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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.

##### MSC:
 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 57M25 Knots and links in the $$3$$-sphere (MSC2010)
##### Keywords:
Heegaard splitting; curve complex; genus; knot
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