On symplectic quandles. (English) Zbl 1168.57011
Quandles are non-associative algebraic structures whose axioms may be understood as transcriptions of the Reidemeister moves. A typical example is a group with quandle product \(*\) given by conjugation, i.e., \(x*y =y^{-1}xy\). This paper is about symplectic quandles. These quandles are also modules equipped with an antisymmetric bilinear form; they were already considered in [D. N. Yetter, J. Knot Theory Ramifications 12, No. 4, 523–541 (2003; Zbl 1051.57003)].
The main result of the paper under review shows that every symplectic quandle over a field (of characteristic other than two if it is not finite) is almost connected, that is, it is a disjoint union of a trivial quandle and a connected quandle. Indeed, connected quandles are of particular interest for defining knot invariants since knot quandles are always connected. The paper ends introducing a new family of enhanced quandle counting invariants, based on finite symplectic quandles.
The main result of the paper under review shows that every symplectic quandle over a field (of characteristic other than two if it is not finite) is almost connected, that is, it is a disjoint union of a trivial quandle and a connected quandle. Indeed, connected quandles are of particular interest for defining knot invariants since knot quandles are always connected. The paper ends introducing a new family of enhanced quandle counting invariants, based on finite symplectic quandles.
Reviewer: Pedro M. González Manchón (Madrid)
MSC:
| 57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |
| 55M25 | Degree, winding number |
| 17D99 | Other nonassociative rings and algebras |
Citations:
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