Laboratory for Computer Geometry and Quantum Topology

Scientific results:

1. We have found a close, earlier unknown, connection between Dijkgraaf–Witten invariants and Arf invariants of bilinear forms that correspond to special quadratic forms on first groups of cohomologies of reviewed manifolds. 
2. A method has been developed for computing Dijkgraaf–Witten invariants over a field of order 2 for oriented Seifert manifolds  with oriented bases. 
3. Using a specially built family of Turaev–Viro invariants we found new precise values of the complexity for several infinite series of 3-manifolds. 
4. Our researchers have classified and tabulated all the knots in a thickened torus and a thickened Klein bottle of low complexity. This classification became possible thanks to the discovery of new quantum invariants of the Kauffman bracket type for knots on thickened surfaces by employees of the Laboratory. 
5. A method has been developed for building new topological SU-invariants of virtual knots that are analogs of the HOMFLY polynomial of classical knots. 
6. By explicitly computing the values of the Casson–Walker–Lescop invariant, we obtained a complete classification of an infinite family of 3-manifolds produced as a result of rational rearrangements of a three-dimensional sphere along several special-type entanglements. 
7. We have found several new families of Yang–Baxter operator on a two-dimensional vector space, built invariants of classical knots and entanglements corresponding to them. 
8. Our researchers have found a link between knotoids on a two-dimensional sphere and knots in thickened surfaces. Using it we managed to obtain a complete classification of primary knotoids, whose diagram complexity does not exceed five. 
9. Our researchers have shown that the diagram complexity of each knotoid is not less than double its height. This allowed to obtain new lower estimates of the diagram complexity of knotoids that turn out to be precise. 
10. We have developed and implemented an approach to building polynomial invariants for knots and entanglements in thickened non-oriented surface. This approach allows to  build invariants for knots in non-oriented manifolds as well. 
11. The Laboratory has developed a new family of quantum invariants for 3-manifolds.

Implemented results of research:

• The Laboratory has created and registered the «Atlas of 3-manifolds», which is a database of 3-manifolds and their characteristics (complexities, values of invariants, types of geometries, graphic structures, common names).

• We have created and registered the software program «One-vertex spines of 3-manifolds» designed for building two-dimensional polyhedrons of a special type (one-vertex spines) that generate 3-manifolds.

• We have created and registered the software program «HOMFLY calculator» designed for computing values of the classical HOMFLY polynomial of rational knots. 

• The Laboratory has developed 14 education programs and implemented them into the education process: «Additional chapters of topology (part I)», «Classification of 3-manifolds», «Classification of Seifert manifolds and computing their quantum  invariants», «Low-dimensional topology», «Quantum topology», «Computational topology», «Graph theory», «Theory of homologies», «Theory of the complexity of geometric objects», «Knot theory», «Algebraic topology», «Algorithms on graphs», «Quantum and polynomial invariants of knots», «One-vortex spine theory».

• 11 Candidate of Sciences dissertations have been prepared and defended.

• 8 scientific conferences in our research domain have been organized.

• 7 members of our academic team have been admitted to postgraduate and doctoral schools. 

Organizational and structural changes:

In 2017, the Research Laboratory of Quantum Topology was merged with the Laboratory of Computer Geometry to form the new Research and Education Laboratory of Computer Geometry and Quantum Topology. The new laboratory continues researching quantum topology and computer geometry with support from the Russian Science Foundation and the Russian Foundation for Basic Research. 

Collaborations: 

Indiana University (USA), University of Bologna (Italy), Georgia Institute of Technology (USA): joint research and publications.