Geodesic
High distance knots
Minsky, Yair N1 ; Moriah, Yoav2 ; Schleimer, Saul3
1Department of Mathematics, Yale University, New Haven CT 06520-8283, USA
2Department of Mathematics, Technion, Haifa 32000, Israel
3Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK
Algebraic and Geometric Topology, Tome 7 (2007) no. 3, pp. 1471-1483
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We construct knots in S3 with Heegaard splittings of arbitrarily high distance, in any genus. As an application, for any positive integers t and b we find a tunnel number t knot in the three-sphere which has no (t,b)–decomposition.

DOI : 10.2140/agt.2007.7.1471
Keywords: Heegaard distance, tunnel number, knot, bridge position

Minsky, Yair N  1   ; Moriah, Yoav  2   ; Schleimer, Saul  3

1 Department of Mathematics, Yale University, New Haven CT 06520-8283, USA
2 Department of Mathematics, Technion, Haifa 32000, Israel
3 Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK 
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Minsky, Yair N; Moriah, Yoav; Schleimer, Saul. High distance knots. Algebraic and Geometric Topology, Tome 7 (2007) no. 3, pp. 1471-1483. doi: 10.2140/agt.2007.7.1471

[1] A Abrams, S Schleimer, Distances of Heegaard splittings, Geom. Topol. 9 (2005) 95

[2] F Bonahon, Geodesic laminations on surfaces, from: "Laminations and foliations in dynamics, geometry and topology (Stony Brook, NY, 1998)", Contemp. Math. 269, Amer. Math. Soc. (2001) 1

[3] A J Casson, S A Bleiler, Automorphisms of surfaces after Nielsen and Thurston, London Mathematical Society Student Texts 9, Cambridge University Press (1988)

[4] H Doll, A generalized bridge number for links in 3–manifolds, Math. Ann. 294 (1992) 701

[5] M Eudave-Muñoz, Incompressible surfaces and $(1,2)$–knots,

[6] M Eudave-Muñoz, Incompressible surfaces and $(1,1)$–knots, J. Knot Theory Ramifications 15 (2006) 935

[7] W J Harvey, Boundary structure of the modular group, from: "Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978)", Ann. of Math. Stud. 97, Princeton Univ. Press (1981) 245

[8] A E Hatcher, Measured lamination spaces for surfaces, from the topological viewpoint, Topology Appl. 30 (1988) 63

[9] J Hempel, 3–manifolds as viewed from the curve complex, Topology 40 (2001) 631

[10] J Johnson, Bridge number and the curve complex,

[11] J Johnson, A Thompson, On tunnel number one knots which are not $(1,n)$,

[12] T Kobayashi, Pseudo-Anosov homeomorphisms which extend to orientation reversing homeomorphisms of $S^3$, Osaka J. Math. 24 (1987) 739

[13] T Kobayashi, Heights of simple loops and pseudo-Anosov homeomorphisms, from: "Braids (Santa Cruz, CA, 1986)", Contemp. Math. 78, Amer. Math. Soc. (1988) 327

[14] T Kobayashi, Y Rieck, Knot exteriors with additive Heegaard genus and Morimoto's Conjecture,

[15] T Kobayashi, Y Rieck, On the growth rate of tunnel number of knots,

[16] D Mccullough, Virtually geometrically finite mapping class groups of 3–manifolds, J. Differential Geom. 33 (1991) 1

[17] Y Moriah, H Rubinstein, Heegaard structures of negatively curved 3–manifolds, Comm. Anal. Geom. 5 (1997) 375

[18] K Morimoto, M Sakuma, Y Yokota, Examples of tunnel number one knots which have the property “$1{+}1{=}3$”, Math. Proc. Cambridge Philos. Soc. 119 (1996) 113

[19] R C Penner, J L Harer, Combinatorics of train tracks, Annals of Mathematics Studies 125, Princeton University Press (1992)

[20] M Scharlemann, M Tomova, Alternate Heegaard genus bounds distance, Geom. Topol. 10 (2006) 593

[21] W P Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. $($N.S.$)$ 19 (1988) 417

[22] M Tomova, Distance of Heegaard splittings of knot complements,

[23] M Tomova, Multiple bridge surfaces restrict knot distance,

[24] F Waldhausen, Heegaard–Zerlegungen der 3–Sphäre, Topology 7 (1968) 195

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