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Multiple bridge surfaces restrict knot distance. (English) Zbl 1142.57005
Let $$P$$ be a compact, orientable surface. The curve complex of $$P$$ is a graph $$C(P)$$, with vertices corresponding to isotopy classes of essential simples closed curves in $$P$$, and where two vertices are adjacent if the corresponding isotopy classes of simple close curves have disjoint representatives. Suppose now that $$P$$ is properly embedded in a compact orientable 3-manifold $$M$$, separating it into submanifolds $$A$$ and $$B$$, and so that $$P$$ is compressible to both sides. The distance of $$P$$, $$d(P)$$, is defined as the length of the shortest path from a vertex in $$C(P)$$ corresponding to a simple closed curve which bounds a disk in $$A$$ to a vertex corresponding to a simple closed curve which bounds a disk in $$B$$. This distance was first defined by J. Hempel [Topology 40, No. 3, 631–657 (2001; Zbl 0985.57014)] for Heegaard splittings. Let $$K$$ be a knot in a 3-manifold $$M$$; $$P$$ is a bridge surface for $$K$$ if $$P$$ is a Heegaard surface for $$M$$ and $$K$$ intersects each of the components of $$M-P$$ in arcs which are simultaneously parallel to $$P$$. Define $$d(K,P)$$ to be $$d(P_K)$$ in $$M_K=M-N(K)$$, where $$P_K=M_K\cap P$$. It has been shown that the existence of interesting surfaces in a manifold gives bounds for the distance of a Heegaard surface or a bridge surface, for example, in the works of K. Hartshorn [Pac. J. Math. 204, No. 1, 61–75 (2002; Zbl 1065.57021)], M. Scharlemann and M. Tomova [Geom. Topol. 10, 593–617 (2006; Zbl 1128.57022 )], and D. Bachman and S. Schleimer [Pac. J. Math. 219, No. 2, 221–235 (2005; Zbl 1086.57011)].
In the paper under review it is shown that if $$K$$ is a non-trivial knot in an irreducible 3-manifold M, $$P$$ is a bridge surface for $$K$$ that it is not a 4-punctured sphere, and $$Q$$ is also a bridge surface for $$K$$ that is not equivalent to $$P$$, or it is a Heegaard surface for $$M- N(K)$$, then $$d(K,P)$$ is bounded in terms of $$\chi(Q)$$, namely $$d(K,P)\leq 2-\chi(Q-K)$$. The proof consists in taking a 2-parameter sweep-out of $$M-K$$ by the two bridge surfaces , in a similar way as it is done by M. Scharlemann and M. Tomova for two Heegaard surfaces [op. cit.]. As a corollary, it is shown that if $$K$$ is a knot in $$S^3$$ and $$K$$ is in minimal bridge position with respect to a sphere $$P$$ with $$d(K,P) > | P\cap K|$$, then $$K$$ has a unique minimal bridge position.

##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010) 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 57M50 General geometric structures on low-dimensional manifolds
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##### References:
 [1] D Bachman, Thin position with respect to a heegaard surface [2] D Bachman, S Schleimer, Distance and bridge position, Pacific J. Math. 219 (2005) 221 · Zbl 1086.57011 [3] A J Casson, C M Gordon, Reducing Heegaard splittings, Topology Appl. 27 (1987) 275 · Zbl 0632.57010 [4] K Hartshorn, Heegaard splittings of Haken manifolds have bounded distance, Pacific J. Math. 204 (2002) 61 · Zbl 1065.57021 [5] J Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (2001) 631 · Zbl 0985.57014 [6] H Rubinstein, M Scharlemann, Comparing Heegaard splittings of non-Haken $$3$$-manifolds, Topology 35 (1996) 1005 · Zbl 0858.57020 [7] T Saito, M Scharlemann, J Schultens, Lecture notes on generalized heegaard splittings · Zbl 0953.57002 [8] M Scharlemann, Proximity in the curve complex · Zbl 1127.57010 [9] M Scharlemann, Local detection of strongly irreducible Heegaard splittings, Topology Appl. 90 (1998) 135 · Zbl 0926.57018 [10] M Scharlemann, A Thompson, Thinning genus two Heegaard spines in $$S^3$$, J. Knot Theory Ramifications 12 (2003) 683 · Zbl 1048.57002 [11] M Scharlemann, M Tomova, Uniqueness of bridge surfaces for 2-bridge knots · Zbl 1152.57006 [12] M Scharlemann, M Tomova, Alternate Heegaard genus bounds distance, Geom. Topol. 10 (2006) 593 · Zbl 1128.57022 [13] M Tomova, Thin position for knots in a 3-manifold · Zbl 1220.57004
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