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.
Characterizations of knot groups and knot symmetric quandles of surface-links
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abstract
The knot group is the fundamental group of a knot or link complement. A necessary and sufficient conditions for a group to be realized as the knot group of some link was provided. This result was shown using the closed braid method. Gonz\'alez-Acu\~na and Kamada independently extended this characterization to the knot groups of orientable surface-links. Kamada applied the closed 2-dimensional braid method to show this result. In this paper, we generalize these results to characterize the knot groups of surface-links, including non-orientable ones. We use a plat presentation for surface-links to prove it. Furthermore, we show a similar characterization for the knot symmetric quandles of surface-links. As an application, we show that every dihedral quandle with an arbitrarily good involution can be realized as the knot symmetric quandle of a surface-link.
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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.