Laminations and Dehn Surgery on Knots

Boston College, Chestnut Hill MA

Investigators

Abstract

There has been tremendous success in studying Dehn surgery on knots through the use of taut foliations and essential laminations. For example, the property R conjecture, and the property P conjecture for satellite knots, have been proved using the methods of taut foliations. The principal investigator plans to study Dehn surgery on knots and links in 3-manifolds using laminations and branched surfaces. The success of the project will advance the understanding of the topology of 3 and 4 dimensional manifolds and contribute to the ultimate solutions of some central problems in this area. The investigator also plans to study tight laminations in 3-manifolds. A tight lamination is an essential lamination with a very nice transverse structure and many remarkable properties. One goal of the proposed research is to prove that if a 3-manifold contains an essential lamination then it contains a tight lamination. The investigator will also continue his work on the classification of essential laminations in hyperbolic punctured-torus bundles, and on the topological rigidity of laminar 3-manifolds. Three-manifolds are objects modeled on the 3-dimensional space that we are living in. For instance, the universe is a 3-manifold and recent study shows that the universe may have an interesting topological structure. A natural geometric way of studying 3-manifolds, which has been proved extremely fruitful, is to view a globally complicated 3-manifold as a collection of simple 3-dimensional pieces glued together along a 2-dimensional object. The tools used in the proposed research, including laminations and branched surfaces, provide such useful 2-dimensional objects. The proposed research is related to some central questions in low-dimensional topology and knot theory, which impact not only mathematics, but also physics and other scientific research. A better understanding of 3-manifolds may help us understand the shape of the universe, and knot theory is used to study the structure of DNA and its biological effect.