By S. Suzuki

This is often the lawsuits of a world workshop on knot idea held in July 1996 at Waseda college convention Centre. It used to be equipped through the foreign examine Institute of Mathematical Society of Japan. The workshop used to be attended via approximately a hundred and eighty mathematicians from Japan and 14 different international locations. such a lot of them have been experts in knot conception. the amount includes forty three papers, which take care of major present examine in knot conception, low-dimensional topology and comparable subject matters. the amount contains papers via the subsequent invited audio system: G. Burde, T. Deguchi, R. Fenn, L.H. Kauffman, J. Levine, J.M. Montesinos(-A), H.R. Morton, ok. Murasugi, T. Soma and D.W. Sumners.

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QED. 2. Let B and T be two quadrilinear mappings with the properties (a), (b) and (c). If B(vi,v2,vi,v2) = T(v1,v2,v1,v2) for all vi, v2 a V, then B = T. Proof. We may assume that T = 0; consider B - T and 0 instead of B and T. We asses therefore that B(vi,v2,vi,v2) = 0 for all vi, v2 of V. B(vi,v2+v4,vl,v2+v4) = B(vl,v2,vl,v4). 0 F'om this we obtain 0 = B(v1+v3,v2,vl+v3,v4) = B(vl,v2,v3,v4) + B(v3,v2,v1,v4). Now, by applying (d) and then (b), we have 0 = B(v1,v2,v3,v4) + B(vi,v4,v3,v2) = B(v1,v2,v3,v4) - B(vl,v4,v2,v3).

Tangnet bundle: Let GL(n;R) act on Rn as above. The tangent bundle T(M) over M is the bundle associated with L(M) with standard fibre R. The fibre of T(M) over x E M may be considered as Tx(M). 3. Tensor bundle: Let Tr be the tensor space of type (r, s) over the vector space Rn. The group GL(n;R) can be regarded as a group of linear transformations of T. With this standard fibre TS, we obtain the tensor bundle Tr(M) of type (r,s) over M which is associated with L(M). It is easy to see that the fibre of TS(M) over x E M may be considered as the tensor space over TX(M) of type (r,s).

Let (x,y,z,w) be a Cartesian coordinate system in R4. We consider the hypersurface M defined by w = (x2z - y2z - 2xy)/2(z2 + 1) or 2z2w - x2z + y2z + 2w + 2xy = 0, which satisfies the non-linear partial differential equation 56 wX - wy + 2w = 0. Then M is a desired manifold. For Riemannian manifolds satisfying the condition (*) see Sekigawa [ ]. D. EQUATION OF 1T1AURER-CARTAN: A differential form w on a Lie group G is called left invariant if (La)*w = w for every a e G. The vector space 7* formed by all left invariant 1-forms is the dual space of the Lie algebra "T: if A E °l and w e 7*, then the function w(A) is constant on G.