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We prove that certain polynomials previously introduced by the author can beidentified with tau functions of Painlev\'e VI, obtained from one of Picard'salgebraic solutions by acting with a four-dimensional lattice of B\"acklundtransformations. For particular lines in the lattice, this proves conjecturesof Bazhanov and Mangazeev. As applications, we describe the behaviour of thecorresponding solutions near the singular points of Painlev\'e VI, and obtainseveral new properties of our polynomials.

Special polynomials related to the supersymmetric eight-vertex model. III. Painlev\'e VI equation