Qualgebras and knotted 3-valent graphs
This paper is devoted to new algebraic structures, called qualgebras and squandles. Topologically, they emerge as an algebraic counterpart of knotted $3$-valent graphs, just like quandles can be seen as an “algebraization” of knots. Algebraically, they are modeled after groups with conjugation and multiplication/squaring operations. We discuss basic properties of these structures, and introduce and study the notions of qualgebra/squandle $2$-cocycles and $2$-coboundaries. Knotted $3$-valent graph invariants are constructed by counting qualgebra/squandle colorings of graph diagrams, and are further enhanced using $2$-cocycles. A classification of size $4$ qualgebras/squandles and a description of their second cohomology groups are given.