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Meridional surfaces and $ (1,1)$-knots

Authors: Mario Eudave-Muñoz and Enrique Ramírez-Losada
Journal: Trans. Amer. Math. Soc. 361 (2009), 671-696
MSC (2000): Primary 57M25, 57N10
Published electronically: September 9, 2008
MathSciNet review: 2452820
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Abstract: We determine all $ (1,1)$-knots which admit an essential meridional surface, namely, we give a construction which produces $ (1,1)$-knots having essential meridional surfaces, and show that if a $ (1,1)$-knot admits an essential meridional surface, then it comes from the given construction.

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  • [CM] A. Cattabriga and M. Mulazzani, $ (1,1)$-knots via the mapping class group of the twice punctured torus, Adv. Geom. 4 (2004), 263-277. MR 2055680 (2005e:57013)
  • [CK] Doo Ho Choi and Ki Hyoung Ko, Parameterizations of $ 1$-bridge torus knots, J. Knot Theory Ramifications 12 (2003), 463-491. MR 1985906 (2004c:57010)
  • [CGLS] M. Culler, C. McA. Gordon, J. Luecke and P. B. Shalen, Dehn surgery on knots, Annals of Mathematics 125 (1987), 237-300. MR 881270 (88a:57026)
  • [E1] M. Eudave-Muñoz, Incompressible surfaces in tunnel number one knot complements, Topology Appl. 98 (1999), 167-189. MR 1719999 (2000h:57010)
  • [E2] -, Meridional essential surfaces for tunnel number one knots, Bol. Soc. Mat. Mex. (3) 6 (2000), 263-277. MR 1810854 (2001m:57011)
  • [E3] -, Incompressible surfaces and $ (1,1)$-knots, J. Knot Theory Ramifications 15 (2006), no. 7, 935-948. MR 2251034 (2007g:57007)
  • [F] E. Finkelstein, Closed incompressible surfaces in closed braid complements, J. Knot Theory Ramifications 7 (1998), 335-379. MR 1625363 (99i:57013)
  • [GMM] H. Goda, H. Matsuda and T. Morifuji, Knot Floer homology of $ (1,1)$-knots, Geom. Dedicata 112 (2005), 197-214. MR 2163899 (2006e:57014)
  • [GL] C. McA. Gordon and R. A. Litherland, Incompressible surfaces in branched coverings, The Smith Conjecture (New York, 1979), Pure Appl. Math., 112, pp. 139-152. MR 758466
  • [GR] C. McA. Gordon and A. Reid, Tangle decompositions of tunnel number one knots and links, J. Knot Theory Ramifications 4 (1995), 389-409. MR 1347361 (96m:57016)
  • [HT] A, Hatcher and W. Thurston, Incompressible surfaces in $ 2$-bridge knot complements, Invent. Math. 79 (1985), 225-246. MR 778125 (86g:57003)
  • [LP] M.T. Lozano and J. Przytycki, Incompressible surfaces in the exterior of a closed $ 3$-braid. I. Surfaces with horizontal boundary components, Math. Proc. Cambridge Philos. Soc. 98 (1985), 275-299. MR 795894 (87a:57013)
  • [M] W. Menasco, Closed incompressible surfaces in alternating knot and link complements, Topology 23 (1984), 37-44. MR 721450 (86b:57004)
  • [MS] K. Morimoto and M. Sakuma, On unknotting tunnels for knots, Math. Ann. 289 (1991), 143-167. MR 1087243 (92e:57015)
  • [O] U. Oertel, Closed incompressible surfaces in complements of star links, Pacific J. Math. 111 (1984), 209-230. MR 732067 (85j:57008)

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Additional Information

Mario Eudave-Muñoz
Affiliation: Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, 04510 México D.F., Mexico

Enrique Ramírez-Losada
Affiliation: Centro de Investigación en Matemáticas, Apdo. Postal 402, 36000 Guanajuato, Gto., Mexico

Keywords: $(1,1)$-knot, essential meridional surface
Received by editor(s): February 10, 2005
Received by editor(s) in revised form: August 7, 2006
Published electronically: September 9, 2008
Article copyright: © Copyright 2008 American Mathematical Society

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