Entropic magmas, their homology and related invariants of links and graphs (Q371274)

The ``entropic magmas'' of the title are algebraic structures that were studied by several authors in the early twentieth century; see for instance [\textit{A. Suschkewitsch}, Transactions A. M. S. 31, 204-214 (1929; JFM 55.0085.01); \textit{K. Toyoda}, Proc. Imp. Acad. Tokyo 17, 221--227 (1941; Zbl 0061.02403); \textit{D.C. Murdoch}, Trans. Am. Math. Soc. 49, 392--409 (1941; Zbl 0025.10003)]. The definition involves a binary operation \(\ast\) that satisfies the entropic identity \((a\ast b)\ast(c\ast d)=(a\ast c)\ast(b\ast d)\). It turns out that every such structure is determined by an abelian group with a distinguished element \(c\) and a distinguished pair of commuting automorphisms \(f\) and \(g\); the operation \(\ast\) is then given by \(a\ast b=f(a)+g(b)+c\).NEWLINENEWLINEIn the paper under review the authors observe that analogues of some well-known constructions can be carried out using entropic magmas. The first of these constructions begins with a sequence of elements that satisfy certain algebraic identities that correspond to Reidemeister moves of the second type, and produces a link type invariant analogous to the Kauffman bracket. The second construction begins with a sequence of elements, and produces an analogue of the Tutte polynomial of a signed graph. In the third construction, \(2^{n}\)-tuples of elements of an entropic magma are used to form \(n\)-dimensional chain groups, and these chain groups are assembled into a chain complex by defining differentials.