Studies in Braids, Knots and Three-Manifolds

9705019 Bman这项研究主要有两个项目。第一个涉及结点理论中的算法问题，并从开发用于识别结点的具体计算机算法开始。推测这个问题是NP类问题。这项工作将依赖于一个相关的项目：开发一种快速算法来解决辫子群中的单词和共轭问题。非结点的算法应该扩展到用于区分其他结点的相关算法。第二个问题涉及三维流形的有限类型不变量。研究者希望能够将Ng关于带结群的定理推广到具有平凡Rohlin不变量的同调3球群上。这项研究的第一个主要项目是开发一种算法，用于检测一个表面上打结的圆(可以被认为是打结的圆线)何时是真正的非结，即它何时可以变形为位于平面上的圆，而不是自交点。相关问题涉及确定两个结何时具有相同的结类型。第二个略有不同的项目是关于三维流形的新的Ohtsuki或有限类型不变量。研究者希望能够描绘出它们与更古老和更经典的三维流形不变量的关系，特别是与Rohlin不变量的关系。***

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. ***