It became an important question in low dimensional topology whether one can use finite type invariants of knots and homology 3-spheres to get more insights into classical problems like the determination of the hyperbolic volume or the Property P quest. One of the goals of this research project is to apply and extend ideas and methods which are used in the theory of 3-manifold invariants, mainly quantum invariants for knots and homology spheres, towards progress in the Property P quest. Furthermore, these methods are combined with tools coming from the theory of Legendrian and Transversal knots. The aim is to get combinatorial obstructions, that are structurally new, ensuring Property P for a knot. In turn, tools that are natural from a more classical topological point of view are applied to the study of quantum invariants.
It has been a huge step in the history of mankind to realize that the earth has the shape of a sphere, i.e. the surface of a ball, rather than it being a disk. Another possibility, for example, would have been that the earth has the shape of the surface of a doughnut. Surfaces are 2-dimensional objects, called 2-manifolds. One dimension higher, one has to deal with similar problems. The understanding of how our 3-dimensional universe could look like is much less developed. The aim of three-dimensional topology is to study and classify the possibilities how the whole universe might be shaped. One way to generate these, infinitely many, possible 3-manifolds is through knotted circles. One gets a new 3-manifold by taking out a small neighborhood of a knot in a 3-manifold and inserting it in a different way. The problem remains, how to distinguish two different 3-manifolds that arise that way. The work in this proposal studies the effect that combinatorial and geometric properties of the knot have on the structure of the resulting 3-manifold. Topology has been shown to be a natural and very fruitful source for applications in fields outside of mathematics. For example, knot theory has applications in the study of DNA functions, in cryptography and in the development of models for quantum computing.
