Knots and 3-Manifolds

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AbstractAward: DMS-0104126Principal Investigator: Abigail A. ThompsonThe focus of the proposed research is the study of knots and3-dimensional spaces. We will examine questions about surgery onknots, knotting of graphs in 3-space dimensional spaces, andHeegaard decompositions of 3-manifolds. We describe threespecific goals and their connections to some of the basicquestions in the field. The first goal is to develop the toolsand techniques of multi-parameter thin position and sweep-outs sothat they can be used to tackle some of the long-standingproblems in knot theory and 3-manifold theory. A second is toextend work of the proposer and M. Scharlemann on unknottedplanar graphs in the 3-sphere to a more general setting.Finally, we aim to obtain a clearer understanding of generalquestions about 3-manifolds by obtaining a deeper understandingof 3-manifolds of Heegaard genus two. We will examine specificquestions about tunnel number one knots, special cases of largerquestions about what genus two manifolds can be obtained bysurgery on a knot in the 3-sphere, which of these manifoldscontain immersed surfaces with injective fundamental groups, andhow information from the Heegaard diagrams of these manifoldstranslates into geometric and algebraic information about themanifold itself.We live in what is apparently a universe with three spatialdimensions. Exploring the possible forms that our universe mighthave is not only a deep problem for physicists but also formathematicians. Different mathematical possibilities lead tovery different expected physical properties. As an example, wedo do not know if, were we able to program a very fast rocket togo "straight" into space, it would eventually return to itsstarting point, like a ship on the surface of the earth, or if itwould continue to travel away from us forever. There are aninfinite number of possibilities for the shape of the universe.This project proposes to explore some of them using thetechniques of low-dimensional topology and knot theory.