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Free medial quandles  [PDF]
Premysl Jedlicka,Agata Pilitowska,Anna Zamojska-Dzienio
Mathematics , 2015,
Abstract: This paper brings the construction of free medial quandles as well as free $n$-symmetric medial quandles and free $m$-reductive medial quandles.
Alexander quandles of order 16  [PDF]
Gabriel Murillo,Sam Nelson
Mathematics , 2004,
Abstract: Isomorphism classes of Alexander quandles of order 16 are determined, and classes of connected quandles are identified. This paper extends the list of known distinct connected finite Alexander quandles.
Hom Quandles  [PDF]
Alissa S. Crans,Sam Nelson
Mathematics , 2013,
Abstract: If $A$ is an abelian quandle and $Q$ is a quandle, the hom set $\mathrm{Hom}(Q,A)$ of quandle homomorphisms from $Q$ to $A$ has a natural quandle structure. We exploit this fact to enhance the quandle counting invariant, providing an example of links with the same counting invariant values but distinguished by the hom quandle structure. We generalize the result to the case of biquandles, collect observations and results about abelian quandles and the hom quandle, and show that the category of abelian quandles is symmetric monoidal closed.
Classification of Finite Alexander Quandles  [PDF]
Sam Nelson
Mathematics , 2002,
Abstract: Two finite Alexander quandles with the same number of elements are isomorphic iff their Z[t,t^-1]-submodules Im(1-t) are isomorphic as modules. This yields specific conditions on when Alexander quandles of the form Z_n[t,t^-1]/(t-a) where gcd(n,a)=1 (called linear quandles) are isomorphic, as well as specific conditions on when two linear quandles are dual and which linear quandles are connected. We apply this result to obtain a procedure for classifying Alexander quandles of any finite order and as an application we list the numbers of distinct and connected Alexander quandles with up to fifteen elements.
Flat connected finite quandles  [PDF]
Yoshitaka Ishihara,Hiroshi Tamaru
Mathematics , 2015,
Abstract: Quandles can be regarded as generalizations of symmetric spaces. In the study of symmetric spaces, the notion of flatness plays an important role. In this paper, we define the notion of flat quandles, by referring to the theory of Riemannian symmetric spaces, and classify flat connected finite quandles.
On the classification of quandles of low order  [PDF]
L. Vendramin
Mathematics , 2011, DOI: 10.1142/S0218216512500885
Abstract: Using the classification of transitive groups we classify indecomposable quandles of size <36. This classification is available in Rig, a GAP package for computations related to racks and quandles. As an application, the list of all indecomposable quandles of size <36 not of type D is computed.
Subdirectly irreducible medial quandles  [PDF]
Premysl Jedlicka,Agata Pilitowska,Anna Zamojska-Dzienio
Mathematics , 2015,
Abstract: We classify subdirectly irreducible medial quandles. We show that in the finite case they are either connected (and therefore affine) or reductive. Moreover, we give an explicit description of all subdirectly irreducible reductive medial quandles.
The structure of medial quandles  [PDF]
P?emysl Jedli?ka,Agata Pilitowska,David Stanovsky,Anna Zamojska-Dzienio
Mathematics , 2014,
Abstract: Medial quandles are represented using a heterogeneous affine structure. As a consequence, we obtain numerous structural properties, including enumeration of isomorphism classes of medial quandles up to 13 elements.
The origins of involutory quandles  [PDF]
David Stanovsky
Mathematics , 2015,
Abstract: We present an overview of some older papers on involutory quandles, mostly from the times before the term "quandle" was born. It is meant as a reference guide, not (yet) as an expository article explaining what the involutory quandles are and what they are good for.
Small connected quandles  [PDF]
F. J. B. J. Clauwens
Mathematics , 2010,
Abstract: It is the purpose of this note to classify connected quandles up to order 14, and in particular to show that there is no connected quandle of order 14.
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