Special polynomials related to the supersymmetric eight-vertex model. III. Painlevé VI equation

We prove that certain polynomials previously introduced by the author can be identified with tau functions of Painleve VI, obtained from one of Picard's algebraic solutions by acting with a four-dimensional lattice of Bäcklund transformations. For particular lines in the lattice, this proves conjectures of Bazhanov and Mangazeev. As applications, we describe the behaviour of the corresponding solutions near the singular points of Painleve VI, and obtain several new properties of our polynomials