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Bridge decompositions with distances at least two. (English) Zbl 1250.57016
Summary: For $$n$$-bridge decompositions of links in $$S^3$$, we propose a practical method to ensure that the Hempel distance is at least two.

##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010) 57N10 Topology of general $$3$$-manifolds (MSC2010)
##### Keywords:
knot distance; bridge position; Hempel distance
Full Text:
##### References:
 [1] D. Bachman and S. Schleimer, Distance and bridge position , Pacific J. Math. 219 (2005), no. 2, 221-235. · Zbl 1086.57011 [2] J. Berge, A closed orientable $$3$$-manifold with distinct distance three genus two Heegaard splittings , [3] J. S. Birman, Plat presentations for link groups , Collection of articles dedicated to Wilhelm Magnus. Comm. Pure Appl. Math. 26 (1973), 673-678. · Zbl 0266.20027 [4] A. Casson and C. Gordon, Manifolds with irreducible Heegaard splittings of arbitrary large genus , Unpublished. · Zbl 0632.57010 [5] R. Crowell and R. Fox, Introduction to knot theory, Graduate Texts in Mathematics, 57 , Springer-Verlag, New York-Heidelberg. · Zbl 0126.39105 [6] K. Hartshorn, Heegaard splittings of Haken manifolds have bounded distance , Pacific J. Math. 204 (2002), no. 1, 61-75. · Zbl 1065.57021 [7] J. Hempel, 3-manifolds as viewed from the curve complex , Topology 40 (2001), no. 3, 631-657. · Zbl 0985.57014 [8] J. H. Lee, Rectangle condition for irreducibility of Heegaard splittings , [9] M. Lustig and Y. Moriah, High distance Heegaard splittings via fat train tracks , Topology Appl. 156 (2009), no. 6, 1118-1129. · Zbl 1214.57021 [10] H.A. Masur and Y.N. Minsky, Geometry of the complex of curves. I. Hyperbolicity , Invent. Math. 138 (1999), no. 1, 103-149. · Zbl 0941.32012 [11] H.A. Masur and Y.N. Minsky, Geometry of the complex of curves. II. Hierarchical structure , Geom. Funct. Anal. 10 (2000), no. 4, 902-974. · Zbl 0972.32011 [12] H. Masur and S. Schleimer, The geometry of the disk complex , · Zbl 1272.57015 [13] M. Scharlemann and M. Tomova, Alternate Heegaard genus bounds distance , Geom. Topol. 10 (2006), 593-617. · Zbl 1128.57022 [14] M. Tomova, Multiple bridge surfaces restrict knot distance , Algebr. Geom. Topol. 7 (2007), 957-1006. · Zbl 1142.57005
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