Seminario di Algebra: The structure and enumeration of quandles (David Stanovský)

Title: The structure and enumeration of quandles

Prof. David Stanovský (Charles Univesity, Prague)

Abstract: Efficient enumeration of small objects in an abstract class may require deep structure theory (for instance, for groups) or smart combinatorial tricks (for instance, for quasigroups). I will talk about two important classes of quandles, where both approaches find their place. Certain configurations in transitive groups are used to describe algebraically connected quandles (these are the ones used for coloring knots). Then the deep theory of transitive groups can be exploited to enumerate connected quandles, currently up to size 47 and of sizes p, p^2, p^3, 2p for a prime p. Another class, medial quandles, is described using a heterogeneous affine structure over the orbit decomposition. A few combinatorial tricks then allow for enumeration of small medial quandles. For example, we can calculate that the number of isomorphism classes of 15-element medial quadles is 563753074951.