Qualgebras and knotted 3-valent graphs

Victoria Lebed (1)

1 OCAMI - Osaka City university Advanced Mathematical Institute [Osaka]

This paper is devoted to qualgebras and squandles, which are quandles enriched with a compatible binary/unary operation. Algebraically, they are modeled after groups with conjugation and multiplication/squaring operations. Topologically, qualgebras emerge as an algebraic counterpart of knotted 3-valent graphs, just like quandles can be seen as an ''algebraization'' of knots; squandles in turn simplify the qualgebra algebraization of graphs. Knotted 3-valent graph invariants are constructed by counting qualgebra/squandle colorings of graph diagrams, and are further enhanced using qualgebra/squandle 2-cocycles. Some algebraic properties and the beginning of a cohomology theory are given for both structures. A classification of size 4 qualgebras/squandles is presented, and their second cohomology groups are completely described.

Mots clés: quandles; knotted 3-valent graphs; qualgebras; squandles; colorings; counting invariants; Boltzmann weight; cocycle invariants; qualgebra cohomology

Domaines: Topologie algébrique [math.AT]

Victoria Lebed. Qualgebras and knotted 3-valent graphs. 2014. ⟨hal-00951712⟩