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. ***
