Entwining tetrahedron maps

TL;DR

This paper introduces three distinct methods to generate entwining tetrahedron maps, expanding the toolkit for constructing solutions to the tetrahedron equation in mathematical physics.

Contribution

It presents three novel procedures for deriving entwining tetrahedron maps, including symmetry-based, composition-based, and companion map-based methods.

Findings

Three non-equivalent procedures for entwining tetrahedron maps.

New classes of entwining tetrahedron maps from compositions with pentagon maps.

Set-theoretical solutions to the tetrahedron equation derived from companion maps.

Abstract

We present three non-equivalent procedures to obtain entwining (non-constant) tetrahedron maps. Given a tetrahedron map, the first procedure incorporates its underlying symmetry group. With the second procedure we obtain several classes of entwining tetrahedron maps by considering certain compositions of pentagon with reverse-pentagon maps which satisfy certain compatibility relations the so-called ten-term relations. Using the third procedure, provided that a given tetrahedron map admits at least one companion map (partial inverse), we obtain entwining set theoretical solutions of the tetrahedron equation.