With the problem of coloring diagrams ternary groups \((G,[\ ])\) are associated satisfying the following two axioms: \([[abc]cd]=[[ab[bcd]] [bcd]d]\) and \([ab[bcd]]=[a[abc][[abc]cd]]\). Such ternary groups are called knot-theoretic ternary groups. Each knot-theoretic ternary group \((G,[\ ])\) can be presented in the form \([xyz]=x-y+z+a\), where \((G,+)\) is an abelian group and \(a\) is a fixed element of order one or two in \((G,+)\). So such ternary groups are a special case of flocks and Vagner’s heaps.
The first part (Sections 1–5) of this paper contains elementary properties of such groups. All these properties are a consequence of the Hosszú-Gluskin theorem (see for example [the reviewer and K. Głazek, Discrete Math. 308, No. 21, 4861–4876 (2008; Zbl 1153.20050)]) and results obtained earlier for flocks [the reviewer, Algebras Groups Geom. 16, No. 3, 329–354 (1999; Zbl 0996.20055)].
In the last section, knot-theoretic ternary groups are used to study of curves immersed in compact surfaces.