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October 2016
by Eugene Gorsky

Eugene Gorsky is an assistant professor who joined the Department in 2014, researching knot theory and representation theory. This year he won the 2016 Hellman Fellowship, an award whose goal is to support the research of promising early-career scholars who exhibit the potential for great academic distinction.

The research of Eugene Gorsky focuses on the study of various invariants of knots and links using the methods from algebraic geometry, representation theory and combinatorics. A knot is a closed curve in three-dimensional space, whereas a link is a collection of several such curves. Knot theory can be applied to the study of long molecules: the 2016 Nobel Prize in Chemistry was awarded to researchers who constructed a new type of molecules called “catenanes” consisting of several linked loops. Although the loops are not connected by chemical bonds, being linked, they can move around freely without separating. These properties eventually enabled the development of molecular machines in which different parts can move independently in a controlled way. Although we will not elaborate it here, knot theory is also related to the 2016 Nobel Prize in Physics, concerning the entanglement of quantum particles on the plane.

Two knots or links are equivalent if one could be transformed to another one by a continuous deformation and stretching (without tearing). For example: a link with two components. Each component separately is a trivial knot (equivalent to a circle), but they cannot be untangled. However, if one modifies the central crossing, the components can be easily separated. This simple example motivates the following basic questions in knot theory: Is a given knot or link trivial? Could one untangle its components? If not, what is the minimal number of crossings that should be changed so that the components can be separated?

All these questions are usually answered with the help of link invariants. An invariant is a quantity that does not change under the deformations of a link. If two links have different invariants, they cannot be deformed into each other. The number of components is the most obvious invariant, but in the last decades a whole zoo of link invariants have been found: Alexander, Jones and HOMFLY-PT polynomials are just a few. Heegaard-Floer homology and Khovanov-Rozansky homology are more recently developed tools that have been successful in producing lower bounds for the unknotting numbers. Despite having a lot of common features, they are defined in a completely different way: Heegaard-Floer homology is defined in terms of the differential geometry of the 3-space and a knot in it, while Khovanov-Rozansky homology has a purely combinatorial definition.

Heegaard-Floer homology was originally defined using analytic tools such as counting solutions of some partial differential equations. Later explicit formulas for these invariants were found for various classes of knots. They enabled topologists to compute unknotting numbers for such knots and to prove a lot of long-standing conjectures in topology. However, the mere computation of these invariants for links with several components is not an easy task. In a series of joint works with Jennifer Hom (Georgia Tech) and Andras Nemethi (Renyi Mathematical Institute, Hungary) Gorsky found an explicit description of these invariants for several classes of links with arbitrary number of components. This opens a possibility to study the topological properties of links with the same tools that were already developed for knots, and their generalizations. A joint paper of Gorsky with Maciej Borodzik (University of Warsaw, Poland) studies the splitting numbers of links. Imagine that the components of a link are rigid, and one is allowed only to change crossings between different components. What is the minimal number of crossings needed to be changed to separate the components? In general, only a few bounds for a splitting number are known, and they are usually far from being optimal.

Despite having a more combinatorial definition, Khovanov-Rozansky homology is much harder to compute. Even for torus knots (the simplest class of knots) no closed formula for Khovanov-Rozansky homology is known. In a series of joint papers with Andrei Negut, (MIT), Alexei Oblomkov (UMass Amherst), Vivek Shende (UC Berkeley) and Jacob Rasmussen (Cambridge University) Gorsky have proposed several conjectures relating the Khovanov-Rozansky homology to the geometry of the configuration space of points in a 4-dimensional space. The structures appearing in these conjectures are closely related to combinatorics of lattice paths and representation theory of double affine Hecke algebras. If true, they would give a very effective way of computing these knot invariants. Some of these conjectures have been recently proved by Matt Hogancamp (USC) and his collaborators. It is hopeful that this line of investigation will yield the full proof of the remaining conjectures.