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Alternating links and cobordism

交替链接和共边

基本信息

批准号：
EP/I033754/1
负责人：
Brendan Edward Owens
金额：
$ 10.19万
依托单位：
University of Glasgow
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2011
资助国家：
英国
起止时间：
2011 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FI033754%2F1
关键词：
Alternating links cobordism

项目摘要

A knot in mathematics is a closed loop in space - this is the same thing as a knot in a piece of string except we stipulate that the ends of the string should be joined together after tying the knot. This research programme will use 21st century mathematics to solve problems in knot theory that have defied solution for over a hundred years.Mathematical knot theory began in the 19th century as an attempt to compose a table of elements based on Lord Kelvin's theory of knotted vortices in the aether. The aether theory proved incorrect but started a rich mathematical study that now has many important applications. In particular DNA molecules exhibit knotting behaviour and the mathematical properties of the knots involved have important biological implications. In order to replicate, knotted DNA needs to become unknotted by a sequence of crossing changes . A crossing change or strand passage is when one strand of the knot is cut, and another strand passes through the cut which is then repaired.Edinburgh physicist P. G. Tait was the first to study these crossing changes, also in the 19th century. He defined a measure of complexity of a knot called the unknotting number, which counts how many crossing changes are needed to completely undo the knot. Computing these numbers is a notoriously difficult problem to this day; in fact there is no known algorithm for deciding if a knot has unknotting number equal to one.Knots are also used in giving mathematical descriptions of 3 and 4 dimensional spaces (or manifolds) such as the universe we live in. Another notoriously difficult and important problem is to decide if a knot is slice; thinking of time as the fourth dimension, a slice knot is a snapshot of a two-dimensional sphere in spacetime. (This problem is only around fifty years old.)In the last two decades of the 20th century new techniques pioneered by Donaldson, Witten and others transformed mathematicians' understanding of 4 dimensional geometry and topology. This new mathematical gauge theory was derived from the quantum field theories of theoretical physics.In the last ten years, a new version of mathematical gauge theory due to Ozsvath and Szabo has made major progress on problems in knot theory and 3-dimensional topology.Owens will combine gauge theory results of Donaldson with the new theory of Ozsvath and Szabo to attack the unknotting number and slice problems for a major class of knots known as alternating knots (these include well-known knots such as the granny knot and reef or square knot) This class of knots is known to be prevalent in knotted DNA. Part of the goal is to find a complete solution to the unknotting number one and slice recognition problems for these knots. Further crossing change information of interest to biologists will also be discovered as well as new insights into the mysterious mathematical nature of this very familiar class of knots.

数学中的结是空间中的一个闭环——这和绳子上的结是一样的，只是我们规定绳子的两端在打结后应该连在一起。这个研究项目将使用21世纪的数学来解决结理论中一百多年来无法解决的问题。数学结论始于19世纪，当时人们试图根据开尔文勋爵关于以太中结涡的理论来组成一个元素表。以太理论被证明是不正确的，但它开启了一项丰富的数学研究，现在有许多重要的应用。特别是DNA分子表现出打结的行为，所涉及的结的数学性质具有重要的生物学意义。为了复制，打结的DNA需要通过一系列的交叉变化来解开。交叉改变或线通道是指结的一条线被剪断，另一条线穿过被剪断的地方，然后被修复。爱丁堡物理学家p·g·泰特是第一个研究这些交叉变化的人，也是在19世纪。他定义了一种测量结的复杂性的方法，称为解结数，它计算了需要多少次交叉变化才能完全解开结。直到今天，计算这些数字仍是一个出了名的难题；事实上，没有已知的算法来决定一个结的解结数是否等于1。结也被用于给出三维和四维空间（或流形）的数学描述，比如我们生活的宇宙。另一个众所周知的困难和重要的问题是确定一个结是否是切片；将时间视为第四维空间，切片结是一个二维球体在时空中的快照。（这个问题只有大约50年的历史。）在20世纪的最后二十年里，由唐纳森、威滕和其他人开创的新技术改变了数学家对四维几何和拓扑的理解。这个新的数学规范理论是从理论物理的量子场论中衍生出来的。在过去的十年里，由Ozsvath和Szabo提出的数学规范理论的新版本在结理论和三维拓扑问题上取得了重大进展。Owens将把Donaldson的规范理论结果与Ozsvath和Szabo的新理论结合起来，研究交替结（包括众所周知的“奶奶结”和“礁结”或“方结”）这类结在结型DNA中普遍存在的解结数和切片问题。部分目标是找到一个完整的解决方案，解开第一个结和切片识别问题，为这些结。生物学家感兴趣的进一步交叉变化信息也将被发现，以及对这类非常熟悉的结的神秘数学性质的新见解。