Cohomology State-Sum Invariants in Dimensions 3 and 4

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Proposal number: 9988107Title: Cohomology state-sum invariants in dimensions 3 and 4PI: J. Scott Carter, University of South AlabamaAbstract:New state-sum invariants for knots in 3-dimensional space andknotted surfaces in 4-dimensional space are defined by theprincipal investigator and collaborators as follows.A finite quandle is chosen. Its elements are assigned to arcsof 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 tocrossings (or triple points), the product of weights are takenover all crossings (or triple points), and the sum is taken overall possible colorings. The resulting expression is the state-suminvariant. The state-sum invariant can detect non-invertibilityof knotted surfaces. Similar state-sum invariants are defined fortriangulated 4-manifolds, using colors and weights from a cohomologytheory 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 inrelation to topological quantum field theories.A knot is a circle situated in space. Knot theory studies differencesamong such knotted circles, and has applications to DNA theory andphysics. When a knot is drawn on a piece of paper with self-crossingpoints (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 weightson crossings, and compute a number called the state-sum, by takingsum 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-sumwhich can also be defined for higher dimensional knots --- knottedsurfaces in 4-dimensional space. They also discovered a similar state-sumfor 4-dimensional geometric objects, that are divided into small 4-dimensionaltetrahedra. The project is to compute, interprete, and apply thesenew state-sums. The investigation requires developing an intricateunderstanding of the algebraic structures that are used as colors,and the geometric study of properties of the state-sums. Relations toother physical theories are expected.