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[Parity, Vassiliev Invariants, Framed Chord diagrams, Legendrian Knots, and Flat-Virtual Knots ] Â. Î. Ìàíòóðîâ |
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Àííîòàöèÿ: In the present talk, I will mention the following two topics: 1) How to get parity for classical knots. For a classical knot K we take its double-cabling L_{2}(K)= K_{1}\sqcup K_{2} and consider the knot K_{2} lying in the complement to K_{1}. The space R^{3}\backslash K_{1} has non-trivial homology, hence the theory of Vassiliev invariants for K_{1} in the complement to K_{2} has some features of “parity”; we formulate many problems concerning framed chord diagram, framed Vassiliev invariants, Kontsevich integral, etc. 2) In 2022, in two joint papers with I.M.Nikonov, we constructed a map from classical knot theory in the full torus S_{1}\times R^{2} to the so-called “flat-virtual knot theory” which has many “virtual features.” Plane curves and fronts naturally lift to Legendrian knots in the spherized bundle S_{*} T R^{2} which is topologically a torus. This leads to a nice interplay between the theory of Legendrian knots, fronts, virtual knots, and flat-virtual knots. ßçûê äîêëàäà: àíãëèéñêèé Website: https://us02web.zoom.us/j/81866745751?pwd=bEFqUUlZM1hVV0tvN0xWdXRsV2pnQT09 |
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