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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.

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
Full Text: DOI arXiv
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