Research in Knots, Links and 3-Manifolds

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AbstractAward: DMS-0102231Principal Investigator: Xiao-Song LinThe central theme of this project is to explore as thoroughly aspossible the significance of the Jones-Witten and Vassilievinvariants to knots, links, and 3-manifolds as topologicalstructures. To be more specific, we will study the thermodynamiclimit of the colored Jones polynomial using various probabilisticmodels; to explore beyond a simple formulation of the Cassoninvariant we found; to understand the congruence relation amongOhtsuki's invariants through congruence subgroups of the modulargroup; to find the normal form of a degree 1 map from a knotcomplement to another knot complement; to study the cohomology ofknot complexes; to explore the applicability of classicaltechniques in 3-manifold topology to the study of Vassilievinvariants; and to continue the study of value and rootdistributions of the Jones polynomial.The phenomenon of knotting is a fundamental feature of the spacethat we live in, and knot theory is thus an important part ofmankind's scientific knowledge because it is aimed atunderstanding the interplay of mathematical formulae and spacestructures. It is no wonder that concepts and tools originatedfrom knot theory have been used in many areas of mathematics, aswell as in chemistry, biology, physics, and computer science. Forexample, geneticists utilize knot theory to understand theprocesses of DNA replication and the function of enzymes that"unknot" DNA strands, and chemists use knot theory to understandand distinguish between different types of molecules. Our abilityto discern different knots could well be served as a test of ourscientific understanding of space structures.