Download Differential and Symplectic Topology of Knots and Curves by S. Tadachnikoz, Serge Tabachnikov PDF

By S. Tadachnikoz, Serge Tabachnikov

This booklet offers a set of papers on comparable issues: topology of knots and knot-like gadgets (such as curves on surfaces) and topology of Legendrian knots and hyperlinks in three-d touch manifolds. Featured is the paintings of foreign specialists in knot concept ("quantum" knot invariants, knot invariants of finite type), in symplectic and make contact with topology, and in singularity concept. The interaction of various tools from those fields makes this quantity distinctive within the research of Legendrian knots and knot-like items resembling wave fronts. a very engaging characteristic of the quantity is its overseas importance. the amount effectively embodies a great collaborative attempt by means of around the globe specialists from Belgium, France, Germany, Israel, Japan, Poland, Russia, Sweden, the U.K., and the U.S.

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Extra info for Differential and Symplectic Topology of Knots and Curves

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We are going to exploit this independence and group together the terms which correspond to one and the same generator of such type so that their individual divergences kill one another. The grouping is mainly based on the following example. 4. Consider the family of all possible pairings on K U KE which have only two pairs involving points from a neighbourhood of some local maximum of the function t on K (see Figure 10), with the upper pair being dangerous and the lower one not. VASSILIEV INVARIANTS OF KNOTS IN lR 3 AND IN A SOLID TORUS FIGURE 49 10.

In Section 5, we obtain the similar result for framed knots in a solid torus. We also show that all the coefficients of the version of the HOMFLY polynomial for framed knots in a solid torus are in fact Vassiliev invariants of finite order (see [7]). 1. The paper [10] establishes the isomorphism between the theory of Vassiliev invariants for framed knots in a solid torus and that for regular plane curves with no direct self-tangencies. 2. The constructions in the present paper are very convenient to construct spectral sequences (similar to Vassiliev's one in [17]) to calculate cohomology of spaces of framed knots in IR3 and unframed and framed knots in a solid torus.

L of the front is the algebraic number of cusps. , the rotation number of the co-orienting covector. It follows from [Arl] that these two integers are the only invariants of Legendre cobordism in ST*]R2, and it follows from the preceding section that they are also the only invariants of J+ -cobordisms. Denote by la the Legendre submanifold obtained from I by antipody in the fibers of ST*]R2 ---+ ]R2. The wave front of la coincides with the one of I, but has the opposite co-orientation. By a J- -singularity, we mean a transverse intersection between I and la.

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