Knotted 3-Valent Graphs, Branched Braids, andMultiplication-Conjugation Relations in a Group

This survey is devoted to a new algebraic structure called qualgebra. Our topologicalmotivation is the study of knotted 3-valent graphs and closely related branched braids via combinatorially defined coloring invariants. From an algebraic viewpoint, our structure a part of an alternative axiomatization of groups, describing the properties of conjugation operation and its interactions with the group multiplication. Qualgebras can thus be metaphorically seen as a widening of the bridge between algebra and topology formed by the quandle structure, popular among knot theorists; see Table 1 to better understand how this bridge works. Only a brief and rather informal exposition of different facets of qualgebras is given here. For more details, comments, and proofs, see [20, 15]. However, Sections 2.1, 2.3, 2.4, and 3.1 contain some recent unpublished results, which will be thoroughly treated elsewhere.