Studies in Braids, Knots and Three-Manifolds

辫子、结和三流形的研究

• 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

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 这项研究有两个主要项目。 第一个涉及结理论中的算法问题，并从开发用于识别结的具体计算机算法开始。 推测该问题属于NP类问题。 这项工作将依赖于一个相关项目：开发一种快速算法来解决辫子组中的单词和共轭问题。 绳结的算法应该对相关算法进行扩展，以区分其他绳结。 第二个问题涉及 3 流形的有限类型不变量。 研究人员希望能够将 Ng 关于带结群的定理推广到关于具有平凡 Rohlin 不变量的同源 3-球群的类似定理。 这项研究的第一个主要项目是开发一种算法，用于检测明显打结的圆（可以将其视为打结的圆绳）何时实际上是无结的，即何时它可以变形，而不会自相交，到位于平面上的圆。 相关问题涉及确定两个结何时具有相同的结类型。 第二个项目有些不同，涉及新的 Ohtsuki 或 3 流形的有限类型不变量。 研究人员希望能够描述它们与更古老、更经典的 3 流形不变量的关系，特别是与 Rohlin 不变量的关系。 ***