The project has three components, all concerning interactions between low-dimensional topology and quantum algebra. The first main theme of the proposed research is multivariable link invariants. The Principal Investigator and Patureau-Mirand have shown that Lie superalgebras naturally give rise to multivariable link invariants. The project continues this work with an aim of further understanding these invariants and how they relate to other link invariants. This initial work suggests that these multivariable link invariants are related to the Volume Conjecture. The proposed research will also focus on the categorification of a quantum invariant arising from a particular Lie superalgebra. This could give a topological categorification of the Alexander polynomial, in the sense of Bar-Natan's topological interpretation of Khovanov homology. A final theme in the proposed research is the construction of a 3-manifold invariant associated to a Lie superalgebra.
Knots appear in many places ranging from everyday items such as tied shoe laces to string theory and protein folding. In mathematics, knots and more generally links are fundamental basic objects in topology. Knots have been studied since the 19th century. More recently, with the quantum field theory point of view moving into low-dimensional topology, the late 20th century saw the emergence of a revolution in the theory of knots and links. In particular, the theory of quantum groups associated to Lie algebras has been widely and productively used in low-dimensional topology. Not as well developed is the theory of quantum groups associated to Lie superalgebras and its applications. The theory of Lie superalgebras has particular properties that do not arise in the theory of Lie algebras. The focus of this proposal is to use these unique properties to gain new information about well known problems and objects in low-dimensional topology.
Knots appear in many places ranging from everyday items such as tied shoe laces to string theory and protein folding. In mathematics, knots and more generally links are fundamental basic objects in topology. Knots have been studied since the 19th century. More recently, with the quantum field theory point of view moving into low-dimensional topology, the late 20th century saw the emergence of a revolution in the theory of knots and links. In particular, the theory of quantum groups associated to Lie algebras has been widely and productively used in low-dimensional topology. Not as well developed is the theory of quantum groups associated to Lie superalgebras and its applications. The theory of Lie superalgebras has particular properties that do not arise in the theory of Lie algebras. The focus of this project was to use these unique properties to gain new information about well known problems and objects in low-dimensional topology. Surprisingly, this line of research led to a general theory of the so called re-normalized link invariants, which are actively being studied by the principal investigator and other mathematicians. This general theory contains a large class of examples. Moreover, these re-normalized link invariants have connections with the Volume Conjecture and 3-manifold invariants.
