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Qualgebras and knotted 3-valent graphs

Volume 230 / 2015

Victoria Lebed Fundamenta Mathematicae 230 (2015), 167-204 MSC: Primary 05C25; Secondary 17D99, 57M27, 20N99. DOI: 10.4064/fm230-2-3


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.


  • Victoria LebedOCAMI, Osaka City University
    3-3-138 Sugimoto-cho, Sumiyoshi-ku
    Osaka, 558-8585, Japan

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