Abstract:
The article discusses and expands the known elements of the eight-vertex model, paying special attention to
the parameterization of the matrix. The matrix values are interconnected with the knot through the braids and this
model is valid on finite square lattices in two-dimensional space. A new solution of the parametrized eight-vertex
model of free fermions with a complex version of elliptic functions, which is valid on a finite lattice, will be
constructed. The range of applicability of the eight-vertex model with elements of the Jacobi elliptic function and
the construction of a knot invariant on its basis is discussed by comparing the results obtained analytically for the
model. The construction of the knot invariant using the Clebsch-Gordan coefficients and the main tool of
statistical mechanics of the Yang-Baxter equation will be studied in detail
