scharlemann, Martin},
      title = {Smooth spheres in {${\bf R}^4$} with four critical points are standard},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {79},
      year = {1985},
      number = {1},
      pages = {125--141},
      issn = {0020-9910},
      mrclass = {57M25 (57R52)},
      mrnumber = {0774532},
      mrreviewer = {Lee Rudolph},
      doi = {10.1007/BF01388659},
      url = {https://doi.org/10.1007/BF01388659},
      zblnumber = {0559.57019},
      }
  • [SilverRibbon] Go to document D. S. Silver, "On knot-like groups and ribbon concordance," J. Pure Appl. Algebra, vol. 82, iss. 1, pp. 99-105, 1992.
    @ARTICLE{SilverRibbon,
      author = {Silver, D. S.},
      title = {On knot-like groups and ribbon concordance},
      journal = {J. Pure Appl. Algebra},
      fjournal = {Journal of Pure and Applied Algebra},
      volume = {82},
      year = {1992},
      number = {1},
      pages = {99--105},
      issn = {0022-4049},
      mrclass = {57M25 (57N70)},
      mrnumber = {1181096},
      mrreviewer = {Mark E. Kidwell},
      doi = {10.1016/0022-4049(92)90013-6},
      url = {https://doi.org/10.1016/0022-4049(92)90013-6},
      zblnumber = {0766.57006},
      }
  • [RapaportKnotLike] Go to document E. R. Strasser, "Knot-like groups," in Knots, Groups, and 3-Manifolds, , 1975, vol. 84, pp. 119-133.
    @INCOLLECTION{RapaportKnotLike,
      author = {Strasser, Elvira Rapaport},
      title = {Knot-like groups},
      booktitle = {Knots, Groups, and 3-Manifolds},
      titlenote = {papers dedicated to the memory of {R}.{H}. {F}ox)},
      pages = {119--133},
      series = {Ann. of Math. Studies},
      volume = {84},
      year = {1975},
      mrclass = {55A25 (20F05)},
      mrnumber = {0440531},
      mrreviewer = {J. S. Birman},
      zblnumber = {0323.55003},
      doi = {10.1515/9781400881512-011},
      url = {https://doi.org/10.1515/9781400881512-011},
      }
  • [ZemQuasi] Go to document I. Zemke, "Quasistabilization and basepoint moving maps in link Floer homology," Algebr. Geom. Topol., vol. 17, iss. 6, pp. 3461-3518, 2017.
    @ARTICLE{ZemQuasi,
      author = {Zemke, Ian},
      title = {Quasistabilization and basepoint moving maps in link {F}loer homology},
      journal = {Algebr. Geom. Topol.},
      fjournal = {Algebraic \& Geometric Topology},
      volume = {17},
      year = {2017},
      number = {6},
      pages = {3461--3518},
      issn = {1472-2747},
      mrclass = {57M25 (57M27 57R58)},
      mrnumber = {3709653},
      mrreviewer = {Jianfeng Lin},
      doi = {10.2140/agt.2017.17.3461},
      url = {https://doi.org/10.2140/agt.2017.17.3461},
      zblnumber = {1387.57020},
      }
  • [ZemAbsoluteGradings] Go to document I. Zemke, "Link cobordisms and absolute gradings on link Floer homology," Quantum Topol., vol. 10, iss. 2, pp. 207-323, 2019.
    @ARTICLE{ZemAbsoluteGradings,
      author = {Zemke, Ian},
      title = {Link cobordisms and absolute gradings on link {F}loer homology},
      journal = {Quantum Topol.},
      fjournal = {Quantum Topology},
      volume = {10},
      year = {2019},
      number = {2},
      pages = {207--323},
      issn = {1663-487X},
      mrclass = {57R58 (57M27)},
      mrnumber = {3950650},
      doi = {10.4171/QT/124},
      url = {https://doi.org/10.4171/QT/124},
      zblnumber = {07068343},
      }
  • [ZemCFLTQFT] Go to document I. Zemke, "Link cobordisms and functoriality in link Floer homology," J. Topol., vol. 12, iss. 1, pp. 94-220, 2019.
    @ARTICLE{ZemCFLTQFT,
      author = {Zemke, Ian},
      title = {Link cobordisms and functoriality in link {F}loer homology},
      journal = {J. Topol.},
      fjournal = {Journal of Topology},
      volume = {12},
      year = {2019},
      number = {1},
      pages = {94--220},
      issn = {1753-8416},
      mrclass = {57M27 (57M25 57R56)},
      mrnumber = {3905679},
      mrreviewer = {Daniel D. Moskovich},
      doi = {10.1112/topo.12085},
      url = {https://doi.org/10.1112/topo.12085},
      zblnumber = {07055381},
      }

Authors

Ian Zemke

Princeton University Princeton, NJ