Bridge number, Heegaard genus and non-integral Dehn surgery
HTML articles powered by AMS MathViewer
- by Kenneth L. Baker, Cameron Gordon and John Luecke PDF
- Trans. Amer. Math. Soc. 367 (2015), 5753-5830 Request permission
Abstract:
We show there exists a linear function $w \colon \mathbb {N} \to \mathbb {N}$ with the following property. Let $K$ be a hyperbolic knot in a hyperbolic $3$–manifold $M$ admitting a non-longitudinal $S^3$ surgery. If $K$ is put into thin position with respect to a strongly irreducible, genus $g$ Heegaard splitting of $M$, then $K$ intersects a thick level at most $2w(g)$ times. Typically, this shows that the bridge number of $K$ with respect to this Heegaard splitting is at most $w(g)$, and the tunnel number of $K$ is at most $w(g) + g-1$.References
- Kenneth L. Baker, Small genus knots in lens spaces have small bridge number, Algebr. Geom. Topol. 6 (2006), 1519–1621. MR 2253458, DOI 10.2140/agt.2006.6.1519
- Kenneth L. Baker, Cameron McA. Gordon, and John Luecke, Bridge number and integral Dehn surgery, arXiv:1303.7018 [math.GT].
- Kenneth L. Baker, Cameron Gordon, and John Luecke, Obtaining genus 2 Heegaard splittings from Dehn surgery, Algebr. Geom. Topol. 13 (2013), no. 5, 2471–2634. MR 3116298, DOI 10.2140/agt.2013.13.2471
- John Berge, Some knots with surgeries yielding lens space, unpublished manuscript.
- Ryan Blair, Marion Campisi, Jesse Johnson, Scott A. Taylor, and Maggy Tomova, Bridge distance, Heegaard genus, and Exceptional Surgeries, arXiv:1209.0197 [math.GT]
- S. Boyer and X. Zhang, On Culler-Shalen seminorms and Dehn filling, Ann. of Math. (2) 148 (1998), no. 3, 737–801. MR 1670053, DOI 10.2307/121031
- Marc Culler, C. McA. Gordon, J. Luecke, and Peter B. Shalen, Dehn surgery on knots, Ann. of Math. (2) 125 (1987), no. 2, 237–300. MR 881270, DOI 10.2307/1971311
- Mario Eudave-Muñoz, Non-hyperbolic manifolds obtained by Dehn surgery on hyperbolic knots, Geometric topology (Athens, GA, 1993) AMS/IP Stud. Adv. Math., vol. 2, Amer. Math. Soc., Providence, RI, 1997, pp. 35–61. MR 1470720, DOI 10.1090/amsip/002.1/03
- Mario Eudave-Muñoz, On hyperbolic knots with Seifert fibered Dehn surgeries, Proceedings of the First Joint Japan-Mexico Meeting in Topology (Morelia, 1999), 2002, pp. 119–141. MR 1903687, DOI 10.1016/S0166-8641(01)00114-6
- David Gabai, Foliations and the topology of $3$-manifolds. III, J. Differential Geom. 26 (1987), no. 3, 479–536. MR 910018
- Hiroshi Goda and Masakazu Teragaito, Dehn surgeries on knots which yield lens spaces and genera of knots, Math. Proc. Cambridge Philos. Soc. 129 (2000), no. 3, 501–515. MR 1780501, DOI 10.1017/S0305004100004692
- C. McA. Gordon, Combinatorial methods in Dehn surgery, Lectures at KNOTS ’96 (Tokyo), Ser. Knots Everything, vol. 15, World Sci. Publ., River Edge, NJ, 1997, pp. 263–290. MR 1474525, DOI 10.1142/9789812796097_{0}010
- C. McA. Gordon and R. A. Litherland, Incompressible planar surfaces in $3$-manifolds, Topology Appl. 18 (1984), no. 2-3, 121–144. MR 769286, DOI 10.1016/0166-8641(84)90005-1
- C. McA. Gordon and J. Luecke, Only integral Dehn surgeries can yield reducible manifolds, Math. Proc. Cambridge Philos. Soc. 102 (1987), no. 1, 97–101. MR 886439, DOI 10.1017/S0305004100067086
- C. McA. Gordon and J. Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989), no. 2, 371–415. MR 965210, DOI 10.1090/S0894-0347-1989-0965210-7
- C. McA. Gordon and J. Luecke, Dehn surgeries on knots creating essential tori. I, Comm. Anal. Geom. 3 (1995), no. 3-4, 597–644. MR 1371211, DOI 10.4310/CAG.1995.v3.n4.a3
- C. McA. Gordon and John Luecke, Non-integral toroidal Dehn surgeries, Comm. Anal. Geom. 12 (2004), no. 1-2, 417–485. MR 2074884
- A. E. Hatcher, On the boundary curves of incompressible surfaces, Pacific J. Math. 99 (1982), no. 2, 373–377. MR 658066
- John Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (2001), no. 3, 631–657. MR 1838999, DOI 10.1016/S0040-9383(00)00033-1
- J. Johnson, Bridge number and the curve complex, arXiv:math/0603102 [math.GT].
- Yair N. Minsky, Yoav Moriah, and Saul Schleimer, High distance knots, Algebr. Geom. Topol. 7 (2007), 1471–1483. MR 2366166, DOI 10.2140/agt.2007.7.1471
- Justin Malestein, Igor Rivin, and Louis Theran, Topological designs, Geom. Dedicata 168 (2014), 221–233. MR 3158040, DOI 10.1007/s10711-012-9827-9
- Yoav Moriah and Hyam Rubinstein, Heegaard structures of negatively curved $3$-manifolds, Comm. Anal. Geom. 5 (1997), no. 3, 375–412. MR 1487722, DOI 10.4310/CAG.1997.v5.n3.a1
- John Kirkpatrick Osoinach Jr, Manifolds obtained by Dehn surgery on infinitely many distinct knots in S(3), ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D.)–The University of Texas at Austin. MR 2697691
- John K. Osoinach Jr., Manifolds obtained by surgery on an infinite number of knots in $S^3$, Topology 45 (2006), no. 4, 725–733. MR 2236375, DOI 10.1016/j.top.2006.02.001
- Yo’av Rieck, Heegaard structures of manifolds in the Dehn filling space, Topology 39 (2000), no. 3, 619–641. MR 1746912, DOI 10.1016/S0040-9383(99)00026-9
- Yo’av Rieck and Eric Sedgwick, Finiteness results for Heegaard surfaces in surgered manifolds, Comm. Anal. Geom. 9 (2001), no. 2, 351–367. MR 1846207, DOI 10.4310/CAG.2001.v9.n2.a5
- Yo’av Rieck and Eric Sedgwick, Persistence of Heegaard structures under Dehn filling, Topology Appl. 109 (2001), no. 1, 41–53. MR 1804562, DOI 10.1016/S0166-8641(99)00147-9
- Martin Scharlemann, Heegaard splittings of compact 3-manifolds, Handbook of geometric topology, North-Holland, Amsterdam, 2002, pp. 921–953. MR 1886684
- Masakazu Teragaito, A Seifert fibered manifold with infinitely many knot-surgery descriptions, Int. Math. Res. Not. IMRN 9 (2007), Art. ID rnm 028, 16. MR 2347296, DOI 10.1093/imrn/rnm028
- Masakazu Teragaito, Toroidal Dehn surgery on hyperbolic knots and hitting number, Topology Appl. 157 (2010), no. 1, 269–273. MR 2556104, DOI 10.1016/j.topol.2009.04.037
- Abigail Thompson, Thin position and bridge number for knots in the $3$-sphere, Topology 36 (1997), no. 2, 505–507. MR 1415602, DOI 10.1016/0040-9383(96)00010-9
- Maggy Tomova, Distance of Heegaard splittings of knot complements, Pacific J. Math. 236 (2008), no. 1, 119–138. MR 2398991, DOI 10.2140/pjm.2008.236.119
- Cynthia Janet Verjovsky Marcotte, Essential surfaces after Dehn filling, ProQuest LLC, Ann Arbor, MI, 2000. Thesis (Ph.D.)–The University of Texas at Austin. MR 2701527
Additional Information
- Kenneth L. Baker
- Affiliation: Department of Mathematics, University of Miami, Coral Gables, Florida 33146
- MR Author ID: 794754
- Email: kb@math.miami.edu
- Cameron Gordon
- Affiliation: Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712
- MR Author ID: 75435
- Email: gordon@math.utexas.edu
- John Luecke
- Affiliation: Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712
- Email: luecke@math.utexas.edu
- Received by editor(s): February 1, 2012
- Received by editor(s) in revised form: September 6, 2013
- Published electronically: October 22, 2014
- Additional Notes: In the course of this work the first author was partially supported by NSF Grant DMS-0239600, by the University of Miami 2011 Provost Research Award, and by a grant from the Simons Foundation (#209184). The first author would like to thank the Department of Mathematics at the University of Texas at Austin for its hospitality during his visits. These visits were supported in part by NSF RTG Grant DMS-0636643
The second author was partially supported by NSF Grant DMS-0906276 - © Copyright 2014 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 367 (2015), 5753-5830
- MSC (2010): Primary 57M27
- DOI: https://doi.org/10.1090/S0002-9947-2014-06328-9
- MathSciNet review: 3347189