In [13], R.M. Kashaev defined a family of complex-valued link invariants indexed by integers N>~2 using the quantum dilogarithm. Later he calculated the asymptotic behavior of his invariant, and observed that for the three simplest hyperbolic knots it grows as exp(Vol(K)N/2rr) when N goes to infinity, where Vol(K) is the hyperbolic volume of the complement of a knot K [14]. This amazing result and his conjecture that the same also holds for any hyperbolic knot have been almost ignored by mathematicians since his definition of the invariant is too complicated (though it uses only elementary tools).The aim of this paper is to reveal his mysterious definition and to show that his invariant is nothing but a specialization of the colored Jones polynomial. The colored Jones polynomial is defined for colored links (each component is decorated with an irreducible representation of the Lie algebra sl(2, C)). The original Jones polynomial corresponds to the case that all the colors are identical to the 2-dimensional fundamental representation.We show that Kashaev's invariant with parameter N coincides with the colored Jones polynomial in a certain normalization with every color the N-dimensional representation, evaluated at the primitive Nth root of unity. (We have to normalize the colored Jones polynomial so that the value for the trivial knot is one, for otherwise it always vanishes.) On the other hand, there are other colored polynomial invariants, such as the generalized multivariable Alexander polynomial defined by Y. Akutsu, T. Deguehi and T. Ohtsuki [1]. They used the same Lie algebra sl(2, C) but a different hierarchy of representations. Their invariants are parameterized by c+l parameters: an integer N This research is supported in part by Sumitomo Foundation and Grand-in-Aid for Scientific Research, The Ministry of Education, Science, Sports and Culture. 86 H. MURAKAMI AND J. MURAKAMI and complex numbers pi (i= 1, 2, ..., c) decorating the components, where c is the number of components of the link. In the case where N=2, their invariant coincides with the multivariable Alexander polynomial, and their definition is the same as that of the second author [22]. Using the Akutsu-Deguchi-Ohtsuki invariants we have another coincidence. We will show that if all the colors are 89 then the generalized Alexander polynomial is the same as Kashaev's invariant since it coincides with the specialization of the colored Jones polynomial as stated above. Therefore the set of colored Jones polynomials and the set of generalized Alexander polynomials of Akutsu-Deguchi-Ohtsuki intersect at Kashaev's invariants. The paper is organized as follows. In the first section we recall the definition of the link invariant defined by Yang-Baxter operators. In w we show that the AkutsuDeguchi-Ohtsuki invariant coincides with the colored Jones polynomial when the colors 1 (N-1) by showing that their representation becomes the usual representation are all corresponding to the irreducible N-dimensional representation of sl(2, C). In w weshow that ...

In order to improve the accuracy of the in-plane response of the shear, deformable laminated composite plate theory, a new laminated plate theory has been developed based upon a new variational principle proposed by Reissner (1984). The improvement is achieved by including a zigzag-shaped C0 function to approximate the thickness variation of in-plane displacements. The accuracy of this theory is examined by applying it to a problem of cylindrical bending of laminated plates which has been solved exactly by Pagano (1969). The comparison of the in-plane response with the exact solutions for symmetric three-ply and five-ply layers has demonstrated that the new theory predicts the in-plane response very accurately even for small span-to-depth ratios.

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In order to improve the accuracy of in-plane responses of shear deformable composite plate theories, a new laminated plate theory was developed for arbitrary laminate configurations based upon Reissner’s (1984) new mixed variational principle. To this end, across each individual layer, piecewise linear continuous displacements and quadratic transverse shear stress distributions were assumed. The accuracy of the present theory was examined by applying it to the cylindrical bending problem of laminated plates which had been solved exactly by Pagano (1969). A comparison with the exact solutions obtained for symmetric, antisymmetric, and arbitrary laminates indicates that the present theory accurately estimates in-plane responses, even for small span-to-thickness ratios.