Quandles of cyclic type with several fixed points

A quandle of cyclic type of order $n$ with $f\geq 2$ fixed points is such that each of its permutations splits into $f$ cycles of length $1$ and one cycle of length $n-f$. In this article we prove that there is only one such connected quandle, up to isomorphism. This is a quandle of order $6$ and $2$ fixed points, known in the literature as octahedron quandle. We prove also that, for each $f\geq 2$, the non-connected versions of these quandles only occur for orders $n$ in the range $f+2 \leq n \leq 2f$ and that, for each $f>1$, there is only one such quandle of order $2f$ with $f$ fixed points, up to isomorphism. Still in the range $f+2 \leq n \leq 2f$, we present sufficient conditions for the existence of such quandles, writing down their permutations; we also show how to obtain new quandles form old ones, leaning on the notion of common fixed point.

The algebraic structure known as quandle appeared first in the literature in 1982, due to Joyce [10] and Matveev [14], independently (see also [8] and [9]). It was designed to constitute the algebraic counterpart of the Reidemeister moves [12]. As such it turned out to be an important tool in telling knots apart [7], [2], [5]. Algebraists also find it interesting in the domain of Hopf algebras [1]. It thus seem relevant to study the structure of quandles. In the current article we take another step in this direction by investigating and almost fully classifying a family of quandles. In [13] quandles are regarded as sequences of permutations and based on the features of permutations conclusions are drawn. In particular, in [13] quandles of the following sort are looked into. Given a positive integer n we consider a quandle of order n, each of whose permutations split into a cycle of length n−1 and a fixed point; this fixed point complies with one of the quandles axioms -this is all detailed below. These quandles were subsequently called "quandles of cyclic type". They were also studied in [11] and [16]. In this article we work in the spirit of [13] i.e., quandles as sequences of permutations, and we look into the classification of a generalization of "quandles of cyclic type" which we call "quandles of cyclic type with several fixed points". Details are supplied below in the text.

Basic Definitions and Results
The algebraic structure known as quandle, introduced independently in [10] and [14], is defined as follows. • Let G be a group and let * be the binary operation on G given by a * b = bab −1 , for every a, b ∈ G, where the juxtaposition on the right-hand side denotes group multiplication. Then, the pair (G, * ) is a quandle; • For each n ≥ 2, (R n , * ) denotes the quandle whose underlying set is Z/nZ and whose operation is a * b = 2b − a mod n. This is called the dihedral quandle (of order n); • For each n ≥ 1, (T n , * ) denotes the quandle whose underlying set is {1, . . . , n} and whose operation is i * j = i, ∀i, j ∈ {1, . . . , n}. This is called the trivial quandle (of order n); • Q 2 6 is the quandle whose multiplication table is displayed in Table 1 Table 1: Q 2 6 multiplication table. An alternative description of the structure of a quandle is the one given in the following theorem ( [3], [9]).

Definition of Quandle of Cyclic Type with Several Fixed Points
As stated in Theorem 1.1, a quandle of order n is uniquely determined by a set of n permutations, where each one of these permutations can be decomposed into a set of disjoint cycles. The lengths of these cycles define the pattern of each permutation. We can collect the information relative to the patterns of the n permutations defining a quandle of order n in order to define its profile.
Definition 1.5. The profile of a quandle of order n is the list of the patterns of the n permutations defining the quandle.
We now introduce the notion of connected quandle in order to state an important proposition. Proof. Let (Q, * ) be a connected quandle. Then, given i, j ∈ Q, ∃ k 1 , k 2 , . . . , k n ∈ Q : by Theorem 1.1, and since conjugate permutations have the same pattern, the result follows.
Remark. Note that quandles with constant profile do not have to be connected. For example, the trivial quandle of order n, (T n , * ), has constant profile and it is not connected. The same for dihedral quandles of even order.
Finally, we introduce the key notion of quandles of cyclic type with several fixed points.
Definition 1.7. Given n, f ∈ N, n − 2 ≥ f > 1, a quandle of cyclic type of order n with f fixed points is a quandle of order n with constant profile given by When there is no need to refer to its order or to its number of fixed points we refer to each of these quandles as quandle of cyclic type with several fixed points.
The previous definition means that each one of the n permutations defining a quandle of cyclic type of order n with f fixed points splits into the following types and numbers of disjoint cycles. One cycle of length n − f (> 1) and f cycles of length 1.
In passing, we note, by inspection of Table 1, that Q 2 6 is a connected quandle. Hence, by Proposition 1.2, it has a constant profile. But more than that, Q 2 6 is, in fact, a quandle of cyclic type of order 6 with 2 fixed points. In this article, we show that this is the only connected quandle of cyclic type with several fixed points.
In the sequel, we use the following notation.   Assume further there is a g 0 ∈ Q such that µ i (g 0 ) = g 0 , for any i ∈ Q. Then, the set Q ′ = Q\ {g 0 } along with the sequence of permutations µ ′ i = µ i | Q ′ for each i ∈ Q ′ defines a quandle of cyclic type of order n − 1 with f − 1 fixed points. We call this the extraction of the common fixed point g 0 . Theorem 1.5. Let n be an integer greater than 2. Let Q be the underlying set of a quandle whose permutations are denoted µ i , for each i ∈ Q. Let g 0 Q and consider the set Q ′ = Q ∪ {g 0 }. Suppose there is a permutation, µ, of the elements of Q, such that µµ i = µ i µ, for each i ∈ Q. Then, Q ′ along with the permutations is a quandle with a common fixed point, g 0 . We call this the adjoining of a common fixed point g 0 .
Corollary 1.1. Let (Q, µ) be a quandle of cyclic type of order n and f fixed points with (n − f ) | f as in Theorem 1.3. Then any two permutations are either equal or move points from disjoint sets. So adjoining a common fixed point g 0 is accomplished by taking µ ′ g 0 = (g 0 )µ i 0 by picking any i 0 ∈ Q and for any j ∈ Q, µ ′ j = (g 0 )µ j . Furthermore, this procedure can be iterated indefinitely, giving rise to an infinite sequence of quandles Q k of cyclic type of order n + k and f + k fixed points such that f + k + 2 ≤ n + k ≤ 2( f + k).

Organization
The Sections below are devoted to the proofs of these facts. In Section 2 we prove that quandles of cyclic type of order n with f fixed points in the range n > 2 f are connected (assertion 1.(a) in Theorem 1.2) and that for n = 2 f there is only one such quandle, up to isomorphism and that this quandle is not connected (assertion 2.(b) and assertion 2.(a) (for n = 2 f ) in Theorem 1.2). In Subsection 2.3 we prove that quandles of cyclic type of order n and f fixed points in the range f + 2 ≤ n ≤ 2 f are not connected; this is the 2.(a) part (for n < 2 f ) in Theorem 1.2. In Section 3 we prove that, up to isomorphism, there is only one quandle of cyclic type of order n with f fixed points in the range n > 2 f . This quandle occurs for n = 6 and f = 2 and is known as the octahedron quandle (assertion 1.(b) in Theorem 1.2). This completes the proof of Theorem 1.2. In Section 4 we prove Theorems 1.3, 1.4, and 1.5. Finally, in Section 5 we collect a few questions for further research.  In this Section, we state and prove a theorem about the structure of quandles of cyclic type with several fixed points. This theorem provides a number of conditions quandles of cyclic type of order n with f fixed points such that n ≥ 2 f must verify. This theorem is key to proceed to a classification of quandles of cyclic type.

Associate Indices
Quandles of cyclic type of order n ≥ 2 f have a very useful property, which is a consequence of the following proposition.
which is a contradiction. Hence, the result follows.
We introduce the notions of associate indices and associate permutations. Definition 2.1. Let Q be a quandle of order n with permutations denoted by µ k , k ∈ {1, . . . , n}.
• If i and j are different indices such that µ i ( j) = j and µ j (i) = i, we say that i and j are associate indices; • If i and j are associate indices then µ i and µ j are said associate permutations; In the sequel, we assume the order n of any quandle of cyclic type to be greater than or equal to 2 f , unless otherwise stated. Therefore, Proposition 2.1 always applies.
We now prove the main result of this section, which is a consequence of the previous results. Proof. We show that if there is an index s such that µ s (k) = k, for some k, then s ∈ F k . For each g ∈ F s \ {k}, that is, µ s fixes µ k (g) and hence µ k (g) ∈ F s \ {k}, as µ k (k) = k. Therefore, the restriction of µ k to F s \ {k} is a bijection from this set to itself. Arguing as in the proof of Proposition 2.1, we must have µ k (g) = g, ∀g ∈ F s \ {k}, which implies that F s = F k , and in particular, s ∈ F k . Thus, the sets of fixed points corresponding to two permutations are either equal or disjoint. Therefore, the order n of a quandle of cyclic type with f fixed points, with n ≥ 2 f , has to be a multiple of f .
The following proposition is now an immediate consequence of our previous considerations.
Proposition 2.2. Given a quandle of cyclic type of order n with f fixed points and n ≥ 2 f > 2, "i is associate to j" generates an equivalence relation on the underlying set of the quandle.
Proof. The equivalence relation is "i is associate to j or i = j". Since i is a fixed point of µ i , then i ∈ F i . Moreover, any two sets F i and F j are either equal or disjoint, thus the result follows.
Example 2.1. R 4 , the dihedral quandle of order 4, whose multiplication table is displayed in Table 2, is a quandle of cyclic type of order 4 with 2 fixed points.  By Proposition 2.2, "i is associate to j" generates an equivalence relation on R 4 , which is also a congruence relation on this set, as it respects the binary operation of the quandle. In Table 3, we see the quotient of R 4 by this congruence relation, which we denote by ∼. This quotient is clearly isomorphic to T 2 . In particular, R 4 is not simple since it admits a non-trivial quotient.

Connected Quandles of Cyclic Type of Order n with f Fixed
Points in the Range n ≥ 2 f -First Properties and Example.
From this point on, we are only working with connected quandles of cyclic type of order n ≥ 2 f . The two following propositions tell us whether a quandle of cyclic type of order n ≥ 2 f is connected or not.
If Q is a quandle of cyclic type of order n with f fixed points such that n = 2 f , Q is not connected.
Proof. Suppose Q is a quandle of cyclic type of order n with f fixed points such that n = 2 f and let i, j ∈ Q be two non-associate indices.
Therefore, we conclude that Q is not connected.
Proposition 2.4. Every quandle of cyclic type of order n with f fixed points such that n > 2 f is connected.
Proof. Suppose Q is a quandle of cyclic type of order n with f fixed points such that n = c f , with c ≥ 3 by Corollary 2.2. Let i, j ∈ Q, with i j. If i and j are associate indices, then for any k F i , i, j ∈ C k . Hence, there exists an integer a ∈ {1, . . . , n− f −1} such that µ a k (i) = j. Now, assume i and j are not associate indices. Since there are at least three distinct sets of associate indices, there is at least one set F k 0 such that both i and j do not belong to F k 0 . Therefore i, j ∈ C k 0 and so there exists an integer a ∈ {1, . . . , n − f − 1} such that µ a k 0 (i) = j. We conclude that Q is connected. Remark. In the sequel, we assume the order n of any quandle to be greater than 2 f , unless otherwise stated. In this condition, all our quandles of cyclic type are connected.
We now use some of the equalities µ µ i ( j) = µ i µ j µ −1 i the permutations defining these quandles have to verify to derive a number of conditions these quandles have to satisfy in order to be cyclic. Theorem 2.1. Consider a quandle of cyclic type of order n with f fixed points such that n > 2 f . Modulo isomorphism, its sequence of permutations satisfies the following conditions.

if h and h ′ are associate indices then
We can assume that µ n is given by (1 2 3 · · · n − f )(n − f + 1) · · · (n − 1)(n) without loss of generality. If necessary, we may relabel the indices. This expression for µ n will be assumed in the sequel -except for Subsection 2.3.

Suppose h and h ′ are associate indices and let
(where we read the free indices modulo n − f ), whence we proved 3. by induction.

4.
Let a ∈ F n and assume µ n− f (a) F n . Let i F n . On one hand, we have, by assertions 2. and 3., On the other hand, again by assertion 3., which implies that Combining 1 and 2, we get We now prove the following lemma, which completes the proof of assertion 4..
This would force the pairs of associate permutations from µ 1 to µ n− f to be equal to each other. In fact, by assertion 2., if two associate permutations have the same image at a point belonging to their non-singular cycles, these permutations have to be the same.
Suppose, now, that for a certain index b ∈ F n , µ n− f (b) F n . Then, for 1 ≤ k ≤ n − f , which would force the pairs of permutations whose indices are associate from µ 1 to µ n− f to be all different from each other, which is a contradiction. Therefore, µ n− f (F n ) = F n . But µ n− f does not fix any element from F n since F n ∩ F n− f = ∅. Then, this implies that µ n− f has a cycle of length at most f , which is again a contradiction, since n− f > f . Hence, µ n− f (a) F n .

5.
Let a ∈ F n and let i F n be the index such that µ n− f (i) = a. Note that otherwise i would take up the role of a in Lemma 2.1 and a would not belong to F n . Then, which is equivalent to having µ n− f µ m n and µ n− f commute in S n . The number of elements in the centralizer of µ n− f in S n , |C S n (µ n− f )|, is given by see [15], for instance, where |µ S n n− f | denotes the number of elements of S n with the same pattern as µ n− f . In fact, we have and since |S n | = n!, we conclude that |C S n (µ n− f )| = (n − f ) f !. However, we know exactly what these (n − f ) f ! permutations are. Let τ be the cycle of length n − f of µ n− f . Indeed, τ k commutes with µ n− f , ∀k ∈ {1, . . . , n − f }. Moreover, any permutation of the f fixed points of µ n− f commutes with µ n− f . The former type of permutation τ k only moves elements within C n− f whereas the latter type of permutation only moves elements within F n− f . Composing permutations from these two commuting types of permutations, we get a total of (n − f ) f ! permutations commuting with µ n− f , which is precisely the number of permutations we found before. Therefore, we may conclude Proof. Let f be as in the statement. We first note that, should it exist, the indicated quandle is not connected via Proposition 2.3. We will next prove (1.) that there is such a quandle; and then (2.) that any such quandle is isomorphic to the one in 1.

Consider the sequence of permutations
Note that µ i 's and µ j 's commute among themselves and with one another, since they are either equal or move points from disjoint sets. Then, for any i, i ′ ∈ {1, 2, . . . , f } and j, j ′ ∈ { f + 1, f + 2, . . . , 2 f }, we have Therefore, the indicated sequence of permutations defines a quandle. Moreover, this is a quandle of cyclic type of order 2 f with f fixed points.
2. Now consider a quandle of cyclic type of order 2 f and f fixed points, along with its permutations, µ ′ i for i = 1, 2, . . . , 2 f . According to 1. in Theorem 2.1 and Corollary 2.3 whose set of fixed points is There are two distinct sets of fixed points (see proof of Proposition 2.3), so the other one is Therefore, Also, for any i ∈ {1, 2, . . . , f }, we have Therefore, Finally, consider the permutation, α, of {1, 2, . . . , f, f + 1, . . . , 2 f } given by Thus, α is a quandle isomorphism between the quandle here and the quandle in 1. Proof. By Proposition 2.3, we know this is true for n = 2 f . Now, let Q be a cyclic quandle of order n with f fixed points such that f + 2 ≤ n ≤ 2 f − 1. We assume, without loss of generality, that that is, µ n fixes µ k ( j). Therefore, we have that µ k ( j) ∈ F n for any j, k ∈ F n . Thus, F n is a subquandle of Q. Now, if this subquandle has constant profile, then the common pattern is that of µ n F n = (n − f + 1)(n − f + 2) . . . (n), hence F n as a quandle is the trivial quandle on f elements. In particular, it is not connected. If F n as a quandle has not constant profile, then, by Proposition 1.2, it is not connected. In either case, this subquandle is not connected and hence there is a finite family of minimal disjoint sets F i n , i ∈ {1, 2, . . . , d}, such that i F i n = F n and µ g (F i n ) = F i n , ∀i, ∀g ∈ F n , which correspond to the (minimal) connected components of F n , as a quandle. We also note that C n ∩ F n = ∅, C n ∪ F n = Q and µ g (C n ) = C n , ∀g ∈ F n . Now, since |C n | = n − f < f = |F 1 |, µ 1 must fix some point a 0 ∈ F n , i.e., a 0 ∈ F n ∩ F 1 .  Thus, the a 0 ∈ F 1 ∩ F n above, satisfies, thanks to Lemma 2.2, Furthermore, for j, j ′ ∈ {n − f + 1, n − f + 2, . . . , n} such that µ j (a 0 ) µ j ′ (a 0 ), then µ i (µ j (a 0 )) µ i (µ j ′ (a 0 )), for any i ∈ {1, 2, . . . , n − f }. Then, for any i ∈ {1, 2, . . . , n − f }, µ i restricted to A is a bijection.
1. If A = {a 0 }, then µ s (a 0 ) = a 0 , for any s ∈ {1, 2, . . . , n}, so A is a connected component of Q. Since Q has more than one element then Q is not connected.
(a) Assume further that, for each i ∈ {1, 2, . . . , n − f }, µ i moves at least on element from A, say µ j 0 (a 0 ), for some j 0 ∈ {n − f + 1, n − f + 2, . . . , n}recall Lemma 2.2. Then, Since µ i has a cycle of length n− f , Since C n ∩ F 1 n = ∅ and C n ⊂ Q, then Q is not connected. i. If A = F 1 n , then we are done, arguing that F 1 n is a connected component inside Q which has fewer elements than Q. ii. If A F 1 n , then since F 1 n is a minimal component of F n , there exist j 0 , j 1 ∈ {n − f + 1, n − f + 2, . . . n} such that µ j 1 (µ j 0 (a 0 )) ∈ F 1 n \ A.
The proof is complete.
3 Classification of Quandles of Cyclic Type of order n with f Fixed Points in the Range n > 2 f .
In this Section, we classify quandles of cyclic type of order n with f fixed points such that n > 2 f . Specifically, we prove that there is only one such quandle such that n > 2 f , up to isomorphism. In this range, this quandle occurs only for n = 6 and f = 2. This quandle is Q 2 6 , up to isomorphism. The proof of this fact establishes Assertion 1.(b) in Theorem 1.2. This is the main goal of the current Section.
In Subsection 3.1, we prove a number of propositions and lemmas that we use in subsequent subsections. In Subsection 3.2, we show that there are no quandles of cyclic type of order n with f fixed points such that n = 3 f for f > 2 and we prove that the only quandle of cyclic type of order 6 with 2 fixed points, up to isomorphism, is Q 2 6 . In Subsection 3.3, we show that there are no quandles of cyclic type of order n with f fixed points such that n = c f for c > 3. Finally, in Subsection 3.4, we collect the results from the preceding subsections to prove Assertion 1.(b) in Theorem 1.2. We also show that Q 2 6 is not simple. In this Section, the results apply only to quandles of cyclic type of order n with f fixed points such that n > 2 f .

Auxiliary Results
In this Subsection, we state and prove a number of results about the structure of quandles of cyclic type of order n with f fixed points such that n > 2 f . These results are used in the following Subsections.
. . , f − 1}. In particular, we have g f n− f = n − f . Suppose these indices are not equally spaced modulo n − f . Therefore, there is an index . . , f }, and there is another index We now have all the results we need to prove there are no quandles of cyclic type of order n with f fixed points such that n = 3 f for f > 2. This is the result we state in the following proposition.

Quandles of Cyclic Type of Order n with f Fixed Points such
that n > 3 f In this Subsection, we use the results from the previous Subsections to show that there are no quandles of cyclic type of order n with f fixed points such that n = c f for c > 3. This is the result we state in the following proposition, where in its proof the free indices are read modulo n − f .
, implying that the two associate permutations µ µ n− f (b)−µ n− f (a) and µ n− f (see Proposition 3.1 and Corollary 3.1) have the same image at a point that is not a fixed point of these permutations, as µ n− f (b) F n− f . Hence, these permutations must be equal to each other, which is a contradiction, as these permutations are different from each other by Corollary 3.4. Therefore, there are no quandles of cyclic type of order n with f fixed points such that n = c f , for c > 3, and the result follows. Example 3.1. Q 2 6 , whose multiplication table is displayed in Table 1, is the only quandle of cyclic type of order n with f fixed points such that n > 2 f , up to isomorphism. In particular, Q 2 6 is not a simple quandle (since it admits a non-trivial congruence). Indeed, by Proposition 2.2, for n ≥ 2 f , "i is associate to j" generates an equivalence relation on Q 2 6 , which is also a congruence relation on this set, as it respects the binary operation of the quandle. In Table 4, we see the quotient of Q 2 6 by this congruence relation, which we denote by ∼. This quotient is clearly isomorphic to R 3 , since "the product" of any two distinct elements equals the other element.
This sequence of permutations defines a quandle of cyclic type of order n with f fixed points over the set {1, 2, . . . , n}.
Proof. The proof of this Theorem is basically a rearrangement of the argument for the proof of the first statement of Corollary 2.4, the existence of a quandle of cyclic type of order 2 f with f fixed points. We add it here for completeness. Let Then, Also, This completes the proof.

Extracting -Adjoining a Common Fixed Point.
Definition 4.1. Let Q be a quandle of cyclic type with several fixed points. If g 0 ∈ Q is such that it is a fixed point for any of the permutations of Q, g 0 is called a common fixed point of Q.
Example 4.1. In Table 5 we provide the multiplication table of a quandle Table 5: Quandle of cyclic type of order 5 and 3 fixed points with a common fixed point: 1. See [6], page 176.
The next two Theorems show us when we can extract a common fixed point (Theorem 1.4) and when we can adjoin a common fixed point (Theorem 1.5). We repeat their statements here for completeness. Once Theorem 1.5 is proved, combining it with Theorem 4.1 and iterating the procedure, provides an infinite sequence of quandles of cyclic type with several fixed points within the range f + 2 ≤ n ≤ 2 f where n is the order and f the number of fixed points. This is the content of Corollary 1.1.
Theorem 4.2. Suppose f is an integer strictly greater than 2 and n a positive integer such that f +2 ≤ n ≤ 2 f . Consider a quandle of cyclic type of order n and f fixed points over the set Q = {1, 2, . . . , n} with sequence of permutations µ i with i ∈ {1, 2, . . . , n}. Assume further g 0 ∈ Q is a common fixed point of Q. Then, the set Q ′ = Q \ {g 0 } along with the sequence of permutations µ ′ i = µ i | Q ′ for each i ∈ Q ′ defines a quandle of cyclic type of order n − 1 with f − 1 fixed points. We call this the extraction of the common fixed point g 0 .
Then, g i+1 = µ g 0 (g i ) =⇒ µ g i+1 = µ g 0 µ g i µ −1 g 0 = µ g i . So the associate permutations to the permutations corresponding to the elements moved by µ g 0 , are all equal to one another.
Consider now the set Q ′ = Q \ {g 0 } along with the sequence of permutations µ ′ i = µ i | Q ′ for each i ∈ Q ′ . For each i, j ∈ Q ′ , we have which completes the proof.
Theorem 4.3. Let n be an integer greater than 2. Let be the underlying set of a quandle whose permutations are denoted µ i , for each i ∈ Q. Let g 0 Q and consider the set Q ′ = Q ∪ {g 0 }. Suppose there is a permutation, µ, of the elements of Q, such that µµ i = µ i µ, for each i ∈ Q. Then, Q ′ along with the permutations µ ′ i = (g 0 )µ i for each i ∈ Q and µ ′ g 0 = (g 0 )µ is a quandle with a common fixed point, g 0 .
This completes the proof.

Further Research
In this article we looked into the classification of quandles of cyclic type of order n with f fixed points. We realize that these quandles split into three sorts according to the ranges their (n, f )'s lie in. If n > 2 f , then these quandles are connected. As a matter of fact, there is only one such quandle which occurs for n = 6 and f = 2; it is the octahedron quandle. For each integer f > 2, there is exactly one such quandle of order n = 2 f and it is not connected. Finally, in the range 2 < f + 1 < n < 2 f , such quandles are not connected and there seem to be plenty of them.
With the techniques developed in this article, we plan on looking into the classification of other families of quandles like those with constant profile with f fixed points and two non-singular cycles, to begin with. We also plan on taking a fresh look at quandles of cyclic type i.e., when f = 1.