ИСТИНА

In the present talk we aim to relate various objects in topology, algebra, and combinatorics. We deal with braids which act on knots. The algebraic construction closely connected with this concept is the \Gamma_n^k groups which stem from Delaunay triangulations of surfaces (and similar partitionings of higher- dimensional spaces). Along the lines of tiling a surface with triangles, one may consider the {\em domino tilings} — breaking the surfaces into $2 \times 1$ quadrilateral (or, more generally, into $k \times m$ quadrilaterals). Such tilings may undergo certain transformations (closely related to Ptolemy flips, Matveev-Piergalliny moves, Pachner moves and other well-known geometric transformations) and that naturally leads to the definition of new interesting groups generated by those moves. We shall discuss the connections between the domino tilings theory with the Legendrian knots theory and with Thompson groups (which are known to be closely related to knots and their diagrams). The talk shall also contain an overview of currently unsolved problems and research ideas in the field.