RUI: Polygonal Knot Theory, Controlled Topology and Topology of Homology Manifolds

Vassar College, Poughkeepsie NY

Investigators

Abstract

Abstract Award: DMS-0104111 Principal Investigator: Heather M. Johnston Polygonal knot theory studies the isotopy classes of embedded polygons in three space where the number and length of edges are fixed throughout the isotopy. The PI and co-author have found the first examples of polygons which are topologically unknotted, but for which the isotopy class of embeddings is nontrivial. The PI and student collaborators will investigate questions such as whether or not there are any such examples for equilateral polygons. Surgery theory studies the set of manifold structures within a given homotopy type. The Bryant-Ferry-Mio-Weinberger surgery exact sequence for homology manifolds has been used by the PI to prove that up to s-cobordism many of the geometric properties of manifolds also hold for homology manifolds. The Novikov conjecture and related coarse Novikov conjecture are keys to our understanding of surgery theory. The PI has developed some techniques for attacking the coarse Novikov conjecture for unusual non uniformly contractible spaces. Controlled and bounded surgery theory will be used to further investigate homology manifolds and the coarse Novikov conjecture. Up to now, topologists have studied well-behaved spaces such as manifolds. Yet singular spaces arise more and more frequently in subjects such as analysis, algebraic geometry and physics. The non-resolvable homology manifolds studied in this project are so singular that they have no points whatsoever with Euclidean neighborhoods. Perhaps these strange spaces will someday account for the extra dimensions of the universe predicted by string theory. Coarse geometry, and the coarse homology of John Roe are ways of separating the large scale behavior of spaces from the local information. The invariants of coarse geometry and topology depend only on the large scale behavior of the space. Better understanding of these objects and their invariants will help topologists to classify the different types of singular spaces, which appear throughout modern physics. In polygonal knot theory, a new twist on the classical study of knots, a different type of singular spaces is studied. This new theory is modeled by sticks joined end to end by universally flexible joints (rubber tubing perhaps) to form a closed loop. The PI has produced the first examples of configurations which are stuck (cannot be unraveled), but only because they are made of sticks. If the same configurations were made of string they could be unraveled. Knot theory of strings has been applied to the study of protein and DNA molecules. For a small number of atoms each bond can be represented by a stick in the model discussed here, This is a richer and more appropriate model for small molecules than the string which has been used in the past.