Skip to main content
SpringerLink
Log in

Knot Floer homology of satellite knots with (1, 1) patterns

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

Abstract

For pattern knots admitting genus-one doubly pointed bordered Heegaard diagrams, we give an immersed-curve approach to compute the knot Floer chain complexes of the corresponding satellite knots. In particular, this approach provides a convenient way to compute the knot concordance invariant \(\tau \). For patterns P obtained from two-bridge links b(p, q), we derive a formula for the \(\tau \)-invariant of \(P(T_{2,3})\) and \(P(-T_{2,3})\) in terms of (p, q). We use this formula to study whether such patterns induce homomorphisms on the knot concordance group, providing a glimpse at a conjecture due to Hedden.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29
Fig. 30
Fig. 31
Fig. 32
Fig. 33
Fig. 34
Fig. 35
Fig. 36
Fig. 37
Fig. 38
Fig. 39
Fig. 40
Fig. 41
Fig. 42
Fig. 43
Fig. 44
Fig. 45
Fig. 46
Fig. 47
Fig. 48
Fig. 49
Fig. 50
Fig. 51
Fig. 52
Fig. 53
Fig. 54

Similar content being viewed by others

Notes

  1. Strictly speaking, we need a version of Lemma 35 of [5] that respects the Alexander filtration; this is a straightforward generalization.

  2. In Ording’s notation, a two-bridge link \(b(p,\epsilon q)\) is equivalent to \(K_1\cup K_2\), where \(K_1\) is a (1, 1) knot that has a (1, 1) normal form \({\mathfrak {d}}(\frac{q-1}{2},\frac{\epsilon (p-2q)}{2},0,0)\) and \(K_2\) is the meridian of the decomposition torus. Our pattern knot P admits a (1, 1)-normal form \({\mathfrak {d}}(|r|-1,\epsilon (r)(s+1),0,0)\).

References

  1. Chen, W.: An infinite-rank summand from iterated Mazur pattern satellite knots. arXiv:2010.11277 (2020)

  2. Chen, W.: On two-bridge knots and a conjecture of Hirasawa–Murasugi. J. Knot Theory Ramif. 30(2), 2150007 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  3. Doll, H.: A generalized bridge number for links in \(3\)-manifolds. Math. Ann. 294(4), 701–717 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. Greene, J.E., Lewallen, S., Vafaee, F.: \((1,1)\) L-space knots. Compos. Math. 154(5), 918–933 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  5. Hanselman, J., Rasmussen, J., Watson, L.: Bordered Floer homology for manifolds with torus boundary via immersed curves. arXiv:1604.03466 (2016)

  6. Hanselman, J., Rasmussen, J., Watson, L.: Heegaard Floer homology for manifolds with torus boundary: properties and examples. Proc. Lond. Math. Soc. (3) 125, no. 4, 879–967 (2022)

  7. Hanselman, J., Watson, L.: Cabling in terms of immersed curves. Geom. Topol. 27(3), 925–952 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hedden, M.: Knot Floer homology of Whitehead doubles. Geom. Topol. 11, 2277–2338 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hedden, M.: On knot Floer homology and cabling. II. Int. Math. Res. Not. IMRN 12, 2248–2274 (2009)

    MathSciNet  MATH  Google Scholar 

  10. Hedden, M., Ording, P.: The Ozsváth–Szabó and Rasmussen concordance invariants are not equal. Am. J. Math. 130(2), 441–453 (2008)

    Article  MATH  Google Scholar 

  11. Hedden, M., Pinzon-Caicedo, J.: Satellites of infinite rank in the smooth concordance group. Invent. Math. 225(1), 131–157 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hom, J.: Bordered Heegaard Floer homology and the tau-invariant of cable knots. J. Topol. 7(2), 287–326 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hom, J., Lidman, T., Vafaee, F.: Berge–Gabai knots and L-space satellite operations. Algebr. Geom. Topol. 14(6), 3745–3763 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kawauchi, A.: A Survey of Knot Theory. Birkhäuser Verlag, Basel (1996). Translated and revised from the 1990 Japanese original by the author

  15. Levine, A.S.: Nonsurjective satellite operators and piecewise-linear concordance. Forum Math. Sigma 4, e34 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lipshitz, R., Ozsvath, P.S., Thurston, D.P.: Bordered Heegaard Floer homology. Memoirs of the American Mathematical Society, vol. 254(1216), p. viii+279 (2018)

  17. Lipshitz, R., Ozsvath, P.S., Thurston, D.P.: Bordered Heegaard Floer Homology, vol. 254. American Mathematical Society, Providence (2018)

    MATH  Google Scholar 

  18. Miller, A.N.: Homomorphism obstructions for satellite maps. Trans. Amer. Math. Soc. Ser. B 10, 220–240 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ording, P.J.P.: On knot Floer homology of satellite (1, 1) knots. ProQuest LLC, Ann Arbor (2006). Thesis (Ph.D.)–Columbia University

  20. Ozsváth, P., Szabó, Z.: Holomorphic disks and genus bounds. Geom. Topol. 8, 311–334 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ozsváth, P., Szabó, Z.: Holomorphic disks and knot invariants. Adv. Math. 186(1), 58–116 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ozsváth, P., Szabó, Z.: Holomorphic disks and topological invariants for closed three-manifolds. Ann. Math. (2) 159(3), 1027–1158 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Petkova, I.: Cables of thin knots and bordered Heegaard Floer homology. Quantum Topol. 4(4), 377–409 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Piccirillo, L.: Shake genus and slice genus. Geom. Topol. 23(5), 2665–2684 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  25. Rasmussen, J.A.: Floer homology and knot complements. ProQuest LLC, Ann Arbor (2003). Thesis (Ph.D.)–Harvard University

  26. Roberts, L.P.: Some bounds for the knot Floer \(\tau \)-invariant of satellite knots. Algebr. Geom. Topol. 12(1), 449–467 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author thank his PhD advisor Matt Hedden for his enormous help, and thank Abhishek Mallick for informing him of Ording’s result used in this paper. The exposition of this paper benefited greatly from the referee’s valuable feedback. The author was supported by the NSF Grant DMS-1709016 and the Max Planck Institute for Mathematics in Bonn during part of the research, and is currently supported by the Pacific Institute for the Mathematical Sciences. The research and findings may not reflect those of the institutes.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, 1984 Mathematics Rd, Vancouver, BC, V6T 1Z2, Canada

    Wenzhao Chen

Authors
  1. Wenzhao Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wenzhao Chen.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, W. Knot Floer homology of satellite knots with (1, 1) patterns. Sel. Math. New Ser. 29, 53 (2023). https://doi.org/10.1007/s00029-023-00859-6

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00029-023-00859-6

Mathematics Subject Classification

Search

Navigation