The authors construct the quantum \(\check R\) matrices for the quantum groups of type \(E_ 7\) and \(F_ 4\), respectively. They compute explicitly the quantum \(\check R\) matrix for the \(E_ 7\) group in the 56-dimensional fundamental representation and that for the \(F_ 4\) group in the 26-dimensional representation. Also, they obtained the Skein relations, the Markovian move invariant link polynomials and the spectral parameter-dependent solutions of the Yang-Baxter equation for these quantum \(\check R\) matrices. It is interesting that the authors observe that the \(\check R\) matrix for the matrix elements related by the Weyl reflection has different values.
This work is the continuation of papers by Jimbo, Reshetikhin and the authors about constructing the quantum \(\check R\) matrix for the quantum group.