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

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AbstractAward: DMS-0104111Principal Investigator: Heather M. JohnstonPolygonal knot theory studies the isotopy classes of embeddedpolygons in three space where the number and length of edges arefixed throughout the isotopy. The PI and co-author have foundthe first examples of polygons which are topologically unknotted,but for which the isotopy class of embeddings is nontrivial. ThePI and student collaborators will investigate questions such aswhether or not there are any such examples for equilateralpolygons. Surgery theory studies the set of manifold structureswithin a given homotopy type. The Bryant-Ferry-Mio-Weinbergersurgery exact sequence for homology manifolds has been used bythe PI to prove that up to s-cobordism many of the geometricproperties of manifolds also hold for homology manifolds. TheNovikov conjecture and related coarse Novikov conjecture are keysto our understanding of surgery theory. The PI has developedsome techniques for attacking the coarse Novikov conjecture forunusual non uniformly contractible spaces. Controlled and boundedsurgery theory will be used to further investigate homologymanifolds and the coarse Novikov conjecture.Up to now, topologists have studied well-behaved spaces such asmanifolds. Yet singular spaces arise more and more frequently insubjects such as analysis, algebraic geometry and physics. Thenon-resolvable homology manifolds studied in this project are sosingular that they have no points whatsoever with Euclideanneighborhoods. Perhaps these strange spaces will someday accountfor the extra dimensions of the universe predicted by stringtheory. Coarse geometry, and the coarse homology of John Roe areways of separating the large scale behavior of spaces from thelocal information. The invariants of coarse geometry andtopology depend only on the large scale behavior of thespace. Better understanding of these objects and their invariantswill help topologists to classify the different types of singularspaces, which appear throughout modern physics. In polygonal knottheory, a new twist on the classical study of knots, a differenttype of singular spaces is studied. This new theory is modeled bysticks joined end to end by universally flexible joints (rubbertubing perhaps) to form a closed loop. The PI has produced thefirst examples of configurations which are stuck (cannot beunraveled), but only because they are made of sticks. If thesame configurations were made of string they could be unraveled.Knot theory of strings has been applied to the study of proteinand DNA molecules. For a small number of atoms each bond can berepresented by a stick in the model discussed here, This is aricher and more appropriate model for small molecules than thestring which has been used in the past.