Studies in Braids, Knots and Three-Manifolds

辫子、结和三流形的研究

基本信息

批准号：
9705019
负责人：
Joan Birman
金额：
$ 6万
依托单位：
Barnard College
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
1997
资助国家：
美国
起止时间：
1997-08-01 至 2000-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=9705019&HistoricalAwards=false
关键词：
Studies Braids Knots Three Manifolds

项目摘要

9705019 Birman There are two main projects in this research. The first concerns algorithmic problems in knot theory and begins with the development of a concrete computer algorithm for recognizing the unknot. It is conjectured that this problem is class NP. That work will rest upon a related project: the development of a fast algorithm for solving the word and conjugacy problems in the braid group. The algorithm for the unknot should have extensions to related algorithms for distinguishing other knots. The second problem concerns finite type invariants of 3-manifolds. The investigator hopes to be able to generalize Ng's theorem on groups of ribbon knots to a similar theorem on groups of homology 3-spheres with trivial Rohlin invariant. The first principal project in this research is the development of an algorithm for detecting when an apparently knotted circle (which can be thought of as a knotted circular string) is really the unknot, i.e., when it can be deformed, without self-intersections, to a circle that lies in a plane. Related questions concern deciding when two knots have the same knot type. A second, and somewhat different, project concerns the new Ohtsuki or finite type invariants of 3-manifolds. The investigator hopes to be able to delineate their relationship to older and more classical invariants of 3-manifolds, in particular, to the Rohlin invariant. ***

9705019 Birman在这项研究中有两个主要项目。第一个涉及结理论中的算法问题，首先是开发用于识别未结的具体计算机算法。猜想这个问题是NP类。这项工作将取决于一个相关项目：开发用于解决编织组中的单词和共轭问题的快速算法。 UNENENGOT的算法应与相关算法具有区分其他结的相关算法。第二个问题涉及3个manifolds的有限类型不变。研究者希望能够将NG定理对色带结组概括为与琐碎的Rohlin不变的同源性3 spheres组的类似定理。这项研究中的第一个主要项目是开发用于检测算法时，何时明显打结的圆圈（可以将其视为打结的圆形弦）确实是无调的，即何时可以将其变形，而无需自我交流，而不是在平面中的一个圆圈。相关问题涉及决定何时两个结具有相同结的类型。第二个项目却有所不同，涉及3个manifolds的新的Ohtsuki或有限类型的不变式。调查人员希望能够描绘出他们与3个Manifolds的年龄较大，更古典的不变性的关系，尤其是与Rohlin不变的。 ***