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The category $\mathbf{Rack}$ of racks is Barr-exact since it is a variety of universal algebras, but it is not protomodular. Indeed, the category of sets is equivalent to the category of racks satisfying the identity $a\triangleleft b =a$, so it is a full epireflective subcategory of $\mathbf{Rack}$. In particular, there is an inclusion functor $\mathbf{Set}\... 11 See Example 3.6 of Ellis and Fragnaud, Computing with knot quandles. It says, The following commands use the quandle invariant ColQ(K) to establish that the granny knot is not equivalent to the square knot. We take Q to be the seventeenth quandle in our enumeration of connected quandles of order 24. gap> Q:=ConnectedQuandle(24,17,"import");; gap> K:=... 11 Q1b Joyce in A classifying invariant of knots, the knot quandle. J. Pure Appl. Algebra 23 (1982) proves that the equational theory of quandles and that of groups endowed with a quandle structure via conjugation is the same. So conjugation quandles are not a subvariety of quandles. There is however one more axiom which is necessary (I follow your convention ... 9 I do not believe that the word problem for quandles was first formulated by the team of Rena Levitt and Sam Nelson, if this formulation was made while Levitt was at Pomona. I had been working on this question well before 2011, which is when I assume Levitt arrived in Pomona and I am certain that it had been asked before. This is a standard question that ... 7 The word problem for quandles was formulated by Rena Levitt and Sam Nelson. In Levitt's research statement the problem was formulated as follows: Word Problem for Quandles. Given a finitely generated quandle$Q$, is there an algorithm to determine if two words in the generators represent the same element of$Q$? Levitt and Nelson solved the word ... 7 You have two excellent expert answers. Here is a very simple answer to some of your questions. First a slight overview. Given a group one has the conjugation operation$a*b=bab^{-1}$This is just$a*b=a$when$G$is an abelian group. A group with$n$elements gives a quandle with$n$elements which satisfies the axioms you listed. A finite quandle can be ... 6 Let me point out a sort of characterization of conjugation quandles. (This is too long for a comment.) Let$X$be a finite quandle. The enveloping group$G_X$is the group with generators$x\in X$and relations$x(x*y)=yx$for all$x,y\in X$. The enveloping group has the following universal property: For any group$G$and any map$f:X\to G$... 5 The short answer is that I think this is an open question. This is stated as Conjecture 3.4 here, proved for knots up to 12 crossings: Clark, W. Edwin; Elhamdadi, Mohamed; Saito, Masahico; Yeatman, Timothy, Quandle colorings of knots and applications, J. Knot Theory Ramifications 23, No. 6, Article ID 1450035, 29 p. (2014). ZBL1302.57016. MR3253967. ... 5 The notion of a semigroup with conjugation can be formalized by the notions of an LD-monoid, LRD-monoid, and LRDQ-monoid. The information in this answer can be found in the book Braids and Self-Distributivity by Patrick Dehornoy.$\textbf{LD-monoids}$An LD-monoid (LD stands for left-distributive) is an algebra$(M,\cdot,1,\wedge )$such that$(M,\cdot,1)$... 3 Answers to some of your questions can be found in this paper: Quandle Colorings of Knots and Applications http://arxiv.org/abs/1312.3307 3 Let's make sure we agree on definitions. A quandle is an algebraic structure$(A,*)$for which each right multiplication map$S_a(x)= x*a$by an element$a\in A$is an automorphism fixing$a$. These right multiplications are called inner automorphisms of the quandle, and$\textit{Inn}(A)$is the group generated by these right multiplications. Let$G$be a ... 2 Beginning with coloring the$k$th box using${\mathbb{Z}_k[t^{\pm1}]}/_{(t-1)^2}$, we conclude that if the link is homotopically trivial, then$p_k$divides all$p_i$'s. It seems that there is a typo here: "${\mathbb{Z}_k[t^{\pm1}]}/_{(t-1)^2}$" should be "${\mathbb{Z}_{p_k}[t^{\pm1}]}/_{(t-1)^2}$". By Corollary 5.2, if we start by colouring the top$k$... 1 Not a direct answer but a related one: The granny knot and the square knot are known to have isomorphic$\pi_1$, but they can be detected by quandle coloring. An application of that is the following: fix a knot$K_0$, and then for a general knot$K$consider the quandle colorings of$K\#K_0$. If you take$K_0$to be the treefoil, and$K$equal to the ... 1 Baer in Representations of groups as quotient groups. I. Trans. Amer. Math. Soc. 58, (1945) defines the notion of commutator quotient (and he says that he takes it from Zassenhaus): if S,T are subsets of a group G he defines: $$S \div T = \lbrace g \in G | [T, g] \subseteq S \rbrace$$ With this notation you have$\$ Z\left( \frac{G}{N} \right) = \frac{...