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AffiliationFederal State Budgetary Educational Institution of Higher Education "Chelyabinsk State University",

Research area 01 - MATHEMATICS, INFORMATICS, AND SYSTEM SCIENCES, 01-105 - Topology

Keywords3-manifold, knot, invariant, topological quantum field theory, Kauffman bracket polynomial, thickened surface

Annotation
The project is devoted to achievement a significant progress in one of the central problems in low-dimensional topology - the classification of 3-dimensional manifolds and knots. It is proposed to construct new invariants both for 3-manifolds and for knots. It is planned to solve two tasks. The first task (task A) is to develop the theory of quantum invariants of 3-manifolds. It includes the categorization of Dijkgraaf-Witten invariants for 3-manifolds (task A.1), and studying of the properties of Turaev - Reshetikhin and Turaev - Viro type invariants arising from fusion categories of small rank (task A.2). The second task (task B) is to construct new invariants for knots and links in 3-manifolds. It includes the construction of polynomial invariants, close to the affine index polynomial for knots in the non-orientable thickening of a non-orientable surface (task B.1), and the construction of invariants of classical links valued in the ring of Laurent polynomials over the skew field of quaternions (task B.2). It is expected that the solution of the listed tasks will make it possible to obtain a significant progress in the topology of 3-manifolds and in the knot theory.

Expected results
Development of the theory of quantum invariants of 3-manifolds. A new class of so-called symmetric multidimensional Dijkgraaf-Witten type invariants will be constructed. In addition, explicit combinatorial formulae will be obtained for invariants of the Turaev-Reshetikhin type and the Turaev-Viro type arising from fusion categories of small rank. Development of knot theory. A new invariant for knots and links in the non-orientable thickenings of a non-orientable surface will be constructed. This invariant is close to the affine index polynomial for virtual knots. A new invariant for classical links in a 3-sphere will be constructed. This invariant is a generalization of the classical Kauffman bracket polynomial which takes values in the ring of Laurent polynomials over the skew field of quaternions. Our research areas closely interact with the most active research in the fields of 3-manifolds topology and the knot theory in the world. In these areas, our results will be in the same level as leading ones. They may be included in courses taught to students of the mathematical departments

Annotation of the results obtained in 2023
The project aims at solving two complex tasks. The first task is to develop the theory of quantum invariants of three-dimensional manifolds, the second task is related to the construction and study of the properties of new invariants of knots and links both in the 3-sphere and in other 3-manifolds. Results obtained in the solution of the first task: 1. A family of symmetric invariants of Dijkgraaff-Witten type for 3-manifolds is constructed. Each such invariant is given by choosing a finite group and a symmetric 3-cocycle for this group. A symmetric 3-cocycles differ from a classical 3-cocycles of finite groups in that they satisfy natural symmetry conditions. The value of the invariant for a 3-manifold is the multiset of values obtained by the product of the values of the chosen 3-cocycle for each real vertex of a standard spine of the 3-manifold. In the case of closed 3-manifolds, the elements of the multiset are characterised by a representations from the fundamental group of the manifold to the chosen finite group. Pairwise non-cohomologous symmetric 3-cocycles are found for all finite cyclic groups up to order 7. 2. We obtained combinatorial formuli which define invariants of the Turaev-Reshetikhin type corresponding to fusion categories of rank two. Among them there are only two non-trivial invariants. In particular, an explicit description of the structural morphisms defining the considered categories was obtained. It was proved that the number of distinct values of these invariants for lens spaces is finite. Results obtained in solving the second task: 3. The construction of an analogue of the Kauffman bracket polynomial for pseudoclassical knots in non-orientable thickenings of a non-orientable surfaces is described. 4. A generalisation of the Goldman-Turaev bialgebra to the space of pseudoclassical curves on a non-orientable surface is described. In particular, an analogue of the Turaev coproduct is constructed for this case. From this, homotopy, homological and polynomial invariants of pseudoclassical knots in a non-orientable thickening of a non-orientable surface are constructed. The aforementioned polynomial invariant is an analogue of the affine index polynomial for the case of pseudoclassical knots.

Publications