ИСТИНА

In low-dimensional topology and graph theory framed chord diagrams are very useful. For example, with the help of them criteria of embedding of four-valent graphs with a cross structure can be formulated. Framed chord diagrams are used in the theory of J-invariants of plane curves and for describing the combinatorics of a non-generic Legendrian knots in some 3-manifolds. The most famous face of (non-framed) chord diagrams is the role they play in the chord diagram algebra. The weight systems (i.e., linear functions on this algebra), due to Vassiliev{Kontsevich theorem, lead to Vassiliev invariants of knots. The multiplication in the chord diagram algebra is dened by taking a connected sum of two chord diagrams. This operation is well dened up to 4T-relation. In the case of framed chord diagrams one can dene 4T-relations and consider a connected sum of two framed diagrams. An attempt to prove that this operation is well dened up to the 4T-relations fails. In our talk we shall give all necessary denitions and construct an additional object, a map from the module of framed chord diagrams to the new object and an invariant. Using this map and the invariant we shall present an example of chord diagrams having different connected sums modulo framed 4T-relations.