AN ESTIMATE OF HEMPEL DISTANCE FOR BRIDGE SPHERES

Tomova (8) gave an upper bound for the distance of a bridge surface for a knot with two different bridge positions in a 3-manifold. In this paper, we show that the result of Tomova (8, Theorem 10.3) can be improved in the case when there are two different bridge spheres for a link in S 3 . Hempel (4) introduced the concept of distance of a Heegaard surface, and it is shown by many authors that it well represents various complexities of 3- manifolds. For example, Hartshorn (2) showed that the Euler characteristic of an incompressible surface in a 3-manifold bounds the distance of its Heegaard splittings, and Scharlemann and Tomova (7) showed that the Euler character- istic of any Heegaard splitting of a 3-manifold similarly bounds the distance of any non-isotopic Heegaard splitting. The above concept and results have been extended to bridge surfaces for knots and links in closed 3-manifolds, and have been studied by several authors. For example, Bachman and Schleimer (1) proved that Hartshorn's results can be extended to the distance of a bridge surface for a knot in a closed orientable 3-manifold, and also Tomova (8) proved that Scharlemann and Tomova's results can be extended to the distance of a bridge surface for a knot in a closed ori- entable 3-manifold. Moreover, Johnson and Tomova (6) proved that Tomova's result can be extended to a bridge surface for a tangle in a compact 3-manifold. Recently Jang (5) showed that for a link in a closed orientable 3-manifold, the result of Bachman and Schleimer (1) can be improved in the case when there exist essential meridional spheres. In this paper, we find a property of essential simple closed curves disjoint from the disk complex of a 3-ball containing trivial arcs (for detail, see Lemma 2.1). This allows us to improve the result of Tomova (8, Theorem 10.3) in the case when there are two different bridge spheres for a link in S 3 .