Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
144 tokens/sec
GPT-4o
8 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Taut foliations from knot diagrams (2402.01225v1)

Published 2 Feb 2024 in math.GT

Abstract: We prove that if a knot $K$ has a particular type of diagram then all non-trivial surgeries on $K$ contain a coorientable taut foliation. Knots admitting such diagrams include many two-bridge knots, many pretzel knots, many Montesinos knots and more generally all arborescent knots defined by weighted planar trees with more than one vertex such that $1)$ all weights have absolute value greater than one, and $2)$ there exists a weight with absolute value greater than two. The ideas involved in the proof can also be adapted to study surgeries on links and as an application we show that for all surgeries $M$ on the Borromean link it holds that $M$ is not an $L$-space if and only if $M$ contains a coorientable taut foliation.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (51)
  1. Foliations, orders, representations, L-spaces and graph manifolds. Advances in Mathematics, 310:159–234, 2017.
  2. On L-spaces and left-orderable fundamental groups. Mathematische Annalen, 356(4):1213–1245, 2013.
  3. Montesinos knots, Hopf plumbings, and L-space surgeries. Journal of the Mathematical Society of Japan, 70(1):95–110, 2018.
  4. Graph manifolds and taut foliations. Journal of Differential Geometry, 45(3):446–470, 1997.
  5. Jonathan Bowden. Approximating C0superscript𝐶0C^{0}italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT-foliations by contact structures. Geometric and Functional Analysis, 26(5):1255–1296, 2016.
  6. New geometric splittings of classical knots, and the classification and symmetries of arborescent knots. preprint, 2010.
  7. Graph manifolds, left-orderability and amalgamation. Algebraic & Geometric Topology, 13(4):2347–2368, 2013.
  8. Taut foliations from double-diamond replacements. Characters in Low-Dimensional Topology, 760:123, 2020.
  9. Persistently foliar composite knots. Algebraic & Geometric Topology, 21(6):2761–2798, 2021.
  10. Transverse foliations of Seifert bundles and self homeomorphism of the circle. Commentarii Mathematici Helvetici, 56(1):638–660, 1981.
  11. Incompressible surfaces via branched surfaces. Topology, 23(1):117–125, 1984.
  12. Constructing Lens Spaces by Surgery on Knots. Math. Z, 175:33–51, 1980.
  13. David Gabai. Foliations and the topology of 3-manifolds. Journal of Differential Geometry, 18(3):445 – 503, 1983.
  14. David Gabai. Foliations and the topology of 3-manifolds. II. Journal of Differential Geometry, 26(3):461–478, 1987.
  15. David Gabai. Foliations and the topology of 3-manifolds. III. Journal of Differential Geometry, 26(3):479–536, 1987.
  16. David Gabai. Taut foliations of 3-manifolds and suspensions of S1superscript𝑆1S^{1}italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT. In Annales de l’institut Fourier, volume 42, pages 193–208, 1992.
  17. Paolo Ghiggini. Knot Floer homology detects genus-one fibred knots. American journal of mathematics, 130(5):1151–1169, 2008.
  18. Essential laminations in 3-manifolds. Annals of Mathematics, 130(1):41–73, 1989.
  19. Matthew Hedden. Notions of positivity and the Ozsváth–Szabó concordance invariant. Journal of Knot Theory and its Ramifications, 19(05):617–629, 2010.
  20. L-spaces, taut foliations, and graph manifolds. Compositio Mathematica, 156(3):604–612, 2020.
  21. András Juhász. A survey of Heegaard Floer homology. In New ideas in low dimensional topology, pages 237–296. World Scientific, 2015.
  22. Taut foliations in surface bundles with multiple boundary components. Pacific Journal of Mathematics, 273(2):257–275, 2014.
  23. C0superscript𝐶0C^{0}italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT approximations of foliations. Geometry & Topology, 21(6):3601–3657, 2017.
  24. Siddhi Krishna. Taut foliations, positive 3-braids, and the L-space conjecture. Journal of Topology, 13(3):1003–1033, 2020.
  25. Siddhi Krishna. Taut foliations, braid positivity, and unknot detection. arXiv preprint arXiv:2312.00196, 2023.
  26. Marc Lackenby. The volume of hyperbolic alternating link complements. Proceedings of the London Mathematical Society, 88(1):204–224, 2004.
  27. Tao Li. Laminar branched surfaces in 3–manifolds. Geometry & Topology, 6(1):153–194, 2002.
  28. Tao Li. Boundary train tracks of laminar branched surfaces. In Proceedings of symposia in pure mathematics, volume 71, pages 269–286. Providence, RI; American Mathematical Society; 1998, 2003.
  29. Tao Li. Taut foliations of 3-manifolds with Heegaard genus two. arXiv preprint arXiv:2202.00737, 2022.
  30. William Bernard Raymond Lickorish. A foliation for 3-manifolds. Annals of mathematics, pages 414–420, 1965.
  31. Pretzel knots with L-space surgeries. Michigan Mathematical Journal, 65(1):105–130, 2016.
  32. L–space knots have no essential Conway spheres. Geometry & Topology, 26(5):2065–2102, 2022.
  33. Taut foliations in knot complements. Pacific Journal of Mathematics, 269(1):149–168, 2014.
  34. On the existence of tight contact structures on Seifert fibered 3-manifolds. Duke Mathematical Journal, 148(2):175–209, 2009.
  35. Yi Ni. Knot Floer homology detects fibred knots. Inventiones mathematicae, 170(3):577–608, 2007.
  36. Sergei Petrovich Novikov. Topology of foliations. Trans. Moscow Math. Soc, 14:248–278, 1965.
  37. Ulrich Oertel. Incompressible branched surfaces. Inventiones mathematicae, 76(3):385–410, 1984.
  38. Holomorphic disks and genus bounds. Geometry & Topology, 8(1):311–334, 2004.
  39. Knot Floer homology and rational surgeries. Algebraic & Geometric Topology, 11(1):1–68, 2010.
  40. Carlos Frederico Borges Palmeira. Open manifolds foliated by planes. Annals of Mathematics, pages 109–131, 1978.
  41. Combinatorics of train tracks. Number 125. Princeton University Press, 1992.
  42. Jessica S Purcell. An introduction to fully augmented links. Interactions between hyperbolic geometry, quantum topology and number theory, 541:205–220, 2011.
  43. Rachel Roberts. Constructing taut foliations. Comment. Math. Helv., 70(4):516–545, 1995.
  44. Rachel Roberts. Taut foliations in punctured surface bundles, II. Proceedings of the London Mathematical Society, 83(2):443–471, 2001.
  45. Harold Rosenberg. Foliations by planes. Topology, 7(2):131–138, 1968.
  46. Infinitely many hyperbolic 3-manifolds which contain no reebless foliation. Journal of the American Mathematical Society, 16(3):639–679, 2003.
  47. Diego Santoro. L-spaces, taut foliations and the Whitehead link. arXiv preprint arXiv:2201.01211. Accepted for publication in Algebraic and Geometric Topology, 2022.
  48. Diego Santoro. L-spaces, taut foliations and fibered hyperbolic two-bridge links. arXiv preprint arXiv:2304.14914, 2023.
  49. William P. Thurston. A norm for the homology of 3-manifolds. Memoirs of the American Mathematical Society, 59(339):99–130, 1986.
  50. Ying-Qing Wu. Dehn surgery on arborescent knots. Journal of Differential Geometry, 43(1):171–197, 1996.
  51. Bojun Zhao. Co-orientable taut foliations in Dehn fillings of pseudo-Anosov mapping tori with co-orientation-reversing monodromy. arXiv preprint arXiv:2310.01368, 2023.

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets