Algebraic & Geometric Topology

Strong Heegaard diagrams and strong L–spaces

Joshua Greene and Adam Levine

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Abstract

We study a class of 3–manifolds called strong L–spaces, which by definition admit a certain type of Heegaard diagram that is particularly simple from the perspective of Heegaard Floer homology. We provide evidence for the possibility that every strong L–space is the branched double cover of an alternating link in the three-sphere. For example, we establish this fact for a strong L–space admitting a strong Heegaard diagram of genus 2 via an explicit classification. We also show that there exist finitely many strong L–spaces with bounded order of first homology; for instance, through order eight, they are connected sums of lens spaces. The methods are topological and graph-theoretic. We discuss many related results and questions.

Article information

Source
Algebr. Geom. Topol., Volume 16, Number 6 (2016), 3167-3208.

Dates
Received: 9 December 2014
Accepted: 2 June 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510841257

Digital Object Identifier
doi:10.2140/agt.2016.16.3167

Mathematical Reviews number (MathSciNet)
MR3584256

Zentralblatt MATH identifier
1361.57019

Subjects
Primary: 57M27: Invariants of knots and 3-manifolds 57R58: Floer homology

Keywords
$3$–manifolds Heegaard diagrams Heegaard Floer homology L–spaces

Citation

Greene, Joshua; Levine, Adam. Strong Heegaard diagrams and strong L–spaces. Algebr. Geom. Topol. 16 (2016), no. 6, 3167--3208. doi:10.2140/agt.2016.16.3167. https://projecteuclid.org/euclid.agt/1510841257


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References

  • C Bankwitz, Über die Torsionszahlen der alternierenden Knoten, Math. Ann. 103 (1930) 145–161
  • J Berge, Some knots with surgeries yielding lens spaces, unpublished (1990)
  • S Boyer, C,M Gordon, L Watson, On L–spaces and left-orderable fundamental groups, Math. Ann. 356 (2013) 1213–1245
  • G,E Bredon, J,W Wood, Non-orientable surfaces in orientable $3$–manifolds, Invent. Math. 7 (1969) 83–110
  • A,J Casson, C,M Gordon, Reducing Heegaard splittings, Topology Appl. 27 (1987) 275–283
  • P,R Cromwell, Knots and links, Cambridge University Press (2004)
  • R,H Crowell, Nonalternating links, Illinois J. Math. 3 (1959) 101–120
  • N,M Dunfield, W,P Thurston, The virtual Haken conjecture: experiments and examples, Geom. Topol. 7 (2003) 399–441
  • C,M Gordon, Dehn surgery and $3$–manifolds, from “Low dimensional topology” (T,S Mrowka, P,S Ozsváth, editors), IAS/Park City Math. Ser. 15, Amer. Math. Soc., Providence, RI (2009) 21–71
  • J,E Greene, Lattices, graphs, and Conway mutation, Invent. Math. 192 (2013) 717–750
  • J,E Greene, A spanning tree model for the Heegaard Floer homology of a branched double-cover, J. Topol. 6 (2013) 525–567
  • J,E Greene, L Watson, Turaev torsion, definite $4$–manifolds, and quasi-alternating knots, Bull. Lond. Math. Soc. 45 (2013) 962–972
  • \relax minus 1pt W Haken, Some results on surfaces in $3$–manifolds, from “Studies in Modern Topology” (P,J Hilton, editor), Math. Assoc. Amer., distributed by Prentice-Hall, Englewood Cliffs, NJ (1968) 39–98
  • J Hanselman, J Rasmussen, S,D Rasmussen, L Watson, Taut foliations on graph manifolds, preprint (2015)
  • J Hanselman, L Watson, A calculus for bordered Floer homology, preprint (2015)
  • M Hedden, On Floer homology and the Berge conjecture on knots admitting lens space surgeries, Trans. Amer. Math. Soc. 363 (2011) 949–968
  • G Hetyei, $2\times 1$-es téglalapokkal lefedhető idomokról (Rectangular configurations which can be covered by $2 \times 1$ rectangles), Pécsi Tanárk. Főisk. Közl. 8 (1964) 351–368\quahookhttp://msp.org/msp_org/media/files/G.Hetyei-Rectangular_hookconfigurations_that_can_be_covered_by_2x1_rectangles.pdf
  • T Homma, M Ochiai, M-o Takahashi, An algorithm for recognizing $S\sp{3}$ in $3$–manifolds with Heegaard splittings of genus two, Osaka J. Math. 17 (1980) 625–648
  • W,H Kazez, R Roberts, Approximating $C^{1,0}$ foliations, preprint (2014)
  • D,A Lee, R Lipshitz, Covering spaces and $\mathbb Q$–gradings on Heegaard Floer homology, J. Symplectic Geom. 6 (2008) 33–59
  • A,S Levine, S Lewallen, Strong L–spaces and left-orderability, Math. Res. Lett. 19 (2012) 1237–1244
  • A,S Levine, D Ruberman, S Strle, Nonorientable surfaces in homology cobordisms, Geom. Topol. 19 (2015) 439–494
  • L Lovász, M,D Plummer, Matching theory, AMS Chelsea Publishing, Providence, RI (2009)
  • B Martelli, C Petronio, F Roukema, Exceptional Dehn surgery on the minimally twisted five-chain link, Comm. Anal. Geom. 22 (2014) 689–735
  • W McCuaig, Pólya's permanent problem, Electron. J. Combin. 11 (2004) art. ID 79
  • W,W Menasco, M,B Thistlethwaite, The Tait flyping conjecture, Bull. Amer. Math. Soc. 25 (1991) 403–412
  • O Morikawa, A counterexample to a conjecture of Whitehead, Math. Sem. Notes Kobe Univ. 8 (1980) 295–298
  • Y Ni, Z Wu, Heegaard Floer correction terms and rational genus bounds, Adv. Math. 267 (2014) 360–380
  • M Ochiai, A counterexample to a conjecture of Whitehead and Volodin–Kuznetsov–Fomenko, J. Math. Soc. Japan 31 (1979) 687–691
  • P Ozsváth, Z Szabó, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Adv. Math. 173 (2003) 179–261
  • P Ozsváth, Z Szabó, Holomorphic disks and genus bounds, Geom. Topol. 8 (2004) 311–334
  • P Ozsváth, Z Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004) 58–116
  • P Ozsváth, Z Szabó, Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. 159 (2004) 1159–1245
  • P Ozsváth, Z Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. 159 (2004) 1027–1158
  • P Ozsváth, Z Szabó, On knot Floer homology and lens space surgeries, Topology 44 (2005) 1281–1300
  • P Ozsváth, Z Szabó, On the Heegaard Floer homology of branched double-covers, Adv. Math. 194 (2005) 1–33
  • P Ozsváth, Z Szabó, Holomorphic triangles and invariants for smooth four-manifolds, Adv. Math. 202 (2006) 326–400
  • J,A Rasmussen, Floer homology and knot complements, PhD thesis, Harvard University, Cambridge, MA (2003) Available at \setbox0\makeatletter\@url http://search.proquest.com/docview/305332635 {\unhbox0
  • J Rasmussen, Lens space surgeries and L–space homology spheres, preprint (2007)
  • J Rasmussen, S,D Rasmussen, Floer simple manifolds and L–space intervals, preprint (2015)
  • N Robertson, P,D Seymour, R Thomas, Permanents, Pfaffian orientations, and even directed circuits, Ann. of Math. 150 (1999) 929–975
  • A Schrijver, Tait's flyping conjecture for well-connected links, J. Combin. Theory Ser. B 58 (1993) 65–146
  • T Usui, Heegaard Floer homology, L–spaces, and smoothing order on links, I, preprint (2012)
  • T Usui, Heegaard Floer homology, L–spaces, and smoothing order on links, II, preprint (2012)
  • V,V Vazirani, M Yannakakis, Pfaffian orientations, $0$–$1$ permanents, and even cycles in directed graphs, Discrete Appl. Math. 25 (1989) 179–190
  • O,J Viro, V,L Kobel'skiĭ, The Volodin–Kuznecov–Fomenko conjecture on Heegaard diagrams is false, Uspekhi Mat. Nauk 32 (1977) 175–176 In Russian
  • I,A Volodin, V,E Kuznetsov, A,T Fomenko, The problem of discriminating algorithmically the standard three-dimensional sphere, Uspekhi Mat. Nauk 29 (1974) 71–168 In Russian; translated in Russian Math. Surveys 29 (1974) 71–172
  • M Voorhoeve, A lower bound for the permanents of certain $(0,\,1)$–matrices, Nederl. Akad. Wetensch. Indag. Math. 41 (1979) 83–86