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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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