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

This quantity is the convention lawsuits of the NATO ARW in the course of August 2001 at Kananaskis Village, Canada on "New thoughts in Topological Quantum box Theory". This convention introduced jointly experts from a few assorted fields all regarding Topological Quantum box thought. The subject matter of this convention used to be to aim to discover new equipment in quantum topology from the interplay with experts in those different fields.

The featured articles contain papers by means of 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 by means of M. Asaeda and J. Przytycki at the torsion conjecture for Khovanov homology by way of Shumakovitch. furthermore, there are articles on extra classical subject matters regarding manifolds and braid teams by means of such popular authors as D. Rolfsen, H. Zieschang and F. Cohen.

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**Example text**

6. Note that the parity of a (generalized) bracket star-diagram is the number of stars plus the number of the exterior product signs. 2). Let A and B be two generalized bracket star-diagrams, such that A can be inserted (∗) in the point t0 of B. Let us deﬁne B|x (∗) =A . To do this we replace xt(∗) in B by A, t 0 0 ˜ an we multiply the obtained expression by (−1)(A− 0 )×(n1 +n2 ) , where 0 is equal to zero (resp. to one) if the point t0 has no star (resp. if the point t∗0 has a star); n1 (resp.

The ﬁltration F0 ⊃ F1 ⊃ F2 . . , with Fi := j≥i Ej , is compatible with the diﬀerential ∂. 4. 5. [39] The Hochschild complex (BV (d−1) , ∂) (d is even) is a direct sum of the complexes (E0 , ∂) and (F1 , ∂). The ﬁrst complex (E0 , ∂) is homology (and even homotopy) equivalent to (BV (d−1) , ∂) ; the second one (F1 , ∂) is acyclic (and even contractible). An analogous statement holds for the complexes (POISS (d−1) , ∂), d being any integer number. 11) corresponding to partitions non-containing singletons will be called the normalized Hochschild orm complexes and denoted by (POISS N (d−1) , ∂).

A#A , and the derivative φ is equal to 0 at all the b last points of this conﬁguration. For any (A, b)-conﬁguration the set of maps respecting it is an aﬃne subspace in K of codimension d(|A| − #A + b); the number |A| − #A + b is called the complexity of the conﬁguration. Two (A, b)-conﬁgurations are called equivalent if they can be transformed into one another by an orientation-preserving homeomorphism R1 → R1 . Consider any (A, b)-conﬁguration J of complexity i and with j geometrically distinct points in R1 .