Width complexes for knots and 3-manifolds

Abstract

We define the width complexes for knots and 3-manifolds. We consider basic cases and reformulate the stabilization problem for Heegaard splittings from this point of view. Simplicial complexes, cell complexes and more general complexes have featured prominently in geometric group theory for many years. Many such complexes have proved useful in the study of 3-manifolds. One example is the curve complex, introduced by W. Harvey nearly 30 years ago, see [8] and extensively studied by H. Masur and Y. Minsky, see [16] and [17] and J. Hempel, see [9]. We here define and study a rather different complex: the “width complex” for a knot or a 3-manifold. The construction grows out of a desire to better understand the workings of thin position, both in the context of knots and in the context of 3-manifolds. The notion of thin position for a knot was introduced by D. Gabai, see [7]. He used this notion to prove Property R for knots. It was also used in the seminal work by M. Culler, C. Gordon, J. Luecke and P. Shalen concerning Dehn surgery on knots, see [6]. The related notion of thin position for 3-manifolds was later introduced by M. Scharlemann and A. Thompson, see [20]. It too has become a fundamental tool in the study of 3-manifolds. Roughly speaking, in the case of knots, vertices of the width complex of a knot consist of appropriate equivalence classes of embeddings of the circle with respect to a height function, that is to say, a Morse function with exactly two critical points. In the case of 3-manifolds, they correspond to appropriate equivalence classes of Morse functions. In the case of knots, edges correspond to specified isotopies of embeddings. In the case of 3-manifolds, they correspond to the effect of passing through the singularities of Cerf theory. Higher dimensional simplices can be defined when edges correspond to independent alterations. Research to date provides glimpses of the width complex. Many results can be interpreted or reformulated from this point of view. In effect, this highlights some of the formal similarities and differences between Heegaard splittings and certain aspects of knot theory. Most interestingly, it provides a reformulation of the stabilization problem for Heegaard splittings of 3-manifolds. I wish to thank the Max-Planck-Institut fur Mathematik in Bonn, Germany where this work was begun and the Max-Planck-Institut fur Mathematik in den Naturwissenschaften in Leipzig, Germany, where this work was completed. The work was supported in part by a grant from the NSF.