A longstanding problem in topology is to classify knots (a closed loop formed from a rope winding in space and closing back on itself) by asking how far a given knot is from being unknotted (that is, can be pulled apart to look like an ordinary circle). This problem (the basis of the subject of knot theory) has implications in physics (quantum theory), chemistry (molecular knots) and biology (knotting of DNA). A central tool in classifying knots, and indeed in many topological questions, is to assign an invariant to a knot: two knots are then different if their invariants are different. The goal is find robust invariants which can distinguish different knots. This project explores new types of knot invariants and continues the trend of using tools from algebra to define and investigate knot invariants. The PI will use techniques from the mathematical fields of algebraic geometry, combinatorics, and representation theory. The focus of the project is on the class of knots and links that arise from intersecting an algebraic curve in the plane with a small sphere centered at the singularity of the curve (such knots and links are called algebraic). In this project the PI will study the interaction between the topological invariants of algebraic knots and links and certain algebraic and combinatorial objects associated to the corresponding curve.
Quantum knot invariants have proven to be a powerful tool in low-dimensional topology. To every knot one can associate a polynomial with integer coefficients in one variable (as in the Alexander polynomial or the Jones polynomial) or in two variables (as in the HOMFLY polynomial). It has recently been discovered by the PI and his collaborators that for torus knots all coefficients in the HOMFLY polynomial are in fact nonnegative. To prove this fact, certain representations of the rational Cherednik algebra were studied and it was shown that the dimensions of some graded subspaces match the HOMFLY coefficients. Khovanov and Rozansky introduced another collection of vector spaces, called HOMFLY homology, such that the HOMFLY coefficients are presented as alternating sums of their dimensions. The similarity of the two constructions suggests that for a torus knot Khovanov-Rozansky homology may be isomorphic to a representation of the rational Cherednik algebra, equipped with an extra grading (or filtration). This conjecture has been verified in many examples, but remains open in general. The PI plans to use the representation theory of rational Cherednik algebras for the construction of explicit combinatorial and geometric models for the Khovanov-Rozansky homology of torus knots, and their generalizations to algebraic knots and links. Other knot homology theories, such as Heegaard-Floer homology, will also be studied.
