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Distance and bridge position. (English) Zbl 1086.57011
A knot \(k\) is said to be in bridge position with respect to a Heegaard surface \(F\) if the arcs into which \(F\) divides \(k\) are unknotted in each handlebody.
The distance of \(k\) with respect to \(F\) is defined in terms of a simplicial complex associated with the simple curves on \(F\). H. A. Masur and Y. N. Minsky [Invent. Math. 138, 1, 103–149 (1999; Zbl 0941.32012)].
Let \(S\) be an orientable, properly imbedded essential surface in the complement of a tubular neighborhood of \(k\).
The authors bound the distance of \(k\) with respect to \(F\) by twice the genus of \(S\) plus the number of its boundary components.
The ideas stem from K.Hartshorn’s thesis result relating the distance of a Heegaard splitting to the genus of an essential surface [K. Hartshorn, ”Heegaard splittings: the distance complex and the stabilization conjecture”, PhD thesis, Univ. of Calif., Berkeley, (1999)].
One corollary to the result is that knots with distance \(\geqslant 3\) have hyperbolic complements of finite volume.

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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