Qualgebras and knotted 3-valent graphs

Victoria Lebed¹

¹ OCAMI, Osaka City University 3-3-138 Sugimoto-cho, Sumiyoshi-ku Osaka, 558-8585, Japan

Fundamenta Mathematicae, Tome 230 (2015) no. 2, pp. 167-204

Résumé

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.

Détail

DOI : 10.4064/fm230-2-3

Keywords: paper devoted algebraic structures called qualgebras squandles topologically emerge algebraic counterpart knotted valent graphs just quandles seen algebraization knots algebraically modeled after groups conjugation multiplication squaring operations discuss basic properties these structures introduce study notions qualgebra squandle cocycles coboundaries knotted valent graph invariants constructed counting qualgebra squandle colorings graph diagrams further enhanced using cocycles classification size qualgebras squandles description their second cohomology groups given