Knots and 3-manifolds

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Low-dimensional topology is in a state of flux. The impact of Perelman?s work on the field has not been fully assimilated. The Geometrization Conjecture was the underlying motivation for many outstanding problems, and evaluating which of these remain, and remain interesting, is a process now underway in the community. Certainly there remains much to be done, and indeed there is now an opportunity to understand the structure of the space of all 3-manifolds with greater clarity and depth. The three problems discussed in this proposal will contribute to that understanding. The goal of the first problem is to shed light on the overall structure of the set of irreducible Heegaard splittings of a specified manifold. The second problem is a classical link surgery question. It asks when it is possible to obtain a connect sum of n copies of S^1×S^2 by surgery on a link in the 3-sphere. This is directly related to the Schoenflies conjecture for smooth 3-spheres imbedded in S^4. The third problem is more speculative, and seeks to obtain information about relations among geometric invariants of an imbedded curve in 3-space. The proposer works with students at every level to convey the excitement of research in mathematics and science, including the research outlined in this proposal. She both directs and teaches in the UC Davis Cosmos program, a month-long residential summer program for high school students in math and science, with her course focused on the fields of low-dimensional topology and knot theory, the topic of the proposed research. She also works extensively with PhD students on problems closely related to those described in the proposal.Low-dimensional topology is the study of spaces of 2, 3 and 4 dimensions. It is an exciting moment in the field, as mathematicians begin to assimilate the impact of Perelman?s famous work on the Geometrization Conjecture. Certainly there remains much to be done, and indeed there is now an opportunity to understand the structure of the space of all 3-manifolds with greater clarity and depth. This proposal discusses three specific problems in the field, in the areas of Heegaard splittings, knot surgery, and knot invariants. The solutions to these problems will shed light on the the relationship among all 3-manifolds, and indeed on how 3-dimensional spaces can lie inside 4-dimensional space.