By V. A. Vassiliev (auth.), John M. Bryden (eds.)

This quantity is the convention court cases of the NATO ARW in the course of August 2001 at Kananaskis Village, Canada on "New concepts in Topological Quantum box Theory". This convention introduced jointly experts from a couple of varied fields all with regards to Topological Quantum box conception. The subject of this convention was once to try to discover new equipment in quantum topology from the interplay with experts in those different fields.

The featured articles contain papers through V. Vassiliev on combinatorial formulation for cohomology of areas of Knots, the computation of Ohtsuki sequence by way of N. Jacoby and R. Lawrence, and a paper via M. Asaeda and J. Przytycki at the torsion conjecture for Khovanov homology by way of Shumakovitch. in addition, there are articles on extra classical issues concerning manifolds and braid teams through such popular authors as D. Rolfsen, H. Zieschang and F. Cohen.

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**Example text**

Obviously the same is true for the whole auxiliary spectral sequence. The above description of the cohomology of complexes of connected graphs allows one to describe easily the Vassiliev spectral sequence in the homological case. 10). The proofs in full detail are given in [39], (see also the Russian version [40]). 3. The paper is organized as follows. In Sections 1, 2, 3 we give some preliminaries on linear graded operads. We give a short deﬁnition and examples (that will be useful for us) of linear graded operads (Section 1).

These n mappings are supposed 1) to be injective, preserving the orientation and the ratio of distances; 2) to have disjoint images. The space T Bd (n) can be evidently identiﬁed with the direct product of the n-th power (SO(d))n of the special orthogonal group with the conﬁguration space of n disjoint balls in the unit ball B d . The last space is homotopy equivalent to the n-th component LC d (n) of the little cubes operad. The composition operations, the symmetric group actions and the unit element are deﬁned analogously to the case of the little cubes operad.

As in our case the corresponding graph-complex depends only on the parity of the dimension d. In fact these graph-complexes are quasi-isomorphic to the complexes dual to the complexes of bracket star-diagrams (a description of the complexes dual to BSD is given in [39, 40]) as diﬀerential Hopf algebras. 3. The second approach is to study the spaces of knots (or more generally spaces of embeddings of arbitrary varieties) by means of the calculus of analitic functors developped by T. Goodwillie, cf.