We describe a scheme of constructing classical integrable models in 2 + 1-dimensional discrete space-time, based on the functional tetrahedron equation-equation that makes manifest the symmetries of a model in local form. We construct a very general "block-matrix model" together with its algebro-geometric solutions, study its various particular cases, and also present a remarkably simple scheme of quantization for one of those cases.

Tetrahedral Zamolodchikov algebras are structures that occupy an intermediate place between the solutions of the Yang-Baxter equation and its generalization onto 3-dimensional mathematical physics -the tetrahedron equation. These algebras produce solutions to the tetrahedron equation and, besides, specific "two-layer" solutions to the Yang-Baxter equation. Here the tetrahedral Zamolodchikov algebras are studied that arise from L-operators of the free-fermion case of Baxter's eight-vertex model.

An invariant of three-dimensional orientable manifolds is built on the base of a solution of pentagon equation expressed in terms of metric characteristics of Euclidean tetrahedra.

A parameterization of Grassmann-algebraic relations corresponding to the Pachner move 3-3 is proposed. In these relations, each 4-simplex is assigned a Grassmann weight depending on five anticommuting variables associated with its 3-faces. The weights are chosen to have the "simplest" form -a Grassmann-Gaussian exponent or its analogue (satisfying a similar system of differential equations). Our parameterization works for a Zariski open set of such relations, looks relevant from the algebraictopological viewpoint, and reveals intriguing nonlinear relations between objects associated with simplices of different dimensions.

Abstract. We present an invariant of a three-dimensional manifold with a framed knot in it based on the Reidemeister torsion of an acyclic complex of Euclidean geometric origin. To show its nontriviality, we calculate the invariant for some framed (un)knots in lens spaces. Our invariant is related to a finite-dimensional fermionic topological quantum field theory.