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In this short note, we construct solutions to quantum tetrahedron equation of the kind "with variables on the edges". Each of these variables takes just two values, called sometimes "colors". We propose two different constructions. The first of them involves, in particular, two $\mathcal R$-operators each depending on one parameter, while these parameters are independent from each other; the number of nonvanishing matrix entries in these two $\mathcal R$-operators is either 12 or 14. In the second construction, we use what may be called "modified tetrahedron in a direct sum" to produce, again, quantum tetrahedron solutions. All matrix entries of our $\mathcal R$-operators are nonnegative, if the relevant parameters are chosen properly., Comment: 9 pages; v2: one more construction added

A new version of the self-similarity spin transform on three-dimensional cubic lattices is proposed that makes possible calculation of nontrivial spin correlations in a "combinatorial" model, in which all permitted spin configurations have equal probabilities., Comment: 27 pages, 4 tikz figures

A family of heptagon relations -- algebraic imitations of five-dimensional Pachner moves -- is constructed, parameterized by simplicial 3-cocycles. These relations are shown to possess a nontrivial "heptagon cohomology", which yields invariants of a pair "manifold, its 3rd cohomology class"., Comment: 23 pages

Cubic blocks are studied assembled from linear operators $\mathcal R$ acting in the tensor product of $d$ linear "spin" spaces. Such operator is associated with a linear transformation $A$ in a vector space over a field $F$ of a finite characteristic $p$, like "permutation-type" operators studied by Hietarinta. One small difference is that we do not require $A$ and, consequently, $\mathcal R$ to be invertible; more importantly, no relations on $\mathcal R$ are required of the type of Yang--Baxter or its higher analogues. It is shown that, in $d=3$ dimensions, a $p^n\times p^n\times p^n$ block decomposes into the tensor product of operators similar to the initial $\mathcal R$. One generalization of this involves commutative algebras over $F$ and allows to obtain, in particular, results about spin configurations determined by a four-dimensional $\mathcal R$. Another generalization deals with introducing Boltzmann weights for spin configurations; it turns out that there exists a non-trivial self-similarity involving Boltzmann weights as well., Comment: 33 pages, 10 figures. v3: conjecture about self-similarity in any finite characteristic is now proven

A cohomology theory for "odd polygon" relations -- algebraic imitations of Pachner moves in dimensions 3, 5, ... -- is constructed. Manifold invariants based on polygon relations and nontrivial polygon cocycles are proposed. Example calculation results are presented., Comment: 32 pages. v2: proof of Proposition 13 made more accurate; new calculation results added to Table 2. This article supersedes arXiv:2110.08780

A cohomology theory is proposed for the recently discovered heptagon relation -- an algebraic imitation of a 5-dimensional Pachner move 4--3. In particular, `quadratic cohomology' is introduced, and it is shown that it is quite nontrivial, and even more so if compare heptagon with either its higher analogues, such as enneagon or hendecagon, or its lower analogue, pentagon. Explicit expressions for the nontrivial quadratic heptagon cocycles are found in dimensions 4 and 5., Comment: 15 pages, 2 figures

We construct a family of heptagon relations -- algebraic imitations of five-dimensional Pachner move 3--4, parameterized by simplicial 3-cocycles., Comment: 11 pages

An ansatz is proposed for heptagon relation, that is, algebraic imitation of five-dimensional Pachner move 4--3. Our relation is realized in terms of matrices acting in a direct sum of one-dimensional linear spaces corresponding to 4-faces., Comment: 16 pages, 5 figures. v3: Section 5 significantly expanded and structured

Hexagon relations are algebraic realizations of four-dimensional Pachner moves. `Constant' -- not depending on a 4-simplex in a triangulation of a 4-manifold -- hexagon relations are proposed, and their polynomial-valued cohomology is constructed. This cohomology yields polynomial mappings defined on the so called `coloring homology space', and these mappings can, in their turn, yield piecewise linear manifold invariants. These mappings are calculated explicitly for some examples. It is also shown that `constant' hexagon relations can be obtained as a limit case of already known `nonconstant' relations, and the way of taking the limit is not unique. This non-uniqueness suggests the existence of an additional structure on the `constant' coloring homology space., Comment: 22 pages

A construction of hexagon relations - algebraic realizations of four-dimensional Pachner moves - is proposed. It goes in terms of "permitted colorings" of 3-faces of pentachora (4-simplices), and its main feature is that the set of permitted colorings is nonconstant - varies from pentachoron to pentachoron. Further, a cohomology theory is formulated for these hexagon relations, and its nontriviality is demonstrated on explicit examples., Comment: 26 pages; v2: improvements in Section 6, expression for 4-cocycle corrected

Hexagon relations are algebraic realizations of four-dimensional Pachner moves, and there are hexagon relations admitting nontrivial cohomologies and leading thus to piecewise linear (PL) 4-manifold invariants. We show that some - but not all! - of the known nontrivial cohomologies can be obtained from a single integral bilinear form corresponding to a PL 4-manifold by using a Frobenius homomorphism for a half of `color' variables (or different Frobenius homomorphisms for both halves). This form can be regarded as a sophisticated analogue of the manifold's intersection form., Comment: 9 pages

Hexagon relations are combinatorial or algebraic realizations of four-dimensional Pachner moves. We introduce some simple set-theoretic hexagon relations and then `quantize' them using what we call `polynomial hexagon cohomologies'. Based on this, topological quantum field theories are proposed with polynomial `discrete Lagrangian densities' taking values in finite fields. First calculations of the resulting manifold invariants, arising from polynomial cocycles of degree three and in characteristic two, show their nontriviality., Comment: 25 pages. v4: calculations for some twisted tori added, and other improvements

Gaussian pentachoron weights can be used for constructing algebraic realizations of four-dimensional Pachner moves. Here, we consider a natural `gauge equivalence' for such weights with one and two bosonic - i.e., commuting - variables on 3-faces. For the one-boson case, all generic weights turn out to be gauge equivalent. For the two-boson case, and generic weights, their gauge equivalence classes are parameterized by multiplicative 2-cocycles. Moreover, a generic two-boson weight can be reduced by a gauge transformation to a delta-function form., Comment: 26 pages. v2: a proof of a Pachner 3--3 relation added

This is the second in a series of papers where we construct an invariant of a four-dimensional piecewise linear manifold $M$ with a given middle cohomology class $h\in H^2(M,\mathbb C)$. This invariant is the square root of the torsion of unusual chain complex introduced in Part I (arXiv:1605.06498) of our work, multiplied by a correcting factor. Here we find this factor by studying the behavior of our construction under all four-dimensional Pachner moves, and show that it can be represented in a multiplicative form: a product of same-type multipliers over all 2-faces, multiplied by a product of same-type multipliers over all pentachora., Comment: 20 pages. v2: two of three Experimental Results now proven, and many small improvements

Recently, an algebraic realization of the four-dimensional Pachner move 3--3 was found in terms of Grassmann--Gaussian exponentials, and a remarkable nonlinear parameterization for it, going in terms of a $\mathbb C$-valued 2-cocycle. Here we define, for a given triangulated four-dimensional manifold and a 2-cocycle on it, an `exotic' chain complex intimately related to the mentioned parameterization, thus providing a basis for algebraic realizations of all four-dimensional Pachner moves., Comment: 23 pages. v2: new and better `gauge' for operators proposed, and many small improvements

Recently, a family of fermionic relations were discovered corresponding to Pachner move 3-3 and parameterized by complex-valued 2-cocycles, where the weight of a pentachoron (4-simplex) is a Grassmann-Gaussian exponent. Here, the proportionality coefficient between Berezin integrals in the l.h.s. and r.h.s. of such relations is written in a form multiplicative over simplices., Comment: 19 pages, no figures