Cohomology State-Sum Invariants in Dimensions 3 and 4

University Of South Alabama, Mobile AL

Investigators

Abstract

Proposal number: 9988107 Title: Cohomology state-sum invariants in dimensions 3 and 4 PI: J. Scott Carter, University of South Alabama Abstract: New state-sum invariants for knots in 3-dimensional space and knotted surfaces in 4-dimensional space are defined by the principal investigator and collaborators as follows. A finite quandle is chosen. Its elements are assigned to arcs of knot diagrams (or regions of knotted surfaces) as colors, where the quandle condition holds at every crossing. Weights in the form of quandle cocycles, then, are assigned to crossings (or triple points), the product of weights are taken over all crossings (or triple points), and the sum is taken over all possible colorings. The resulting expression is the state-sum invariant. The state-sum invariant can detect non-invertibility of knotted surfaces. Similar state-sum invariants are defined for triangulated 4-manifolds, using colors and weights from a cohomology theory of quantum double of finite groups. Our project is to compute, interprete, and apply these new invariants. Relations to other theories, such as Seiberg-Witten invariants, spin-foam models of quantum gravity, are expected. Higher categorical structures are also investigated in relation to topological quantum field theories. A knot is a circle situated in space. Knot theory studies differences among such knotted circles, and has applications to DNA theory and physics. When a knot is drawn on a piece of paper with self-crossing points (called crossings), it is called a knot diagram. One of the methods in knot theory is to assign numbers (called colors) to arcs in a knot diagram with certain rules imposed, assign weights on crossings, and compute a number called the state-sum, by taking sum and product of weights with respect to all possible colorings. The idea of state-sums came from statistical mechanics. Instead of numbers, abstract algebraic systems can be used as colors. The principal investigator and collaborators discovered a new state-sum which can also be defined for higher dimensional knots --- knotted surfaces in 4-dimensional space. They also discovered a similar state-sum for 4-dimensional geometric objects, that are divided into small 4-dimensional tetrahedra. The project is to compute, interprete, and apply these new state-sums. The investigation requires developing an intricate understanding of the algebraic structures that are used as colors, and the geometric study of properties of the state-sums. Relations to other physical theories are expected.