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Quandles versus symmetric quandles for oriented links
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Given a quandle, we can construct a symmetric quandle called the symmetric double of the quandle. We show that the (co)homology groups of a given quandle are isomorphic to those of its symmetric double. Moreover, quandle coloring numbers and quandle cocycle invariants of oriented links and oriented surface-links can be interpreted by using symmetric quandles.
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Cited by 2 Pith papers
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Associated groups of symmetric quandles
Symmetric quandle associated groups are characterized: the underlying quandle's group is a central extension of the symmetric one with a free abelian kernel, and embeddability is equivalent.
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Characterizations of knot groups and knot symmetric quandles of surface-links
A group is the knot group of a surface-link, orientable or not, exactly when it admits a (2m,n)-presentation with inverses satisfying the weak ∂-condition, and an analogous statement holds for knot symmetric quandles.
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