Short hierarchies for knot complements

低维拓扑中的一个中心问题是对节点进行分类。在他最后发表的论文中，图灵强调了这一点：“目前还没有一种系统的方法可以用来判断两个结是否相同。现在有几种方法可以解决这个问题，但仍然没有解决的是是否有一个有效的解决方案。具体地说，下面这个著名的问题仍然没有答案：人们能否决定一个纽结图是否在多项式时间内表示解结？这就是所谓的“解结问题”，马克·拉肯比（Marc Lackenby）多年来一直致力于解决这个问题和相关问题。在最近的一项突破中，他发现了一种在准多项式时间内运行的解结识别算法。具体地说，如果输入图有n个交叉点，则对于某个常数k，运行时间至多为k^（（log（n））^3）。他使用的主要方法是层次结构，层次结构的定义如下：首先从结的外部开始，这是一个3-流形M1。然后我们沿着沿着一个适当嵌入的曲面切割这个曲面，得到一个流形M2。重复这个过程，直到我们到达一个流形ML，它是一个3球的集合。这里L是层次结构的长度。事实证明，在Lackenby的算法中，L是一个寻求约束的关键量。粗略地说，他能够将运行时间限制在nL以内。他展示了如何找到长度最多为（log（n））2的层次结构，然后导致运行时间上的准多项式约束。目的：该项目的主要目的是找到和使用长度较短的层次结构。事实上，它是由雅科表明，每一个纽结补语承认一个层次的长度为4。如果人们可以找到这样一个层次的算法，然后将有一个多项式时间的解决方案解开问题！当然，这有点雄心勃勃，但也有值得的中间目标：-找到短层次结构的其他应用程序。特别是，人们可以使用它们来有效地确定一个结是否被解开？它们是否可以用来证明结的双曲性可以有效地得到证明？是否可以为特定类别的结找到短层次结构，例如具有有界索引的闭合编织物？层次结构也被用于其他场合，例如Gabai和Thurston对纽结亏格的确定，以及Waldhausen对拓扑刚性的证明。在这些设置中使用短层次结构有什么好处吗？方法论：Lackenby开发的方法是高度组合的，尽管许多是几何启发的。它们是非常新的，因此在现阶段尝试利用它们是明智的。尽管如此，层次结构的使用已经得到了很好的认可：它是由Haken在20世纪60年代提出的，因此有大量的理论可以作为这个项目的基础。研究领域：这福尔斯EPSRC研究领域“几何与拓扑”。没有公司或合作者参与。

A central problem in low-dimensional topology is to classify knots. In his final published paper, Turing highlighted this: 'No systematic method is yet known by which one can tell whether two knots are the same.' There are now several methods to solve this problem, but what remains unresolved is whether there is an efficient solution. Specifically, the following famous question remains unanswered: can one decide whether a knot diagram represents the unknot in polynomial time? This is known as the 'unknotting problem'.Marc Lackenby has been working on this and related questions for many years. In a recent breakthrough, he has found an algorithm for unknot recognition that runs in quasi-polynomial time. Specifically, if the input diagram has n crossings, the running time is at most k^((log(n))^3) for some constant k. The main method that he used was hierarchies, which are defined as follows.One starts with the exterior of the knot, which is a 3-manifold M1. Then one cuts this along a properly embedded surface, giving a manifold M2. This process is repeated, until we reach a manifold ML which is a collection of 3-balls. Here L is the length of the hierarchy. It turns out that, in Lackenby's algorithm, L is the crucial quantity that one seeks to bound. Roughly, he was able to bound the running time by nL. He showed how to find hierarchies with length at most (log(n))2, which then led to the quasi-polynomial bound on running time.Aim: The main aim of the project is to find and use hierarchies with short length. Indeed, it was shown by Jaco that every knot complement admits a hierarchy with length 4. If one could find such a hierarchy algorithmically, then one would have a polynomial time solution to the unknotting problem!Of course, this is somewhat ambitious, but there are worthwhile intermediate goals:- Find other applications of short hierarchies. In particular, can one use them to determine efficiently whether a knot is fibred? Might they be used to show that knot hyperbolicity can be efficiently certified?- Can short hierarchies be found for specific classes of knots, for example closed braids with bounded index?- Hierarchies have been used in other contexts, for example, the determination of knot genus by Gabai and Thurston, and the proof of topological rigidity by Waldhausen. Is there any advantage to using short hierarchies in these settings?Methodology: The methods developed by Lackenby are highly combinatorial, although many are geometrically inspired. They are very new and so it is sensible to try to exploit them at this stage. Nevertheless, the use of hierarchies is well-established: it was initiated by Haken in the 1960s, and so there is a substantial body of theory upon which this project can be based.Research area: This falls within the EPSRC research area "Geometry & Topology".No companies or collaborators will be involved.