# Short summary of my research

Research field: Topology and knot theory.

Main topics: Quantum and finite type invariants of links and 3-manifolds. Applications of ideas of quantum physics in topology. Feynman diagrams and related configuration spaces in knot theory. Applications to real enumerative geometry. Virtual knots. Curves on surfaces. Deformation quantization.

## Gauss diagrams and finite type invariants

• Jointly with O. Viro, we were one of the pioneers of the Gauss diagrams technique in knot theory. Gauss diagrams present a simple (probably the simplest) combinatorial way to encode a knot. By now, a majority of programs dealing with knots use this or similar encoding. In this language one may recover the classical knot invariants. In particular, we observed that for Vassiliev — a.k.a. finite type — invariants there is an extremely simple type of formulas, given by weighted counting of certain subdiagrams. We then studied some examples of low degrees and studied the relation of this approach to other known approaches (the Kontsevich integral and the configuration spaces integrals) using maps of configuration spaces. In 1998, jointly with M. Goussarov and O. Viro, we have proven that such formulas exist for all Vassiliev knot invariants.
• However, explicit formulas were, until recently, known only for few invariants of low degree, thus a search for explicit formulas remains of interest. In particular, I established tree-like formulas for Milnor’s $\mu$-invariants. A further developments happened last year when, jointly with S. Chmutov, we discovered explicit formulas for some 2-parameter infinite family of invariants arising from the HOMFLY polynomial.
• Some other extensions of this theory of Gauss diagrams were developed, in particular they worked extremely well in Arnold’s theory of invariants of plane curves and were also successfully applied to Legendrian fronts.
• My latest direction of research is an extension of the Gauss diagram technique to invariants of 3-manifolds. In our recent papers with S. Matveev, we applied this technique to the famous Casson-Walker invariant. This allowed us to develop a program for its simple computation and established many new properties of this invariant. Now we are working on perturbative 3-manifold invariants of higher degrees.

## Virtual knots and arrow diagrams

• Another related direction of my resarch stems from the fact that not all diagrams which look like Gauss diagrams are realizable. This became the starting point of a new, now flourishing theory of “virtual knots”. Together with L. Kauffman, we were the first ones to establish the basics of this theory in 1998. It remains extremely popular and we continue to work in this direction.
• Another abstract relative of Gauss diagrams, which I introduced in 2000 — so-called algebra of arrow diagrams — turned out to be related to several other fields, in particular Hopf bialgebras, and deformation quantization. The corresponding $6T$-relations often reappear in various settings. Arrow diagrams also present an ongoing direction of my research.

## Finite type invariants of 3-manifolds

• Around 2000, there appeared several alternative theories of finite type invariants of 3-manifolds, based on 4 or 5 different approaches. Jointly with S. Garoufalidis and M. Goussarov (and independently K. Habiro), we developed a new technique — surgery along trivalent graphs, which are now known as claspres or clovers. Using claspers, we managed to unify all known theories of finite type invariants of 3-manifolds and to show their equivalence.
• While theories of FTI of manifolds are numerous, explicit examples are few — only in the lowest degrees — and effective ways to compute them, as well and an understanding of their topological meaning, are missing. Other approaches, such as the LMO invariant, the Aarhus integral, or perturbative invariants, use either the Kontsevich integral, or integrals over configuration spaces, so are not well-suited for this purpose. A new approach which we currently develop with S. Matveev is meant to fill this void. Instead of a computation of complicated configuration spaces integrals related to graphs, we use a combinatorial count of certain maps of trivalent graphs to a Gauss diagram of the surgery link. This approach is based on a Gauss diagrams technique, see above.

## Cubic complexes and finite type invariants

• There is a number of alternative notions of finite type invariants, both for links and for 3-manifolds. All of them, surprisingly, turn out to be equivalent. Our joint work with S. Matveev resulted from a wish to understand the reasons for this equivalences and include all these theories in a unified generalized setting. We formulated a generalized theory of finite type invariants in the framework of cubical complexes. We also considered a number of examples, including theories for curves, graphs, etc. This topic remains unstudied in many directions and seems to be extremely interesting and promising. Cubic complexes continue to appear in a variety of fields (lately in a fashionable theory of Heegard-Floer homology).
• Currently we study an application of this theory to real enumerative geometry: it turns out, that a natural real analogue of such famous constructions as the Gromov-Witten invariants involve FTI. Our first results in this direction were accepted with a lot of interest at a number of conferences. However, enumerative geometry is a different area with a highly developed machinery, so more time is needed to enter this field and apply our methods.
• The latest application of this approach is a recently started (and still in an early stages) work with D. Aharonov on quantum computation.

## Methods of quantum physics in topology

• My whole research, starting from my PhD thesis on invariants of 3-manifolds from conformal field theories, is deeply motivated by an application of ideas of quantum physics in low dimensional topology.
• In our early works with S. Matveev we applied some ideas of conformal field theory for a new presentation of the mapping class group and a new surgery presentation of 3-manifolds.
• Quantum groups and the Yang-Mills theory led to joint papers with N. Reshetikhin, and also to my latest paper on area-dependent TQFT’s.
• Perturbative technique of Feynman diagrams was reflected in several papers on invariants of 3-manifolds and on deformation quantization (see below).

## Feynman diagrams, configuration spaces, and their applications

• In the last decade the Feynman diagram method of quantum physics was successfully applied to various problems of low-dimensional topology. A topological quantity is expanded in an infinite series of graphs, with a coefficient of each graph being some integral over a configuration space. We made several steps to understand the topological “roots” of the Feynman diagrams technique and to apply it in various areas.
• In our latest work in progress, we propose a new type of Feynman graphs (with vertices only on the know, without usual trivalent vertices) and use some homology intersections to obtain the invariants. It leads to new homological and geometrical meanings of the invariants. Also, it allows us to get rid of the problems related to Fulton-MacPherson compactification.
• Apart from knots, an application of this approach to planar curves leads to a real version of the Gromov-Witten-type invariants (see above). In this way we expect to obtain simple solutions to a variety of enumerative problems in real algebraic geometry.
• This approach is also promising in several another areas, in particular, in deformation quantization. We already successfully applied our method for quantization of Poisson structures. This shed light on the geometric/homological meaning of Kontsevich’s series and the associativity proof.

## Invariants of curves, Whitney formulas, Milnor’s mu-invariants

• I also specialize in Arnold’s theory of plane curves and its extension to fronts and curves on surfaces. In particular, I obtained new formulas for Arnold’s invariants and constructed a number of invariants of higher degrees.
• Another direction of my research related to plane curves deals with the classical Whitney formula. I established various splittings and generalizations of this formula to higher dimensions, curves on surfaces, etc. The latest paper has been just submitted for publication.
I remain one of the actively working specialists in the study of Milnor’s mu-invariants of string links, with a number of papers on the subject. These invariants are related to a variety of other topics: Massey products, Alexander polynomial, theory of link-homotopy, etc. The latest work in progress is on an interpretation of mu-invariants via moduli of maps of binary rooted trees and dialgebra operad.