Description of inverse polynomial images which consist of two Jordan arcs with the help of Jacobi's elliptic functions

First it is shown that four given points from the complex plane are the endpoints of two Jordan arcs representable as the inverse image of [-1,1] under a polynomial mapping if and only if the four endpoints have a certain representation in terms of Jacobi's elliptic functions. The polynomial which generates the two Jordan arcs is given explicitly in terms of Jacobi's theta functions. Then the main emphasis is put on the case where the two arcs are symmetric with respect to the real line. For instance it is demonstrated that the endpoints vary monotone with respect to the modulus k of the associated elliptic functions.