Intelligence of Low Dimensional Topology 2006 (Series on by Editors: J. S. Carter, S. Kamada, L. H. Kauffman, A.

By Editors: J. S. Carter, S. Kamada, L. H. Kauffman, A. Kawauchi, and T. Kohno

This quantity gathers the contributions from the overseas convention "Intelligence of Low Dimensional Topology 2006," which came about in Hiroshima in 2006. the purpose of this quantity is to advertise examine in low dimensional topology with the focal point on knot idea and similar subject matters. The papers comprise complete studies and a few most modern effects.

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Intelligence of Low Dimensional Topology 2006 (Series on Knots and Everything)

This quantity gathers the contributions from the foreign convention "Intelligence of Low Dimensional Topology 2006," which happened in Hiroshima in 2006. the purpose of this quantity is to advertise learn in low dimensional topology with the focal point on knot conception and similar themes. The papers contain finished stories and a few newest effects.

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1 A similar equivariant Khovanov homology can be defined for links in the solid torus. Actually, this equivaraint Khovanov homology is an invariant of links in the solid torus. References 1. M. M. Asaeda, J. H. Przytycki, A. S. Sikora Categorification of the Kauffman bracket skein module of I−bundles over surfaces. Algebraic and Geometric Topology 4, 2004, 1177-1210. 2. G. E Bredon, Introduction to compact transformation groups. Acdemic Press (1972) 3. V. F. R. Jones, A polynomial invariant for knots via von Neumann algebras.

E. τ (S) = {cirles with + sign} − { circles with − sign }. Let i and j be two integers. We define C i (D) to be the free abelian group generated by all enhanced states with i(S) = i. Let C i,j (D) be the subgroup of C i (D) generated by enhanced states with j(S) = j. The Khovanov differential is defined by: di,j : C i,j (D) −→ C i+1,j (D) S −→ (−1)t(S,S ) (S : S )S All states S’ where (S : S ) is • 1 if S and S differ exactly at one crossing, call it v, where S has a +1 marker, S has a −1 marker, all the common circles in DS and DS have the same signs and around v, S and S are as in figure 2, • (S : S ) is zero otherwise and t(S, S ) is the number of −1 markers assigned to crossings in S labelled greater than v.

Here, Ck may be 0. For example, 1849/10044 = [6, 2, 4, −6, −2, −6, 4] is modified to 1 + [−1, 4, −1, 0, −1, 2, −1, −6, +1, 0, +1, −4, +1, 4, −1]. Note that j Cj = B(L) − 1 + j (|Cj | − 1). According to such a modified continued fraction, we have a Conway diagram for L. See Figure 3 (left) depicting the example above. Each horizontal half twist corresponds to an inserted ±1, and the vertical twistings correspond to the entries Cj . The diagram carries a checker board Seifert surface F for L, which is of minimal genus.

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