Abstract. The topological index of a surface was previously introduced by the first author as the topological analogue of the index of an unstable minimal surface. Here we show that surfaces of arbitrarily high topological index exist.Consider a compact, connected, two sided surface S properly embedded in a compact, orientable 3-manifold M . The disk complex Γ(S) is the simplicial complex defined as follows: Vertices of Γ(S) are isotopy classes of compressing disks for S. A collection of n such isotopy classes is an (n − 1)-simplex of Γ(S) if there are representatives of each that are pairwise disjoint.1. Definition. If Γ(S) is non-empty then the topological index of S is the smallest n such that π n−1 (Γ(S)) is non-trivial. If Γ(S) is empty then S will have topological index 0. If H has a well-defined topological index (i.e. Γ(S) = ∅ or some homotopy group of Γ(S) is non-trivial) then we will say that S is topologically minimal. 2. Theorem. There is a closed 3-manifold, M (1), with an index 1 Heegaard surface S, such that for each n, the lift of S to some n-fold cover M (n) of M (1) has topological index n.The manifold M (1) of Theorem 2 is obtained by gluing together the boundary components of the complement of a link in S 3 , which we will construct as follows: We say S 2 ⊂ S 3 is a bridge sphere for a knot or link L ⊂ S 3 if L meets each of the balls bounded by S 2 in a collection of boundary parallel arcs. If the minimum number of such arcs is b, then we say L is a b-bridge knot/link.Throughout the paper, we will assume L ⊂ S 3 is a two component two-bridge link such that for a regular neighborhood N of L the complement M = S 3 \ N contains no essential planar surface with Euler characteristic greater than −3.By [HT85] or [BS05], such links can be constructed by choosing a sufficiently complicated braid to define the link. Let S 2 be a bridge sphere for L that realizes its bridge number, and let B ± be the balls in S 3 bounded by S 2 . Let S = (S 2 \ N ) ⊂ M and C ± = (B ± \ N ) ⊂ M . The manifold M (1) of Theorem 2 is obtained from M by

We define integral measures of complexity for Heegaard splittings based on the graph dual to the curve complex and on the pants complex defined by Hatcher and Thurston. As the Heegaard splitting is stabilized, the sequence of complexities turns out to converge to a non-trivial limit depending only on the manifold. We then use a similar method to compare different manifolds, defining a distance which converges under stabilization to an integer related to Dehn surgeries between the two manifolds.

We describe for each positive integer k a 3‐manifold with Heegaard surfaces of genus 2k and 2k−1 such that any common stabilization of these two surfaces has genus at least 3k−1. We also find, for every k, a 3‐manifold with boundary admitting Heegaard splittings of genus k and k+1 whose stable genus is 2k, and for every positive n, a 3‐manifold that has n pairwise nonisotopic Heegaard splittings of the same genus all of which are stabilized.

A clustering algorithm partitions a set of data points into smaller sets (clusters) such that each subset is more tightly packed than the whole. Many approaches to clustering translate the vector data into a graph with edges reflecting a distance or similarity metric on the points, then look for highly connected subgraphs. We introduce such an algorithm based on ideas borrowed from the topological notion of thin position for knots and 3-dimensional manifolds.

We calculate the bridge distance for $m$-bridge knots/links in the $3$-sphere with sufficiently complicated $2m$-plat projections. In particular we show that if the underlying braid of the plat has $n - 1$ rows of twists and all its exponents have absolute value greater than or equal to three then the distance of the bridge sphere is exactly $\lceil n/(2(m - 2)) \rceil$, where $\lceil x \rceil$ is the smallest integer greater than or equal to $x$. As a corollary, we conclude that if such a diagram has more than $4m(m-2)$ rows then the bridge sphere defining the plat projection is the unique minimal bridge sphere for the knot.Comment: 21 pages 13 figure