Good involutions do not exist on symplectic quandles defined on free R-modules equipped with antisymmetric bilinear forms.
Good involutions of twisted conjugation subquandles and Alexander quandles
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abstract
We completely describe good involutions of free quandles and subquandles of twisted conjugation quandles of groups, including all Alexander quandles. As an application, we enumerate good involutions of linear quandles, and we provide explicit mappings for those up to order 23 via a computer search. Along the way, we completely characterize connected, involutory Alexander quandles, which may be of independent interest.
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2026 1verdicts
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On the nonexistence of good involutions of symplectic quandles
Good involutions do not exist on symplectic quandles defined on free R-modules equipped with antisymmetric bilinear forms.