A generalized bridge number for links in 3-manifolds

Abstract

In [Sch] Schubert introduced a new invariant of knots in the 3-sphere, called the bridge number, and showed that, when reduced by 1, it is an additive invariant under the connected sum operation of knots. Several generalizations of Schubert 's theorem can be imagined. Indeed, from Schubert 's perspective the behaviour of bridge number under addition is only a corollary of a more general theorem about bridge number of satellite knots. Boileau and Lickorish have independently observed [B,L] that the following generalization of Schubert 's theorem would be useful. Bridge number as defined by Schubert begins with a projection of the knot (or link) onto a sphere in S 3. Project instead onto a higher genus unknotted surface in S 3. One advantage of this approach is that it allows bridge number to be defined for a link in a 3-manifold other than 5 '3, by projecting the link onto a Heegaard surface for the 3-manifold. Here we explore how such an invariant behaves under connected sum, using a line of argument similar to Jaco 's proof of Haken 's theorem that Heegaard genus is additive [Ja]. A crucial additional ingredient needed is Lemma4.6. Some of the arguments were motivated by [NO], in which, as an aside, a different generalization of Schubert 's theorem is attempted. They want to show that the standard bridge number behaves well under knot decomposition even when the knot is decomposed by spheres intersecting the knot more than twice (e.g. under tangle sums), l Besides recovering a new proof of Schubert 's theorem, more in the spirit of Jaco 's proof of the Haken lemma, we are also able to prove an analogous theorem for bridge number defined with respect to genus one Heegaard splittings of 5'3 or the Lens space and also with respect to irreducible Heegaard splittings of closed nonHaken 3-manifolds.