Studies in Braids, Knots and Three-Manifolds

9705019 Birman本研究主要有两个项目。第一部分关注结理论中的算法问题，并从开发一种识别解结的具体计算机算法开始。我们推测这个问题属于NP类。这项工作将依赖于一个相关的项目：开发一种快速算法，用于解决辫组中的单词和共轭问题。解结算法应扩展到相关算法以区分其他结。第二个问题涉及3-流形的有限型不变量。研究者希望能够将Ng在带状结群上的定理推广到具有平凡Rohlin不变量的同调三球群上的类似定理。本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（可以认为是打结的圆弦）何时真正解结，即它何时可以变形，没有自交，成为一个平面上的圆。相关问题涉及到确定两个结何时具有相同的结类型。第二个有点不同的项目涉及3流形的新Ohtsuki或有限型不变量。研究者希望能够描绘出它们与更古老、更经典的3流形不变量的关系，特别是与罗林不变量的关系。＊＊＊

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. ***