Quandle coloring and cocycle invariants of composite knots and abelian extensions

Abstract

Quandle colorings and cocycle invariants are studied for composite knots, and applied to chirality and abelian extensions. The square and granny knots, for example, can be distinguished by quandle colorings, so that a trefoil and its mirror can be distinguished by quandle coloring of composite knots. We investigate this and related phenomena. Quandle cocycle invariants are studied in relation to quandle coloring of the connected sum, and formulas are given for computing the cocycle invariant from the number of colorings of composite knots. Relations to corresponding abelian extensions of quandles are studied, and extensions are examined for the table of small connected quandles, called Rig quandles. Computer calculations are presented, and summaries of outputs are discussed.

1. Introduction

Sets with certain self-distributive operations called quandles have been studied since the 1940s in various areas with different names. The fundamental quandle of a knot was defined in a manner similar to the fundamental group [21, 24] of a knot, which made quandles an important tool in knot theory. The number of homomorphisms from the fundamental quandle to a fixed finite quandle has an interpretation as colorings of knot diagrams by quandle elements, and has been widely used as a knot invariant. Algebraic homology theories for quandles were defined [5, 19], and investigated in [22, 25–27]. Extensions of quandles by cocycles have been studied [1, 4, 16], and invariants derived thereof are applied to various properties of knots and knotted surfaces (see [8] and references therein).

Tables of small quandles have been made previously (e.g. [8, 15, 17]). Computations using GAP [34] significantly expanded the list for connected quandles. These quandles may be found in the GAP package Rig [33]. Rig includes all connected quandles of order less than 48. We refer to these quandles as Rig quandles, and use the notation Q(n, i) for the ith quandle of order n in the list of Rig quandles. As a matrix Q(n, i) is the transpose of the quandle matrix SmallQuandle(n, i) in [33]. In this paper, however, we focus on Rig quandles of order less than 36. There are 431 such quandles.

In [11], it was investigated to what extent the number of quandle colorings of a knot by a finite quandle can distinguish the prime oriented knots with at most 12 crossings in the knot table at KnotInfo [14]. It is known that quandle colorings do not distinguish K from its reversed mirror, rm(K). It is also known [10] that the quandle cocycle invariant can distinguish a trefoil 3₁ from its mirror image. Since 3₁ is reversible, it cannot be distinguished from its mirror by quandle colorings. However, we show here that quandle colorings can be used via connected sums to distinguish K from rm(K) for many knots (we conjecture for all knots K such that K ≠ rm(K)). In particular, for some reversible knots, we can distinguish K from m(K) using this technique. For example, by distinguishing the square and granny knots by quandle colorings, we distinguish a trefoil from its mirror image. In this paper, we investigate this phenomenon, and other properties and applications of quandle invariants under connected sum. In particular, we relate quandle colorings of composite knots to quandle 2-cocycle invariants.

We also note that quandle colorings of the connected sum can be used to recover quandle cocycle invariants in many cases. It is well-known that quandle 2-cocycles give rise to abelian extensions of quandles, see for example [4]. We investigate the relations among abelian extensions that result from our computations, and their properties. As a result, several problems arise naturally.

An important part of this work depends on computer calculations. For that reason, we developed algorithms and techniques for computing quandle (co)homology groups and explicit quandle 2-cocycles, abelian extensions of quandles, dynamical cocycles and non-abelian extensions, colorings and quandle cocycle invariants of classical and virtual knots. The algorithms are freely available in the GAP package Rig. Several tables with all these calculations are available online at the Wiki page of Rig: http://github.com/vendramin/rig/wiki.

The paper is organized as follows. Preliminary material necessary for the paper follows this section, and it is shown that the number of quandle colorings by finite quandles can distinguish the unknot in Sec. 3. Quandle colorings of composite knots are studied in Sec. 4. In Sec. 5, quandle colorings of composite knots are applied to distinguish knots from their reversed mirror images, relations to the quandle cocycle invariant are discussed, and computer calculations are presented. In Sec. 6, a method of computing quandle cocycle invariants from colorings of composite knots is studied. Relations to abelian extensions of quandles are examined in Sec. 7. Further considerations regarding extensions of Rig quandles are presented in Sec. 7. For convenience of the reader, we collect problems, questions and conjectures posed all over the text in Sec. 8.

2. Preliminaries

We briefly review some definitions and examples of quandles. More details can be found, for example, in [1, 8, 19].

A quandle X is a set with a binary operation (a, b) ↦ a * b satisfying the following conditions:

1. For any a ∈ X, a * a = a.

2. For any b, c ∈ X, there is a unique a ∈ X such that a * b = c.

3. For any a, b, c ∈ X, we have (a * b) * c = (a * c) * (b * c).

A quandle homomorphism between two quandles X, Y is a map f : X → Y such that f(a*_X b) = f(a)*_Y f(b), where *_X and *_Y denote the quandle operations of X and Y, respectively. A quandle isomorphism is a bijective quandle homomorphism, and two quandles are isomorphic if there is a quandle isomorphism between them.

Example 2.1

Any non-empty set X with the operation a * b = a for any a, b ∈ X is a quandle called a trivial quandle.

Example 2.2

A conjugacy class X of a group G is a quandle with the quandle operation a * b = b⁻¹ab. We call this a conjugation quandle.

Example 2.3

Let X and Y be quandles. Then X × Y is a quandle with (x, y) * (x′, y′) = (x *_X x′, y *_Y y′) for all x, x′ ∈ X and y, y′ ∈ Y.

Example 2.4 (see [21])

A generalized Alexander quandle is defined by a pair (G, f) where G is a group, f ∈ Aut(G), and the quandle operation is defined by x * y = f(xy⁻¹)y. If G is abelian, this is called an Alexander (or affine) quandle.

Example 2.5

A function ϕ : X × X → A for an abelian group A is called a quandle 2-cocycle [5] if it satisfies

$$\varphi (x,y)-\varphi (x,z)+\varphi (x*y,z)-\varphi (x*z,y*z)=0$$

for any x, y, z ∈ X and ϕ(x, x) = 0 for any x ∈ X. For a quandle 2-cocycle ϕ, E = X × A becomes a quandle by

$$(x,a)*(y,b)=(x*y,a+\varphi (x,y))$$

for x, y ∈ X, a, b ∈ A, denoted by E(X, A, ϕ) or simply E(X, A), and it is called an abelian extension of X by A. The set of quandle 2-cocycles of X with coefficients in A is denoted by $Z_Q^2(X,A)$. Two cocycles ϕ₁ and ϕ₂ are cohomologous if there is a function γ : X → A such that

$${\varphi }_2(x,y)=-\gamma (x)+{\varphi }_1(x,y)+\gamma (x*y)$$

for any x, y ∈ X. The set of equivalence classes is a group and it is denoted by $H_Q^2(X,A)$. See [4] for more information on abelian extensions of quandles and [5–7] for more on quandle cohomology.

Example 2.6

In [1], extensions by constant 2-cocycles were defined as follows. For a quandle X and a set S, a constant quandle cocycle is a map

$$\beta :X\times X\to \text{Sym}(S),$$

where Sym(S) is the symmetric group on S, such that X × S has a quandle structure by (x, t) * (y, s) = (x * y, β_x,y(t)) for x, y ∈ X and s, t ∈ S (see [1] for details). This quandle is denoted by X ×_β S. The map β satisfies the constant cocycle condition β_x*y,zβ_x,y = β_x*z,y*zβ_x,z for any x, y, z ∈ X and the quandle condition β_x,x = id for any x ∈ X. Following [1], we also call these extensions non-abelian extensions.

Let X be a quandle. The right translation ℛ_a : X → X, by a ∈ X, is defined by ℛ_a(x) = x * a for x ∈ X. Then ℛ_a is a permutation of X by Axiom (2). The subgroup of Aut(X), the quandle automorphism group, generated by the permutations ℛ_a, a ∈ X, is called the inner automorphism group of X, and is denoted by Inn(X). A quandle is connected if Inn(X) acts transitively on X. A quandle is homogeneous if Aut(X) acts transitively on X. A quandle is faithful if the mapping φ : X → Inn(X) defined by φ(a) = ℛ_a is an injection from X to Inn(X). We note that abelian as well as non-abelian extensions are not faithful. The operation *¯ on X defined by $a\overline{*}b={ℛ}_b^{-1}(a)$ is a quandle operation, and (X, *¯) is called the dual quandle of (X, *). A quandle X is called a kei [31], or involutory, if (x * y) * y = x for all x, y ∈ X.

A coloring of an oriented knot diagram by a quandle X is a map 𝒞 from the set of arcs 𝒜 of the diagram to X such that the image of the map satisfies the relation depicted in Fig. 1 at each crossing. More details can be found in [8, 16], for example. A coloring that assigns the same element of X to all the arcs is called trivial, otherwise non-trivial. The number of colorings of a knot diagram by a finite quandle is known to be independent of the choice of diagram, and hence is a knot invariant. We denote by SCol_X(K) and Col_X(K) the set and the number of colorings of K by X.

Fig. 1.

Colored crossings and cocycle weights.

The fundamental quandle is defined in a manner similar to the fundamental group [21, 24]. A presentation of a quandle is defined in a manner similar to groups as well, and a presentation of the fundamental quandle is obtained from a knot diagram (see, for example, [18]), by assigning generators to arcs of a knot diagram, and relations corresponding to crossings. The set of colorings of a knot diagram K by a quandle X, then, is in one-to-one correspondence with the set of quandle homomorphisms from the fundamental quandle of K to X.

In this paper, all knots are oriented. Let m : 𝕊³ → 𝕊³ be an orientation reversing homeomorphism of the 3-sphere. For a knot K contained in 𝕊³, m(K) is the mirror image of K, and r(K) is the knot K with its orientation reversed. We regard m and r as maps on equivalence classes of knots. We consider the group 𝒢 = {1, r, m, rm} acting on the set of all oriented knots. For each knot K let 𝒢(K) = {K, r(K), m(K), rm(K)} be the orbit of K under the action of 𝒢.

For knots K and K′, we write K = K′ to denote that there is an orientation preserving homeomorphism of 𝕊³ that takes K to K′ preserving the orientations of K and K′. By a symmetry we mean that a knot (type) K remains unchanged under one of r, m, rm. As in the definition of symmetry type in [14] we say that a knot K is

• reversible if the only symmetry it has is K = r(K),

• negative amphicheiral if the only symmetry it has is K = rm(K),

• positive amphicheiral if the only symmetry it has is K = m(K),

• chiral if it has none of these symmetries,

• fully amphicheiral if K = r(K) = m(K) = rm(K), i.e. if K has all three symmetries.

The symmetry type of each knot on at most 12 crossings is given at [14]. Thus each of the 2977 knots K given there represents 1, 2 or 4 knots depending on the symmetry type. Among the 2977 knots, there are 1580 reversible, 47 negative amphicheiral, 1 positive amphicheiral, 1319 chiral, and 30 fully amphicheiral knots.

It is known [21, 24] that the fundamental quandles of K and K′ are isomorphic if and only if K = K′ or K = rm(K′).

Let X be a quandle, and ϕ be a 2-cocycle with coefficient group A, a finite abelian group; we use multiplicative notation. We regard ϕ as a function ϕ : X × X → A. For a coloring of a knot diagram by a quandle X as depicted in Fig. 1 at a positive (left) and negative (right) crossing, respectively, the pair (x_τ, y_τ) of colors assigned to a pair of nearby arcs is called the source colors. The third arc receives the color x_τ * y_τ.

The 2-cocycle (or cocycle, for short) invariant is an element of the group ring ℤ[A] defined by Φ_ϕ(K) = ∑_𝒞∏_τ ϕ(x_τ, y_τ)^ε(τ), where the product ranges over all crossings τ, the sum ranges over all colorings of a given knot diagram, (x_τ, y_τ) are source colors at the crossing τ, and ε(τ) is the sign of τ as specified in Fig. 1.

When ℤ_n is contained as a subgroup in ℤ_m and in $H_Q^2(X,{\mathbb{Z}}_m)$, and if a 2-cocycle ϕ : X × X → ℤ_n is such that [ϕ] is a generator of the subgroup ℤ_n in $H_Q^2(X,{\mathbb{Z}}_m)$, then we say that ϕ is a generating 2-cocycle of the subgroup ℤ_n.

Lemma 2.7

If the second homology group $H_2^Q(X,\mathbb{Z})$ for X satisfies $H_2^Q(X,\mathbb{Z})={\mathbb{Z}}_{n_1}\oplus {\mathbb{Z}}_{n_2}\oplus \cdots \oplus {\mathbb{Z}}_{n_k}$, n_i > 0 for all i, then we have

$$H_Q^2(X,{\mathbb{Z}}_n)\cong {\mathbb{Z}}_{n_1^{\prime }}\oplus {\mathbb{Z}}_{n_2^{\prime }}\oplus \cdots \oplus {\mathbb{Z}}_{n_k^{\prime }},$$

where $n_i^{\prime }=\text{gcd}(n_i,n)$.

Proof

It is known that $H_Q^2(X,A)$ is isomorphic to $\text{Ham}(H_2^Q(X,\mathbb{Z}),A)$ by the universal coefficient theorem and from the fact that $H_1^Q(X,\mathbb{Z})$ is torsion free [7].

The result follows from the standard facts

$$\text{Hom}(A_1\oplus A_2\oplus \cdots \oplus A_k,C)=\text{Hom}(A_1,C)\oplus \text{Hom}(A_2,C)\oplus \cdots \oplus \text{Hom}(A_k,C)$$

and Hom(ℤ_n, ℤ_m) ≅ ℤ_gcd(n,m), for positive integers n and m.

The groups $H_2^Q(X,\mathbb{Z})$ for some Rig quandles are found at [33]. Note that the groups given in [33] are rack homology $H_2^R(X,\mathbb{Z})$, and the relationship is given by $H_2^R(X,\mathbb{Z})\cong H_2^Q(X,\mathbb{Z})\oplus \mathbb{Z}$ [22].

The package Rig [33] includes cohomology groups, 2-cocycles, abelian extensions and cocycle invariants for some Rig quandles and some knots in the KnotInfo table [14]. Multiplication tables of Rig quandles, (co)homology groups, generating 2-cocycles, and abelian extensions of Rig quandles that we used for computations can be obtained online at the Wiki page of Rig: http://github.com/vendramin/rig/wiki.

3. Distinguishing the Unknot by Quandle Colorings

We recall the following conjecture of [11].

Conjecture 3.1

If K and K′ are any two knots such that K′ ≠ K and K′ ≠ rm(K) then there is a finite quandle X such that Col_X(K) ≠ Col_X(K′).

In this section, we prove this conjecture when K′ is the unknot. The idea is somewhat similar to that of Eisermann, see [16, Remark 59].

Proposition 3.1

Let K be a non-trivial knot. Then there exists a finite quandle X such that K admits a non-trivial coloring with X.

First, we recall the facts we need for the proof, see for example [16].

1. Papakyriakopoulos [28] proved that a knot is trivial if and only its longitude is trivial in the fundamental group of the complement of the knot, called the knot group, π₁(𝕊³\K).

2. The Wirtinger presentation of the knot group of an oriented knot K is defined as follows. Label the arcs x₁, x₂, …, x_n. At the end of the arc x_i−1 we undercross the arc x_k(i) and continue on arc x_i. Let ε(i) be the sign of the crossing as in Fig. 1. Then the knot group is

   $${\pi }_1({𝕊}^3\backslash K)\simeq \langle x_1,\dots ,x_n|r_1,\dots ,r_n\rangle ,$$

   where $r_i={x_{k(i)}}^{-\epsilon (i)}{x_{i-1}{x}_{k(i)}}^{\epsilon (i)}{x}_i^{-1}$ for all i.

3. The map ∂ : π₁(𝕊³\K) → ℤ given by ∂(x_i) = 1 for all i is a group homomorphism. By [3], Remark 3.13, the longitude l_K can be written as a word w on all the generators x₁, …, x_n with ∂(w) = 0.

4. Recall that a group G is residually finite if every non-trivial g ∈ G is mapped non-trivially into some finite quotient of G. As a consequence of [32] one obtains that every knot group is residually finite, see [20] for a proof.

Proof of Proposition 3.1

Since K is non-trivial, l_K ≠ 1. Since knot groups are residually finite, there exists a finite group G and a surjective group homomorphism f : π₁(𝕊³\K) → G such that f(l_K) ≠ 1. Then f maps the conjugacy class of x₁ into a non-trivial conjugacy class X of G. From this it follows that the knot K admits a non-trivial coloring with the conjugation quandle X.

4. Quandle Colorings of Composite Knots

In this section, we introduce the concept of end monochromatic, and show that if a knot K₁ or a knot K₂ is end monochromatic with a finite homogeneous quandle X, then |X|Col_X(K₁⋕K₂) = Col_X(K₁)Col_X(K₂).

A 1-tangle is a properly embedded arc in a 3-ball, and the equivalence of 1-tangles is defined by ambient isotopies of the 3-ball fixing the boundary (cf. [13]). A diagram of a 1-tangle is defined in a manner similar to a knot diagram, from a regular projection to a disk by specifying crossing information, see Fig. 2(a). An orientation of a 1-tangle is specified by an arrow on a diagram as depicted. A knot diagram is obtained from a 1-tangle diagram by closing the end points by a trivial arc outside of a disk. This procedure is called the closure of a 1-tangle. If a 1-tangle is oriented, then the closure inherits the orientation.

Fig. 2.

1-tangles.

A 1-tangle is obtained from a knot K as follows. Choose a base point b ∈ K and a small open neighborhood B of b in the 3-sphere 𝕊³ such that (B, K ∩ B) is a trivial ball-arc pair (so that K ∩ B is unknotted in B, see Fig. 2(b)). Then (𝕊³\Int(B), K ∩ (𝕊³\Int(B))) is a 1-tangle called the 1-tangle associated with K. The resulting 1-tangle does not depend on the choice of a base point. If a knot is oriented, then the corresponding 1-tangle inherits the orientation.

A quandle coloring of an oriented 1-tangle diagram is defined in a manner similar to those for knots. We do not require that the end points receive the same color for a quandle coloring of 1-tangle diagrams.

Definition 4.1

Let K be a 1-tangle diagram and X be a quandle. We say that (K, X) is end monochromatic, or K is end monochromatic with X, if any coloring of K by X assigns the same color on the two end points.

Two diagrams of the same 1-tangle are related by Reidemeister moves. The one-to-one correspondence of colorings under each Reidemeister move does not change the colors of the end points. Thus we have the following.

Lemma 4.2

The property of being end monochromatic for a 1-tangle corresponding to a knot K and a base point b does not depend on the choice of the base point b.

Thus, if a diagram of a 1-tangle corresponding to a knot K and some base point b is end monochromatic with X, then we say that a knot K is end monochromatic with X.

Lemma 4.3

Let X be a finite homogeneous quandle, x ∈ X, and Col_(X,x)(K, b) be the number of colorings of a diagram K by X such that the arc that contains the base point b receives the color x. Then

$${\text{Col}}_{(X,x)}(K,b)={\text{Col}}_X(K)/|X|$$

for any x ∈ X.

Proof

First we show that Col_(X,x)(K, b) = Col_(X,y)(K, b) for any x, y ∈ X. Let SCol_(X,x)(K, b) be the set of colorings 𝒞 such that 𝒞(α) = x, where α is the arc that contains b. Since X is homogeneous, there is an automorphism h of X such that h(x) = y. For any coloring 𝒞 ∈ SCol_(X,x)(K, b), h_⋕(𝒞) = h◦𝒞 satisfies h_⋕(𝒞)(α) = y, hence h induces a bijective map h_⋕ : SCol_(X,x)(K, b) → SCol_(X,y)(K, b). Then we have

$${\text{Col}}_X(K)=\sum _{y\in X}{\text{Col}}_{(X,y)}(K,b)=|X|{\text{Col}}_{(X,x)}(K,b)$$

for any x ∈ X.

The following lemma was stated and proved in [29] for the 3-element dihedral quandle Q(3, 1) (and dihedral quandles in [30]) and generalized by Nosaka [27]. The idea of proof is illustrated by Fig. 3, which was taken from [29].

Fig. 3.

End monochromatic tangle.

Lemma 4.4 (see [27])

If a quandle X is faithful, then for any knot K, (K, X) is end monochromatic.

Remark 4.5

There are many examples of knots K and quandles X where X is not faithful, but (K, X) is end monochromatic. For example, Q(8, 1), which is an abelian extension of Q(4, 1), is not faithful, but 5₁ and 8₅ are end monochromatic with Q(8, 1), where 5₁ has only trivial colorings, and 8₅ has non-trivial colorings with Q(8, 1). The smallest non-faithful quandle for which 3₁ is end monochromatic is Q(12, 1), which is an abelian extension of Q(6, 1).

In the following lemma, a formula is given for the number of colorings of composite knots. For a composite knot K₁⋕K₂, we assume that K₁ and K₂ are oriented, and the composite K₁⋕K₂ is defined in such a way that an orientation of the composite restricts to the orientation of each factor, and such an orientation is specified for the composite to make it an oriented knot, see Fig. 4.

Fig. 4.

Taking connected sum.

Lemma 4.6 (cf. [27, 29])

If a knot K₁ or a knot K₂ is end monochromatic with a finite homogeneous quandle X, then

$$|X|{\text{Col}}_X(K_1⋕K_2)={\text{Col}}_X(K_1){\text{Col}}_X(K_2).$$

Proof

Let b₁, b₂ be base points on diagrams of K₁ and K₂, respectively, with respect to which 1-tangles and connected sum are formed. Let x ∈ X. Let SCol_(X,x)(K_i, b_i), and Col_(X,x)(K_i, b_i), i = 1, 2, be the set and the number of colorings of K_i by X such that the arc that contains b_i receives the color x. Let c₁, c₂ be points on a diagram K = K₁⋕K₂ that result from taking a connected sum with respect to b₁ and b₂ by connecting 1-tangles, see Fig. 4.

For colorings 𝒞_i ∈ Col_(X,x)(K_i, b_i), i = 1, 2, a coloring 𝒞 = 𝒞₁⋕𝒞₂ of K is uniquely determined such that the colors of the arcs containing c_i, i = 1, 2, coincide and is x. Conversely, any coloring 𝒞 of K has the property that the color of the arcs containing c_i, i = 1, 2, coincide, since 𝒞 will also be a coloring of the tangles K₁ and K₂. If, say, K₁ is monochromatic with X then the colors of c₁ and c₂ must be the same. Hence there is a bijection

$$\underset{x\in X}{\cup }[\mathrm{S}{\text{Col}}_{(X,x)}(K_1,b_1)\times \mathrm{S}{\text{Col}}_{(X,x)}(K_2,b_2)]\to \mathrm{S}{\text{Col}}_X(K).$$

By Lemma 4.3, we have Col_(X,x)(K_i, b_i) = Col_X(K_i)/|X| for any x ∈ X, hence the left side above has the cardinality

$$|X|({\text{Col}}_X(K_1)/|X|)({\text{Col}}_X(K_2)/|X|),$$

as desired.

Lemmas 4.4 and 4.6 imply the following.

Lemma 4.7 (see [27])

If X is a finite faithful quandle, then

$$|X|{\text{Col}}_X(K_1⋕K_2)={\text{Col}}_X(K_1){\text{Col}}_X(K_2)$$

for knots K₁ and K₂.

Corollary 4.8

If X is a finite faithful quandle and R, K are knots, then

$${\text{Col}}_X(R⋕K)={\text{Col}}_X(R⋕rm(K)).$$

In particular, if X is a finite faithful quandle and K is reversible or positive-amphicheiral, respectively, then either Col_X(R⋕K) = Col_X (R⋕m(K)) or Col_X(R⋕K) = Col_X(R⋕r(K)).

Proof

By Lemma 4.7,

$${\text{Col}}_X(R⋕K)={\text{Col}}_X(R){\text{Col}}_X(K)/|X|={\text{Col}}_X(R){\text{Col}}_X(rm(K))/|X|={\text{Col}}_X(R⋕rm(K)).$$

This completes the proof.

According to this lemma, the situation of quandle colorings of composite knots may differ for non-faithful quandles, and indeed, the computer calculations reveal this. In the following sections we investigate these cases. We used the closed braid form for computer calculations of the number of quandle colorings as in [11]. In computing the number of colorings for composite knots, we formed the closed braid form as depicted in Fig. 5. In the braid notation of [14], an m-braid is represented by [a₁, …, a_s], a_i ∈ ℤ, where a_i represents the braid generator σ_k if a_i = k > 0, and ${\sigma }_k^{-1}$ if k < 0. The sign of a_i, sign(a_i), is defined to be 1 (−1, respectively), if k > 0 (resp. k < 0). If [a₁, …, a_s] ([b₁, …, b_t], respectively) is an m-braid (resp., n-braid) representative for a knot K (resp., K′), then

$$[a_1,\dots ,a_s,b_1+\text{sign}(b_1)(m-1),\dots ,b_t+\text{sign}(b_t)(m-1)]$$

is an (m + n − 1)-braid representative for K⋕K′. For example, for a trefoil 3₁, s = 3, m = 2, t = 3, n = 2, and [1, 1, 1, 2, 2, 2] is a (2+2−1)-braid representative of 3₁⋕3₁. The orientations of each factor and the composite are defined by downward orientation of the braid form. It is known [2] that for the braid index Br, the formula Br(K₁⋕K₂) = Br(K₁) + Br(K₂) − 1 holds.

Fig. 5.

The connected sum of two closed braids.

5. Distinguishing K from rm(K) via Colorings of Composite Knots

Since quandle colorings do not distinguish K from rm(K), they do not distinguish m(K) from r(K). Consequently, in [11], distinguishing K from m(K) by quandle colorings was examined only for chiral and negative-amphicheiral knots.

In this section, we exhibit computational results on distinguishing reversible and chiral knots K from rm(K) using quandle colorings of composite knots R⋕K and R⋕rm(K) for knots R and K.

Proposition 5.1

Conjecture 3.1 implies that for any knot K such that K ≠ f(K) for some f ∈ 𝒢, there is a finite quandle X and a prime knot P (with braid index 2) such that Col_X(P⋕K) ≠ Col_X(P⋕f(K)).

Proof

First, we observe that for any knots K₁ and K₂ and f ∈ 𝒢,

$$f(K_1⋕K_2)=f(K_1)⋕f(K_2),$$

and for any prime knot P and f ∈ 𝒢, f(P) is prime. Let K = P₁⋕⋯⋕P_n be the prime factorization of K. Then

$$f(K)=f(P_1)⋕\cdots ⋕f(P_n)$$

is the prime factorization of f(K). Let P be a prime knot such that P is not in 𝒢(P_i) for i = 1, …, n and P ≠ rm(P) (take, for example, a (2, n)-torus knot, that is, the closure of a 2-braid, of a large crossing number for P). Clearly P⋕K ≠ P⋕f(K). The prime factorization of rm(P⋕K) is

$$rm(P)⋕rm(P_1)⋕\cdots ⋕rm(P_n)$$

and by the definition of P we again have by uniqueness of prime factorization that rm(P⋕K) is not equal to P⋕f(K). By the conjecture it follows that there is a finite quandle X such that Col_X(P⋕K) ≠ Col_X(P⋕f(K))

As a corollary to the proof of Proposition 5.1, we obtain the following.

Corollary 5.2

For any knot K such that K ≠ f(K) for some f ∈ 𝒢, there exists a prime knot P such that the fundamental quandles of P⋕K and P⋕f(K) are not isomorphicitalic.

Recall from Corollary 4.8 that if X is a finite faithful quandle, then we cannot distinguish R⋕K from R⋕rm(K). Thus to apply this technique, we must use nonfaithful quandles.

Remark 5.3

For reversible or chiral prime knots K up to 12 crossings and up to braid index 4, among the Rig quandles E of order less than 36, only the quandles Q(24, 2) and Q(27, 14) distinguished R⋕K and R⋕m(K) for some closed 2-braids R by the condition

$${\text{Col}}_E(R⋕K)\ne {\text{Col}}_E(R⋕rm(K)).$$

We noticed that these are abelian extensions of Q(6, 2) and Q(9, 6) with coefficient groups ℤ₄ and ℤ₃, respectively. In the remainder of the section, we give an interpretation of this method in terms of the quandle cocycle invariant, and extend this method to quandles of order larger than 36. Corollary 4.8 and Proposition 7.1 partly explain why only abelian extensions worked for this purpose among Rig quandles. Remark 5.6 suggests why many abelian extensions do not work.

Let X be a quandle, A be a finite abelian group, and $\varphi \in Z_Q^2(X,A)$ be a 2-cocycle with coefficient group A. Let Φ_ϕ(K) = ∑_g∈A a_gg ∈ ℤ[A] be the cocycle invariant of a knot K. We write C_g(Φ_ϕ(K)) = a_g. In particular, C_e(Φ_ϕ(K)) ∈ ℤ denotes the coefficient of the identity element e ∈ A.

An examination of the proof of Theorem 4.1 in [4] reveals the following two lemmas. For convenience of the reader, we include a proof of Lemma 5.5.

Lemma 5.4 (see [4])

Let E be an abelian extension of X with respect to a 2-cocycle ϕ with coefficient group A. Let K be a knot that is end monochromatic with X. Then Col_E(K) = C_e(Φ_ϕ(K))|A|.

Lemma 5.5

Suppose (K, X) is end monochromatic, and E = E(X, A, ϕ) is an abelian extension of X. Then (K, E) is end monochromatic if and only if Φ_ϕ(K) = Col_X(K) e.

Proof

In [4], an interpretation of the cocycle invariant as an obstruction to extending a coloring of a knot diagram K by X to a coloring by the abelian extension E of X with respect to a 2-cocycle ϕ was given as follows. Let 𝒞 be a coloring of a 1-tangle S of K with initial and terminal end points b₀, b₁, respectively. Suppose (K, X) is end monochromatic, so that 𝒞(b₀) = 𝒞(b₁) = x₀ ∈ X. Let a₀ ∈ A and assign a color (x₀, a₀) ∈ E = X × A to the arc at b₀. By traveling along the diagram from b₀ to b₁, a color of S by E is defined inductively using colors by X; if an under-arc colored by (x, a) goes under an over-arc colored by (y, b) at a positive crossing, then the other under-arc receives a color (x*y, aϕ(x, y)). The color extends at negative crossing as well. Then the coloring thus extended to S has the color (x₀, a₀ d) at the arc at b₁, where d ∈ A is the contribution of the cocycle invariant d = ∏_τ ϕ(x_τ, y_τ)^ε_(τ) ∈ A. Thus, the coloring by X extends to that by E if and only if d is the identity element.

Remark 5.6

The examples mentioned in Remark 4.5 are explained by Lemma 5.5. Among Rig quandles of order less than 36, the following are abelian extensions and end monochromatic for all knots up to nine crossings:

$$Q(12,1),\hspace{1em}Q(20,3),\hspace{1em}Q(24,3),\hspace{1em}Q(24,4),\hspace{1em}Q(24,5),$$

$$Q(24,6),\hspace{1em}Q(24,14),\hspace{1em}Q(24,16),\hspace{1em}Q(24,17),\hspace{1em}Q(30,1),$$

$$Q(30,16),\hspace{1em}Q(32,5),\hspace{1em}Q(32,6),\hspace{1em}Q(32,7),\hspace{1em}Q(32,8).$$

Thus, we conjecture that this is the case for all knots. The corresponding quandle X for these abelian extensions E are found in [12], and they are, respectively:

$$Q(6,1),\hspace{1em}Q(10,1),\hspace{1em}Q(12,6),\hspace{1em}Q(12,5),\hspace{1em}Q(12,8),$$

$$Q(12,9),\hspace{1em}Q(12,7),\hspace{1em}Q(12,8),\hspace{1em}Q(12,8),\hspace{1em}Q(15,2),$$

$$Q(15,7),\hspace{1em}Q(16,4),\hspace{1em}Q(16,4),\hspace{1em}Q(16,5),\hspace{1em}Q(16,6).$$

Duplicates in the list of X are due to non-cohomologous 2-cocycles of the same quandle.

There are non-faithful quandles that are not abelian extensions, see Proposition 7.1, and we do not know any characterization of knots that are end monochromatic with such quandles. All prime knots up to nine crossings are end monochromatic with Q(30, 4).

Definition 5.7 (e.g. [6])

For an element a = ∑_h a_hh ∈ ℤ[A], the element ā = ∑_h a_hh⁻¹ ∈ ℤ[A] is called the conjugate of a.

Lemma 5.8 (see [6])

${\mathrm{\Phi }}_{\varphi }(K)=\overline{{\mathrm{\Phi }}_{\varphi }(rm(K))}$.

Definition 5.9

The value of the quandle cocycle invariant Φ_ϕ(K) of a knot K with respect to a 2-cocycle ϕ of a quandle X is called asymmetric if ${\mathrm{\Phi }}_{\varphi }(K)\ne \overline{{\mathrm{\Phi }}_{\varphi }(K)}$.

Corollary 5.10

If Φ_ϕ(K) is asymmetric, then K ≠ rm(K).

From the above corollary we can sometimes distinguish K from rm(K) using the cocycle invariant for some quandles.

Proposition 5.11 (see [27])

Let ϕ be a 2-cocycle of a finite homogeneous quandle X with coefficient group A. Suppose that K₁ or K₂ is end monochromatic with X. Then

$$|X|{\mathrm{\Phi }}_{\varphi }(K_1⋕K_2)={\mathrm{\Phi }}_{\varphi }(K_1){\mathrm{\Phi }}_{\varphi }(K_2).$$

The following corollary relates the condition

$${\text{Col}}_E(R⋕K)\ne {\text{Col}}_E(R⋕rm(K))$$

to Corollary 5.10 via asymmetry of the cocycle invariant.

Corollary 5.12

Let ϕ be a 2-cocycle of a finite connected faithful quandle X with coefficient group A. Assume that Φ_ϕ(R) = r_ee + r_uu for r_e, r_u ∈ ℕ, the identity element e, and a non-identity element u ∈ A, and that r_e = |X|, that is, any nontrivial coloring contribute u to the cocycle invariant. Suppose a knot K satisfies

$${\mathrm{\Phi }}_{\varphi }(K)=k_{e}e+k_{u}u+k_{u^{-1}}{u}^{-1}+V,$$

where V does not contain terms in e, u or u⁻¹. Then k_u ≠ k_u⁻¹ if and only if

$${\text{Col}}_E(R⋕K)\ne {\text{Col}}_E(R⋕rm(K)),$$

where E is the abelian extension of X by ϕ.

Proof

By Proposition 5.11,

$$C_e({\mathrm{\Phi }}_{\varphi }(R⋕K))=(r_e{k}_e+r_u{k}_{u^{-1}})/|X|,$$

$$C_e({\mathrm{\Phi }}_{\varphi }(R⋕rm(K)))=(r_e{k}_e+r_u{k}_u)/|X|.$$

By Lemma 5.4, k_u ≠ k_u⁻¹ if and only if

$${\text{Col}}_E(R⋕K)=|A|(r_e{k}_e+r_u{k}_{u^{-1}})/|X|\ne |A|(r_e{k}_e+r_u{k}_u)/|X|={\text{Col}}_E(R⋕rm(K)).$$

This completes the proof.

We note that often computing the number of colorings has computational advantage over applying Corollary 5.10 by computing the cocycle invariant, even though Corollary 5.12 theoretically derives the condition

$${\text{Col}}_E(R⋕K)\ne {\text{Col}}_E(R⋕rm(K))$$

from asymmetry of the cocycle invariant in many cases.

Example 5.13

Let X = Q(6, 2) and ϕ be a generating 2-cocycle over ℤ₄ such that the abelian extension of X with respect to ϕ is E = Q(24, 2). Let us take an example of R⋕K and R⋕rm(K) for a trefoil R = 3₁ and K = 6₁. It was found in [10] that there is a multiplicative generator u of ℤ₄ such that the trefoil has the cocycle invariant Φ_ϕ(3₁) = 6 + 24u for Q(6, 2). With the same 2-cocycle, it is computed that Φ_ϕ(K) = 6 + 24u⁻¹. By Corollary 5.12, Col_E(R⋕K) ≠ Col_E(R⋕rm(K)), where E = Q(24, 2). For a more complex knot K, however, it becomes difficult to compute the cocycle invariant, and easier to confirm the condition Col_E(R⋕K) ≠ Col_E(R⋕rm(K)), which then implies that K ≠ rm(K) and K has an asymmetric invariant value.

We summarize outcomes of the methods described in this section, i.e. using Corollary 5.10 and cocycle invariants, or by directly computing

$${\text{Col}}_E(R⋕K)\ne {\text{Col}}_E(R⋕rm(K)).$$

First we summarize our results for prime knots with nine crossings or less using the cocycle invariant. Among 84 knots in the table up to nine crossings, they are all reversible except:

• Fully amphicheiral knots: 4₁, 6₃, 8₃, 8₉, 8₁₂, 8₁₈.

• Negative amphicheiral knot: 8₁₇.

• Chiral knots: 9₃₂, 9₃₃.

The rest are 75 reversible knots. The colorings of 3₁⋕K and 3₁⋕rm(K) or the method described in Corollary 5.12 distinguished the following reversible knots from their mirrors.

• Using Q(24, 2), the following knots are distinguished from mirrors:

  3₁, 6₁, 7₄, 7₇, 8₁₁, 9₁, 9₂, 9₄, 9₆, 9₁₀, 9₁₁, 9₁₅, 9₁₇, 9₂₃, 9₂₉, 9₃₄, 9₃₅, 9₃₇, 9₃₈, 9₄₆, 9₄₇, 9₄₈.

• Using Q(27, 14), the following knots are distinguished from mirrors:

  3₁, 6₁, 7₄, 8₅, 8₁₅, 8₁₉, 8₂₁, 9₂, 9₄, 9₁₆, 9₁₇, 9₂₈, 9₂₉, 9₃₄, 9₃₈, 9₄₀.

Furthermore, computer calculations show that the following knots K in the KnotInfo table up to 12 crossings with braid index less that 4 have the property Col_E(3₁⋕K) ≠ Col_E(3₁⋕m(K)).

• Both E = Q(24, 2) and Q(27, 14) have this property for:

  10₅, 10₉, 10₁₁₂, 10₁₅₉, 12a₀₈₀₅, 12a₀₈₇₈, 12a₁₂₁₀, 12a₁₂₄₈, 12a₁₂₈₃, 12n₀₅₇₁, 12n₀₆₆₆, 12n₀₇₅₀, 12n₀₇₅₁.

• Only E = Q(24, 2) but not Q(27, 14) has this property for:

  11a₃₅₅, 12a₁₂₁₄, 12n₀₅₇₄, 12n₀₈₈₂.

• Only E = Q(27, 14) but not Q(24, 2) has this property for:

  10₆₄, 10₁₃₉, 10₁₄₁, 11a₃₃₈, 12a₁₂₁₂, 12n₀₆₀₄, 12n₀₈₅₀.

Remark 5.14

To distinguish more knots from their mirrors using the property Col_E(R⋕K) ≠ Col_E(R⋕m(K)) for some abelian extensions E and for some R, we further computed abelian extensions of some Rig quandles. We computed cohomology groups for some coefficient groups and found some 2-cocycles for Rig quandles up to order 23, and obtained 40 abelian extensions. This information is available online at http://github.com/vendramin/rig/wiki.

Let ℰ be this set of quandles. It is likely that there are other abelian extensions that are not in this list.

There are 168 chiral, reversible or positive amphicheiral knots with braid index less than 4 and crossing number at most 12. Of these, we computed that 144 knots have the property Col_E(R⋕K) ≠ Col_E(R⋕m(K)) with E ∈ ℰ and for R = 3₁, 5₁, or 9₁.

Remark 5.15

Reversible prime knots K, up to 12 crossings with braid index less than 4, distinguished from their mirror images by a quandle knot pair (X, R) are listed in Table 1. The table shows a quandle X, a knot R and knots K such that Col_X(R⋕K) ≠ Col_X(R⋕m(K)). We recall that Q(24, 2) and Q(27, 14) are also abelian extensions.

Table 1.

Some reversible prime knots K distinguished from their mirror images by a quandle knot pair (X,R).

X	R	K
E(Q(12, 3), ℤ5)	31	12n0472
E(Q(12, 3), ℤ5)	91	1046, 10127, 10155, 12n0466
E(Q(12, 3), ℤ10)	31	87, 810, 10116, 10143, 12a0576 12a1220, 12n0233, 12n0234, 12n0235 12n0570, 12n0722, 12n0830, 12n0887
E(Q(12, 3), ℤ10)	91	52, 102, 10100, 10125, 10152, 11a240 12a0835, 12a1203, 12a1222, 12n0242, 12n0467
E(Q(15, 5), ℤ5)	31	12n0888
E(Q(18, 11), ℤ6)	31	820, 1062, 12a0999, 12n0831
E(Q(20, 1), ℤ3)	51	12a1027, 12a1233, 12n0468, 12n0721
E(Q(20, 1), ℤ6)	51	51, 1047, 1048, 10157, 11a234 12a0869, 12a1114, 12a1176, 12a1199
E(Q(20, 2), ℤ3)	51	73, 12a0146, 12a0369, 12a0722, 12n0822
E(Q(20, 2), ℤ6)	51	62, 816, 93, 10126, 10161, 12a0838, 12a1246, 12a1250 12n0417, 12n0725, 12n0749, 12n0820, 12n0829
Q(24, 2)	31	31, 91, 96, 105, 109, 10112, 10159, 11a355 12a0805, 12a0878, 12a1210, 12a1214, 12a1248, 12a1283 12n0571, 12n0574, 12n0666, 12n0750, 12n0751, 12n0882
Q(27, 14)	31	85, 819, 821, 916, 1064, 10139, 10141, 11a338 12a1212, 12n0604, 12n0850
E(Q(30, 3), ℤ4)	31	12n0821

Remark 5.16

Chiral prime knots K, up to 12 crossings with braid index less than 4, distinguished from rm(K) by a quandle knot pair (X, R) are listed in Table 2. The table shows a quandle X, a knot R and knots K such that such that Col_X(R⋕K) ≠ Col_X(R⋕rm(K)).

Table 2.

Some chiral prime knots K distinguished from rm(K) by a quandle knot pair (X,R).

X	R	K
E(Q(12, 3), ℤ5)	31	10149, 12n0344, 12n0679, 12n0688
E(Q(12, 3), ℤ10)	31	12a0815, 12a0898, 12a0981, 12n0708
E(Q(12, 3), ℤ10)	91	12a1223, 12n0748
E(Q(18, 11), ℤ6)	31	1082, 12a1191, 12a1215, 12a1253, 12n0675
E(Q(20, 1), ℤ3)	51	10148, 12a1047, 12a1227
E(Q(20, 1), ℤ6)	51	12a0850, 12a0859, 12n0113, 12n0114, 12n0345
E(Q(20, 2), ℤ3)	51	12a1227, 12a1235, 12a1258
E(Q(20, 2), ℤ6)	51	12a0920, 12n0709
Q(24, 2)	31	10106, 12a0909, 12a0916, 12a1002, 12a1120, 12a1226 12a1255, 12n0640, 12n0767
Q(27, 14)	31	1085, 12a0864, 12a1219, 12a1221, 12n0674
E(Q(30, 3), ℤ4)	31	12a1011, 12a1051, 12n0191, 12n0684

6. Recovering Cocycle Invariants from Colorings

In this section, we obtain formulas for computing the cocycle invariant from the number of colorings for some cases. The formulas give computational advantage in many cases. To obtain formulas, however, one needs information on concrete non-trivial invariant values for a few knots.

Proposition 6.1

Let X, A, ϕ be as above. Suppose that X is end monochromatic with K. Suppose further that for an elemen υ ∈ A that is not the identity element e, there exists a knot R_υ such that Φ_ϕ(R_υ) = r_ee + r_υυ ∈ ℤ[A]. Then

$$C_{{\upsilon }^{-1}}({\mathrm{\Phi }}_{\varphi }(K))=\frac{1}{r_{\upsilon }|A|}(|X|{\text{Col}}_E(R_{\upsilon }⋕K)-r_e{\text{Col}}_E(K)).$$

Proof

By Proposition 5.11, we have |X|Φ_ϕ(R_υ⋕K) = Φ_ϕ(R_υ)Φ_ϕ(K). By assumption Φ_ϕ(R_υ)Φ_ϕ(K) = (r_ee + r_υυ)(∑_u∈A a_uu). The coefficient of the identity element in the left-hand side is r_ea_e + r_υa_υ⁻¹. Hence we obtain |X|C_e(Φ_ϕ(R_υ⋕K)) = r_ea_e + r_υa_υ⁻¹. Let E be the abelian extension of X with respect to ϕ. Then by Lemma 5.4, we have

$${\text{Col}}_E(R_{\upsilon }⋕K))=C_e({\mathrm{\Phi }}_{\varphi }(R_{\upsilon }⋕K))|A|$$

and Col_E(K) = a_e|A|. By substitution and solving for a_υ⁻¹, we obtain the lemma.

In the following examples, we focus on the Rig quandles of order up to 12 where the second cohomology group is non-trivial when the coefficient group is other than ℤ₂. When the coefficient group A is cyclic of order n, even though we write A = ℤ_n (a notation usually used for the additive group of integers modulo n), we specify a multiplicative generator u, so that A = 〈u〉 where u has order n, and write A multiplicatively.

Example 6.2

Let X = Q(6, 2) and ϕ be a generating 2-cocycle over A = ℤ₄ such that the abelian extension of X with respect to ϕ is E = Q(24, 2). Since X is faithful, any knot is end monochromatic with X.

The cocycle invariants of X = Q(6, 2) using this cocycle are given in the wiki page of Rig at http://github.com/vendramin/rig/wiki, for knots up to 10 crossings. Some of the results are shown in Table 3. Knots that are not listed have the trivial invariant value 6. We abbreviate the identity element in the remaining of the paper. For example, 6 + 24u means 6e + 24u for the identity element e. In particular, in order to use Proposition 6.1, we obtain the following invariant values:

$${\mathrm{\Phi }}_{\varphi }(3_1)=6+24u, {\mathrm{\Phi }}_{\varphi }(8_5)=30+24u^2, {\mathrm{\Phi }}_{\varphi }(9_1)=6+24u^3.$$

Proposition 6.1 implies that

$$C_u({\mathrm{\Phi }}_{\varphi }(K))=(1/(24·4))(6·{\text{Col}}_E(9_1⋕K)-6·{\text{Col}}_E(K)),$$

$$C_{u^2}({\mathrm{\Phi }}_{\varphi }(K))=(1/(24·4))(6·{\text{Col}}_E(8_5⋕K)-30·{\text{Col}}_E(K)),$$

$$C_{u^3}({\mathrm{\Phi }}_{\varphi }(K))=(1/(24·4)(6·{\text{Col}}_E(3_1⋕K)-6·{\text{Col}}_E(K)).$$

We also have

$$C_e({\mathrm{\Phi }}_{\varphi }(K))=(1/|A|){\text{Col}}_E(K)=(1/4){\text{Col}}_E(K)$$

from Lemma 5.4. Therefore we obtain

$${\mathrm{\Phi }}_{\varphi }(K)=\frac{1}{16}[4{\text{ Col}}_E(K)+({\text{Col}}_E(9_1⋕K)-{\text{Col}}_E(K))u+({\text{Col}}_E(8_5⋕K)-5{\text{ Col}}_E(K))u^2+({\text{Col}}_E(3_1⋕K)-{\text{Col}}_E(K))u^3].$$

See the appendix for examples of cocycles invariants computed using this formula.

Table 3.

Some cocycle invariants for the quandle Q(6, 2).

Cocycle invariant	Knot
6 +24u	31, 77, 811, 92, 94, 910, 911, 915
54 + 72u	935
6 +48u2	940
30 + 24u2	85, 810, 815, 819, 820, 821, 916, 924, 928
6 +24u3	61, 74, 91, 96, 917, 923, 929, 934, 938
6 +72u + 48u2	948
6 +48u + 48u3	818
6 +24u + 72u3	947
54 + 24u + 48u3	946
6 +48u + 48u2+ 24u3	937

Remark 6.3

In computing the coloring numbers of knots by quandles, some computational techniques have been developed in [11], such as fixing a color of the first braid strand to reduce the computation time. On the other hand, to compute the cocycle invariant, every coloring must be computed, and the cocycle value must be evaluated for each coloring. The latter increases the computational time significantly. Thus, the formula of Proposition 6.1 is useful in determining invariant values for higher crossing knots with lower braid indices.

Remark 6.4

There are discrepancies of representatives of knots and their mirrors in different notations in [14] for the following knots up to nine crossings: 7₇, 9₁₁, 9₁₇, 9₃₄, 9₄₆, 9₄₇, 9₄₈. Specifically, the diagram of 7₇ listed agrees with the braid notation, but its PD notation seems to represent its mirror. In our first computation up to nine crossings, we used the PD notation in [14], and the second computations for those with braid index less than 4 are performed using the braid notation. For up to nine crossings, these calculations showed discrepancies for the above listed knots. The discrepancies are all related by conjugate values of the invariant. We note that in the following computations, these knots are not used for R in Col_E(R⋕K) in the formulas.

Below we give a summary of the formula in Proposition 6.1 for Rig quandles of order up to 12, as examples to indicate how to use the formula, and to illustrate varieties of actual formulas obtained.

Example 6.5

Let X = Q(9, 6) = ℤ₃[t]/(t²+2t+1) and ϕ be a generating 2-cocycle over A = ℤ₃ such that the abelian extension of X with respect to ϕ is E = Q(27, 14). Since X is faithful, any knot is end monochromatic with X. Computer calculation shows that Φ_ϕ(3₁) = 27 + 54u, where u is a multiplicative generator of A and it also implies that Φ_ϕ(m(31)) = 27 + 54u². Proposition 6.1 implies that

$$C_u({\mathrm{\Phi }}_{\varphi }(K))=(1/(54·3))(9·{\text{Col}}_E(m(3_1)⋕K)-27·{\text{Col}}_E(K)),=(1/18)({\text{Col}}_E(m(3_1)⋕K)-3·{\text{Col}}_E(K)),$$

$$C_{u^2}({\mathrm{\Phi }}_{\varphi }(K))=(1/18)({\text{Col}}_E(3_1⋕K)-9·{\text{Col}}_E(K)).$$

Example 6.6

Let X = Q(12, 3). This quandle is not Alexander, not kei, not Latin, faithful, and $H_Q^2(X,A)={\mathbb{Z}}_{10}$ for A = ℤ₁₀. Let E be the abelian extension corresponding to a cocycle that represents a generator of ℤ₁₀. We obtain the following invariant values:

$${\mathrm{\Phi }}_{\varphi }(3_1)=12+60u,\hspace{1em}{\mathrm{\Phi }}_{\varphi }(8_{19})=12+60u^2,\hspace{1em}{\mathrm{\Phi }}_{\varphi }(5_2)=12+60u^3,$$

$${\mathrm{\Phi }}_{\varphi }(m(9_{29}))=12+60u^6, {\mathrm{\Phi }}_{\varphi }(5_1)=12+60u^5,\hspace{1em}{\mathrm{\Phi }}_{\varphi }(9_{29})=12+60u^6,$$

$${\mathrm{\Phi }}_{\varphi }(5_2)=12+60u^7,\hspace{1em}{\mathrm{\Phi }}_{\varphi }(m(8_{19}))=12+60u^8, {\mathrm{\Phi }}_{\varphi }(8_6)=12+60u^9.$$

One computes

$$C_u({\mathrm{\Phi }}_{\varphi }(K))=(1/(60·12))(12·{\text{Col}}_E(8_6⋕K)-12·{\text{Col}}_E(K))=(1/60)({\text{Col}}_E(8_6⋕K))-{\text{Col}}_E(K))$$

and the other terms are similar with the corresponding knots listed above. We note that the coefficient of every term is computed by these formulas, but we needed to compute the invariant for up to nine crossings for this conclusion, as u⁴ and u⁶ are missing up to eight crossing knots.

Example 6.7

Let X = Q(12, 5). This quandle is not Alexander, not kei, not Latin, faithful, and $H_Q^2(X,{\mathbb{Z}}_4)={\mathbb{Z}}_4$. With a choice of a generating cocycle ϕ, up to eight crossings, all knots have the cocycle invariant of the form Φ_ϕ(K) = a + bu², a, b ∈ ℤ. Thus we conjecture that this is the case for all knots. The trefoil has the invariant value Φ_ϕ(3₁) = 12 + 96u². Hence we obtain

$$C_{u^2}({\mathrm{\Phi }}_{\varphi }(K))=(1/(96·4))(12·{\text{Col}}_E(3_1⋕K)-12·{\text{Col}}_E(K))=(1/32)({\text{Col}}_E(3_1⋕K)-{\text{Col}}_E(K)).$$

If the conjecture does not hold and a knot with the term u or u³ is found, then it can be used to evaluate other terms.

Example 6.8

Let X = Q(12, 6). This quandle is not Alexander, not kei, not Latin, faithful, and $H_Q^2(X,{\mathbb{Z}}_4)={\mathbb{Z}}_4$. With a generating 2-cocycle ϕ of ℤ₄ the invariant values Φ_ϕ(K) for K up to nine crossing knots are listed in Table 4. Thus, we conjecture that the invariant values are of the form

$${\mathrm{\Phi }}_{\varphi }(K)=a+b{u}^2,$$

for a, b ∈ ℤ and for all knots K. One computes

$$C_{u^2}({\mathrm{\Phi }}_{\varphi }(K))=(1/(492·4))(12·{\text{Col}}_E(9_{35}⋕K)-12·{\text{Col}}_E(K))=(1/164)({\text{Col}}_E(9_{35}⋕K)-41·{\text{Col}}_E(K)).$$

We note that we needed to compute the invariant for knots up to nine crossings to obtain this formula.

Table 4.

Some cocycle invariants for Q(12, 6).

Cocycle invariant	Knot
108	31, 61, 74, 77, 811 91, 92, 94, 96, 910, 911, 915, 917, 923, 929, 934, 938
204	85, 810, 815, 819, 820, 821, 916, 924, 928
396	818, 940, 947
492 + 192u2	935, 937, 946, 948
12	otherwise

Remark 6.9

The second cohomology groups for Q(12, 7), Q(12, 9) with coefficient group ℤ₄ are ℤ₂ × ℤ₄ and ℤ₄ × ℤ₄, respectively, and for choices of generating cocycles, the cocycle invariants are non-trivial. Situations and computations are similar to those for Q(6, 2) and Q(12, 5) for each factor, for up to seven crossings.

Example 6.10

Let X = Q(12, 10). This quandle is not Alexander, not kei, not Latin, faithful, and $H_Q^2(X,{\mathbb{Z}}_6)={\mathbb{Z}}_6$. With a generating cocycle ϕ of ℤ₆, we obtain

$${\mathrm{\Phi }}_{\varphi }(3_1)=12+108u^3, {\mathrm{\Phi }}_{\varphi }(8_5)=120+216u^2, {\mathrm{\Phi }}_{\varphi }(8_{15})=120+216u^4.$$

Since we observed, up to eight crossings, one or more of the terms with u², u³ and u⁴ (and no terms of u or u⁵), we conjecture that it is the case for all knots. One computes

$$C_{u^2}({\mathrm{\Phi }}_{\varphi }(K))=(1/(108·6))(12·{\text{Col}}_E(3_1⋕K))-12·{\text{Col}}_E(K)),=(1/54)({\text{Col}}_E(3_1⋕K))-{\text{Col}}_E(K)),$$

$$C_{u^3}({\mathrm{\Phi }}_{\varphi }(K))=(1/(216·6))(12·{\text{Col}}_E(8_{15}⋕K))-120·{\text{Col}}_E(K)),=(1/108)({\text{Col}}_E(8_{15}⋕K))-12·{\text{Col}}_E(K)),$$

$$C_{u^4}({\mathrm{\Phi }}_{\varphi }(K))=(1/108)({\text{Col}}_E(8_5⋕K))-12·{\text{Col}}_E(K)).$$

Remark 6.11

The 2-cocycle invariants discussed in this section are derived from the following invariant: Let R₁, …, R_n be knots and X₁, …, X_m be finite quandles. Then an invariant is defined for a knot K by

$$\mathrm{C}{\mathrm{L}}_{X_1,\dots ,X_m,R_1,\dots ,R_n}(K)={[{\text{Col}}_{X_i}(R_j⋕K)]}_{i=1,\dots ,m,j=1,\dots ,n}.$$

It is, then, a natural question whether for any quandle 2-cocycle invariant Φ_ϕ(K), there is a sequence of knots R₁, …, R_n and quandles X₁, …, X_m such that Φ_ϕ(K) is derived from CL_{X1, …, Xm, R1, …, Rn}(K).

7. Properties of Abelian Extensions

Finding abelian extensions have, for example, the following applications: (1) nontriviality of the second cohomology group can be confirmed, (2) knots and their mirrors may be distinguished by colorings of composite knots as in Sec. 5, (3) they are useful in computing cocycle knot invariants via colorings as in Sec. 6.

We summarize our findings on extensions of Rig quandles in this section. Among the 790 Rig quandles of a order < 48 there are 66 non-faithful quandles. All but 8 are extensions by ℤ₂.

Proposition 7.1

Among the non-faithful Rig quandles (of order less than 48), Q(30, 4),Q(36, 58), and Q(45, 29) are the only quandles that are not abelian extensions.

Proof

Computations show that the only non-trivial quotient of Q(30, 4) is X = Q(10, 1). So it suffices to show that there is no abelian extension of X of order 30. We have $H_2^Q(X,\mathbb{Z})\cong {\mathbb{Z}}_2$ [33]. To get an abelian extension of X of order 30 we would have to have a non-trivial 2-cocycle X × X → ℤ₃ which would give an element of $H_Q^2(X,{\mathbb{Z}}_3)=\text{Hom}({\mathbb{Z}}_2,{\mathbb{Z}}_3)=0$, a contradiction.

The only non-trivial quotients of Q(36, 58) are Q(4, 1) and Q(12, 10). Since H₂(Q(4, 1)) ≅ ℤ₂, a similar argument implies that Q(36, 58) is not an abelian extension of Q(4, 1). We have H₂(Q(12, 10)) ≅ ℤ₆, and let f be a 2-cocycle that generates H²(Q(12, 10), ℤ₆) ≅ ℤ₆. Then 2f and 4f take values in ℤ₃, and computations show that the corresponding abelian extensions are both isomorphic to Q(36, 57). Since cohomologous cocycles give rise to isomorphic quandles, this implies that Q(36, 58) is not an abelian extension of Q(12, 10).

The only non-trivial quotient of Q(45, 29) is Q(15, 7). Since H₂(Q(15, 7)) ≅ ℤ₂, a similar argument implies that Q(45, 29) is not an abelian extension of Q(15, 7).

Then one checks by computer that all the other non-faithful Rig quandles are abelian extensions. We note that many cases satisfy the condition in Lemma 8.1 below.

In [1, Proposition 2.11], it was proved that if Y is a connected quandle and X = φ(Y) ⊂ Inn(Y), then each fiber has the same cardinality, and if S is a set with the same cardinality as a fiber, then there is a constant cocycle β : X × X → Sym(S) such that Y is isomorphic to X ×_β S.

Proposition 7.2

The quandles Q(30, 4),Q(36, 58), and Q(45, 29) are non-abelian extensions of the quandles Q(10, 1), Q(12, 10) and Q(15, 7), respectively, by constant 2-cocycles.

Proof

By calculation we see that the image of the mapping φ from Q(30, 4) (resp., Q(36, 58), Q(45, 29)) to its inner-automorphism group is isomorphic to Q(10, 1) (resp., Q(12, 10), Q(15, 7)). The claim follows from [1, Proposition 2.11],

We noticed that some non-cohomologous cocycles give isomorphic extensions, such as Q(36, 57) over Q(12, 10) as in the proof of Proposition 7.1. We also had the following observation from computer calculations.

Remark 7.3

Let X = Q(15, 2), which has cohomology group $H_Q^2(X,{\mathbb{Z}}_2)\cong {\mathbb{Z}}_2\times {\mathbb{Z}}_2$. Hence there are three 2-cocycles that are non-trivial and pairwise non-cohomologous. There are, however, only two non-isomorphic abelian extensions of X by ℤ₂, Q(30, 1) and Q(30, 5). Then calculations show that two non-cohomologous cocycles define the extension Q(30, 5). Similar examples are found for some 12 element quandles, see below.

Lemma 7.4

For abelian groups B and C and a quandle X, let

$${\varphi }_B:X\times X\to B\mathit{\text{ and }}{\varphi }_C:X\times X\to C$$

be 2-cocycles with abelian extensions E(X, B, ϕ_B) and E(X, C, ϕ_C), respectively. Then for A = B × C, ϕ = (ϕ_B, ϕ_C) : X × X → A is a 2-cocycle with abelian extension E(X,A, ϕ), and E(X, A, ϕ) is an abelian extension of E(X, B, ϕ_B) and E(X, C, ϕ_C).

Proof

Define ${\varphi }_C^{\prime }:E(X,B,{\varphi }_B)\times E(X,B,{\varphi }_B)\to C$ by

$${\varphi }_C^{\prime }((x_1,b_1),(x_2,b_2))={\varphi }_C(x_1,x_2).$$

Then ${\varphi }_C^{\prime }$ is a 2-cocycle of E(X, B, ϕ_B) with coefficient C.

Define f : E(X, A, ϕ) → E(X, B, ϕ_B) × C by f((x, (b, c)) = ((x, b), c), which is clearly bijective. Then one computes

$$f((x_1,(b_1,c_1))*(x_2,(b_2,c_2)))=f((x_1*x_2,(b_1,c_1)+\varphi (x_1,x_2)))=f((x_1*x_2,(b_1+{\varphi }_B(x_1,x_2),c_1+{\varphi }_C(x_1,x_2)))=((x_1*x_2,b_1+{\varphi }_B(x_1,x_2)),c_1+{\varphi }_C(x_1,x_2)),$$

and

$$f((x_1,(b_1,c_1)))*f((x_2,(b_2,c_2))=((x_1,b_1),c_1)*((x_2,b_2),c_2))=((x_1*x_2,b_1+{\varphi }_B(x_1,x_2)),c_1+{\varphi }_C^{\prime }((x_1,c_1),(x_2,c_2)))=((x_1*x_2,b_1+{\varphi }_B(x_1,x_2)),c_1+{\varphi }_C(x_1,x_2)),$$

as desired.

Similarly, we obtain the following.

Lemma 7.5

Let B and C be abelian groups and A = B × C, X be a quandle, and ϕ : X × X → A be a 2-cocycle with abelian extension E(X, A, ϕ). Further, let p_B and p_C be the projections from A onto B and C respectively. Then p_Bϕ : X × X → B is a 2-cocyle giving abelian extension E(X, B, p_Bϕ), and E(X, A, ϕ) is isomorphic to E(E(X, B, p_Bϕ), C, ϕ′), where ϕ′((x₁, b₁), (x₂, b₂)) = p_Cϕ(x₁, x₂) for (x₁, b₁), (x₂, b₂) ∈ X × B.

Lemma 7.5 is generalized as follows.

Proposition 7.6

Let X be a finite quandle, and $0\to C\stackrel{\iota }{\to }A\stackrel{p_B}{\to }B\to 0$ be an exact sequence of finite abelian groups. Let ϕ : X × X → A be a quandle 2-cocycle. Then E(X, A, ϕ) is an abelian extension of E(X, B, p_Bϕ) with coefficient group C.

Proof

Let s : B → A be a section of the map p_B, that is, p_Bs = id_B. Then

$$p_B(s(b_1+b_2)-s(b_1)-s(b_2))=0.$$

Thus, s(b₁ + b₂) − s(b₁) − s(b₂) lies in the kernel of p_B so we can write

$$s(b_1+b_2)-s(b_1)-s(b_2)=\iota \left(c\right)$$

for some c ∈ C. Let η : B × B → C be given by η(b₁, b₂) = c. Then p_B(a−sp_B(a)) = 0 and hence we can write a − sp_B(a) = ι(p_C(a)) where p_C : A → C. This yields

$$\iota p_C(a)+s{p}_B(a)=a$$

for all a ∈ A.

Define ϕ′ : E(X, B, p_Bϕ) × E(X, B, p_Bϕ) → C by

$$\varphi \prime ((x_1,b_1),(x_2,b_2))=p_C\varphi (x_1,x_2)-\eta (b_1,p_B\varphi (x_1,x_2))$$

for (x_i, b_i) ∈ E(X, B, p_Bϕ) = X × B, i = 1, 2. To show that ϕ′ is a 2-cocycle it suffices to show that E(E(X, B, p_Bϕ), C, ϕ′) is a quandle. For this it suffices to show that the mapping

$$f:E(E(X,B,p_B\varphi ),C,\varphi \prime )\to E(X,A,\varphi )$$

defined by f(((x, b), c)) = (x, s(b) + ι(c)) is a bijection and preserves the product. To show that f is a bijection, since the domain and codomain of f have the same cardinality, it suffices to show that f is a surjection. Given (x, a) ∈ X × A we see that

$$f((x,p_B(a)),p_C(a))=(x,s{p}_B(a)+\iota p_C(a))=(x,a).$$

Finally to show that f preservers the product we compute:

$$f(((x_1,b_1),c_1)*((x_2,b_2),c_2))=f(((x_1,b_1)*(x_2,b_2),c_1+\varphi \prime ((x_1,b_1),(x_2,b_2))))=f(((x_1*x_2,b_1+p_B\varphi (x_1,x_2)),c_1+p_C\varphi (x_1,x_2)-\eta (b_1,\varphi (x_1,x_2))))=(x_1*x_2,s(b_1+p_B\varphi (x_1,x_2))+\iota (c_1+p_C\varphi (x_1,x_2)-\eta (b_1,\varphi (x_1,x_2))))=(x_1*x_2,s(b_1)+s{p}_B\varphi (x_1,x_2)+\iota \eta (b_1,p_B\varphi (x_1,x_2))+\iota (c_1)+\iota p_C\varphi (x_1,x_2)-\iota \eta (b_1,p_B\varphi (x_1,x_2)))=(x_1*x_2,s(b_1)+\iota (c_1)+\varphi (x_1,x_2)),$$

and

$$f(((x_1,b_1),c_1))*f(((x_2,b_2),c_2))=(x_1,s(b_1)+\iota (c_1))*(x_2,s(b_2)+\iota (c_2))=(x_1*x_2,s(b_1)+\iota (c_1)+\varphi (x_1,x_2)),$$

as desired.

If we suppress the 2-cocycle in the notation E(X, A, ϕ) and write merely E(X, A) then the above Lemma 7.4 and Proposition 7.6 may be stated more simply.

Corollary 7.7

(i) If E(X, B) and E(X, C) are abelian extensions, then so is E(X, B × C), and

$$E(X,B\times C)=E(E(X,B),C).$$

(ii) If E(X, A) is a finite abelian extension of a quandle X and C is a subgroup of the finite abelian group A then

$$E(X,A)=E(E(X,A/C),C).$$

We note that if E(X, A) is connected, then E(X, A/C) is connected since the epimorphic image of a connected quandle is connected.

We examine some connected abelian extensions of Rig quandles of order up to 12. In the following, we use the notation $E\stackrel{n}{\to }X$ if E = E(X, ℤ_n, ϕ) for some 2-cocycle ϕ such that E is connected. $E_2\stackrel{m}{\Rightarrow }{E}_1\stackrel{d}{\Rightarrow }X$ if there is a short exact sequence 0 → ℤ_m → ℤ_n → ℤ_d → 0 such that ${\mathbb{Z}}_n\subset H_Q^2(X,{\mathbb{Z}}_n)$ and E₁, E₂ are corresponding extensions as in Proposition 7.6. In this case $E_2\stackrel{n}{\to }X$ where n = md. The notation $\varnothing \stackrel{1}{\to }X$ indicates that $H_Q^2(X,A)=0$ for any coefficient group A, and hence there is no non-trivial abelian extension. It is noted to the left when all quandles in question are keis.

$$\varnothing \stackrel{1}{\to }Q(8,1)\stackrel{2}{\to }Q(4,1)$$

$$(\text{Kei})\varnothing \stackrel{1}{\to }Q(24,1)\stackrel{2}{\to }Q(12,1)\stackrel{2}{\to }Q(6,1)$$

$$\varnothing \stackrel{1}{\to }Q(24,2)\stackrel{2}{\Rightarrow }Q(12,2)\stackrel{2}{\Rightarrow }Q(6,2)$$

$$(\text{Kei})\varnothing \stackrel{1}{\to }Q(27,1)\stackrel{3}{\to }Q(9,2)=Q(3,1)\times Q(3,1)$$

$$\varnothing \stackrel{1}{\to }Q(27,6)\stackrel{3}{\to }Q(9,3)={\mathbb{Z}}_3[t]/(t^2+1)$$

$$\varnothing \stackrel{1}{\to }Q(27,14)\stackrel{3}{\to }Q(9,6)={\mathbb{Z}}_3[t]/(t^2+2t+1)$$

$$\varnothing \stackrel{1}{\to }Q(24,8)=Q(3,1)\times Q(8,1)\stackrel{2}{\to }Q(12,4)=Q(3,1)\times Q(4,1).$$

In the following, we list abelian extensions of Rig quandles that contain quandles of order higher than 35. The notation Q(n, −) indicates that it is a quandle of order n > 35 and is not a Rig quandle. The notation ? → Q(n, −) indicates that we do not know if non-trivial abelian extension exists for the quandle Q(n, −) in question. Except for the quandle Q(120, −) in the third line, we have explicit quandle operation tables for the quandles appearing in the list and hence we can prove by computer that such quandles are connected.

$$?\to Q(120,-)\stackrel{6}{\to }Q(20,3)\stackrel{2}{\to }Q(10,1)$$

$$?\to Q(120,-)\stackrel{5}{\Rightarrow }Q(24,7)\stackrel{2}{\Rightarrow }Q(12,3)$$

$$?\to Q(120,-)\stackrel{2}{\Rightarrow }Q(60,-)\stackrel{5}{\Rightarrow }Q(12,3)$$

$$?\to Q(48,-)\stackrel{2}{\Rightarrow }Q(24,4)\stackrel{2}{\Rightarrow }Q(12,5)$$

$$?\to Q(48,-)\stackrel{2}{\Rightarrow }Q(24,3)\stackrel{2}{\Rightarrow }Q(12,6).$$

It is interesting to remark that all quandles appearing in the first and the last lines are keis.

These observations raise the following questions.

• What is a condition on cocycles for abelian, or non-abelian extensions to be connected?

In [1], a condition for an extension to be connected was given in terms of elements of the inner automorphism group

• Is there an infinite sequence of abelian extensions of connected quandles ⋯ → Q_n → ⋯ → Q₁?

We note that sequences of abelian extensions of connected quandles terminate as much as we were able to compute.

• Is any abelian extension of a finite kei a kei?

In relation to this question, below we observe a condition of 2-cocycles that give extensions that are keis.

Lemma 7.8

Let X be a kei, ϕ be a 2-cocycle with coefficient group A, and E be the abelian extension of X with respect to ϕ. Then E is a kei if and only if ϕ(x, y) + ϕ(x * y, y) = 0 ∈ A for any x, y ∈ X, in additive notation.

Proof

One computes, for any x, y ∈ X and a, b ∈ A,

$$[(x,a)*(y,b)]*(y,b)=(x*y,a+\varphi (x,y))*(y,b)=((x*y)*y,a+\varphi (x,y)+\varphi (x*y,y)).$$

For any x, y ∈ X and a, b ∈ A, the right-hand side is equal to (x, a) if and only if ϕ(x, y) + ϕ(x * y, y) = 0 for any x, y ∈ X.

Remark 7.9

Let X = Q(12, 7). Then $H_Q^2(X,{\mathbb{Z}}_4)\cong {\mathbb{Z}}_2\times {\mathbb{Z}}_4$. By computer calculation, there is a particular generating 2-cocycle of the ℤ₂-factor, $Q(24,15)\stackrel{2}{\to }Q(12,7)$. Notice that $H_Q^2(Q(24,15),{\mathbb{Z}}_4)\cong {\mathbb{Z}}_4$. We also note that there are epimorphisms Q(24, 14) → Q(12, 7) and Q(24, 18) → Q(12, 7), where$H_Q^2(Q(24,14),{\mathbb{Z}}_4)\cong {\mathbb{Z}}_4$ and $H_Q^2(Q(24,14),{\mathbb{Z}}_2)\cong {\mathbb{Z}}_2$. Hence there is a quandle of order 48 corresponding to the ℤ₄-factor of $H_Q^2(X,{\mathbb{Z}}_4)$, that has epimorphic image Q(24, 14) or Q(24, 18).

Remark 7.10

Let X = Q(12, 8). Then $H_Q^2(X,{\mathbb{Z}}_2)\cong {({\mathbb{Z}}_2)}^3$. There are three epimorphisms from Rig quandles of order less than 36:

$$Q(24,5)\to X, Q(24,16)\to X, Q(24,17)\to X$$

and their cohomology groups with A = ℤ₂ are (ℤ₂)³, (ℤ₂)², and (ℤ₂)², respectively. We note that there are 7 cocycles that are not cohomologous each other, yet there are only 3 extensions as in Remark 7.3.

Remark 7.11

Let X = Q(12, 9). Then $H_Q^2(X,{\mathbb{Z}}_4)\cong {\mathbb{Z}}_4\times {\mathbb{Z}}_4$. There are two extensions in Rig quandles of order less than 36:

$$Q(24,6)\to Q(12,9), Q(24,19)\to Q(12,9)$$

and with A = ℤ₄ their cohomology groups are ℤ₂ × ℤ₂ × ℤ₄ and ℤ₂ × ℤ₄, respectively. There are three cocycles that give order-two extensions, yet there are two extensions as in Remark 7.3.

Remark 7.12

Let X = Q(12, 10). Then $H_Q^2(X,{\mathbb{Z}}_6)\cong {\mathbb{Z}}_6$. There is one extension among Rig quandles of order less than 36, Q(24, 20) → Q(12, 10) and we have $H_Q^2(Q(24,20),{\mathbb{Z}}_3)\cong {\mathbb{Z}}_3\times {\mathbb{Z}}_3$. One of the order-3 cocycle corresponds to an extension of X of order 6.

8. Finding Extensions of Higher Order

We further investigated extensions among non-faithful quandles over Rig quandles. Extensions of some of the Rig quandles of order greater than 12 and less than 28 can be found in http://github.com/vendramin/rig/wiki. The computations of cocycles become difficult for quandles of order 28. Thus we take an approach of constructing non-faithful connected quandles and identify extensions as follows.

We considered Rig quandles of order less than 36. To find extensions of Rig quandles, we made a list 𝒩ℱ of 315 non-faithful connected generalized Alexander quandles with respect to pairs (G, f) for non-abelian groups G and f ∈ Aut(G) (see Sec. 2). We considered all groups or order n, 36 ≤ n < 128, and for n = 128, only the first 172 groups in GAP Small Groups library (the library contains all the 2328 groups of size 128). All possible automorphisms f ∈ Aut(G) were considered up to conjugacy. For example, there are 39 non-abelian groups of order 108 which give 74 connected non-faithful quandles of order 108.

Lemma 2.3 and Proposition 2.11 in [1] were used to determine abelian extensions and non-abelian extensions by constant cocycles among quandles in 𝒩ℱ over Rig quandles of order less than 36. Specifically, quotient quandles are computed, dynamical cocycles ([1, Lemma 2.3]) are computed, whether the cocycles are constant is determined, and whether the extensions are abelian is determined.

We note that most examples computed for abelian extensions are 2-fold epimorphisms, and observe the following.

Lemma 8.1

Let Y be a finite connected quandle of even order 2n, and assume that φ (Y) = X ⊂ Inn(Y) with |X| = n. Then Y is isomorphic to an abelian extension of X by ℤ₂.

Proof

As in Proposition 7.2, it follows from Proposition 2.11 of [1] that Y is isomorphic to an extension X ×_β S by a constant cocycle β, where a set S consists of two elements. Let S = {0, 1}, and we identify S with ℤ₂. Then Sym(S) consists of two elements, the identity and the transposition of 0 and 1. We define ϕ : X × X → ℤ₂ by ϕ(x, y) = 0 if β_x,y = id and ϕ(x, y) = 1 if β_x,y is the transposition. Then β_x,y(t) = t + ϕ(x, y) for t ∈ ℤ₂ and ϕ is a 2-cocycle.

Remark 8.2

Among Rig quandles of order less than 36, the following have 2-fold extensions among quandles in 𝒩ℱ.

$$Q(18,i), i=1,3,4,5,6,7,8,$$

$$Q(24,i), i=6,10,11,13,18,22,23,$$

$$Q(28,i), i=1,2,3,4,5,6,7,8,9.$$

Other than these, we found that Q(12, 3) has a five-fold abelian extension, and Q(15, 2) has a four-fold non-abelian extension in 𝒩ℱ. We remark that the five-fold extension of Q(12, 3) was predicted by Lemma 7.5, see the list in Sec. 7 for Q(12, 3). Thus this specific extension is found in 𝒩ℱ.

We observe the following generalization of Lemma 8.1. Let Y be a finite connected quandle, and let φ(Y) = X ⊂ Inn(Y) with |X| = n. It follows again from Proposition 2.11 of [1] that Y is isomorphic to an extension X ×_β S by a constant cocycle β.

Lemma 8.3

Let X and Y be as above. If |Y | = kn where k is a prime power, and the subgroup H_β of Sym(S) generated by {β_x,y | x, y ∈ X} is cyclic of order k, then Y is isomorphic to an abelian extension of X.

Proof

Since |Y | = kn, we have |S| = k. Since H_β is cyclic of order k and k is a prime power, it is generated by a k-cycle σ. We can identify S with ℤ_k in such a way that σ = (1, 2, …, k). Then σ(t) = 1+t for any t ∈ ℤ_k. Hence for any x, y ∈ X, β_x,y = σⁱ for some i ∈ ℤ_k, so that β_x,y(t) = t + ϕ(x, y) with ϕ(x, y) = i.

Remark 8.4

Although homology groups of Rig quandles have been computed in [33], as mentioned earlier, explicit 2-cocycles have not been computed for Rig quandles of order greater than 23. The above computations of extensions give rise to explicit 2-cocycles, and also may be used for computations of cocycle invariants as in Sec. 6. Furthermore, the computations identify the pairs (G, f) of generalized Alexander quandles that are abelian extensions of Rig quandles.

9. Problems, Questions and Conjectures

For convenience of the reader, we collect here questions, problems and conjectures discussed all over the text.

Problem 9.1

Compute explicit 2-cocycles and extensions of Rig quandles of order ≥ 24.

In Remark 5.6, we made the following conjecture.

Conjecture 9.1

Let X be one of the following quandles:

$$Q(12,1), Q(20,3), Q(24,3), Q(24,4), Q(24,5),$$

$$Q(24,6), Q(24,14), Q(24,16), Q(24,17), Q(30,1),$$

$$Q(30,16), Q(32,5), Q(32,6), Q(32,7), Q(32,8).$$

Then every knot K is end monochromatic with X.

In Examples 6.7, 6.8 and 6.10 we made the following conjectures.

Conjecture 9.2

Let X = Q(12, 5) and ϕ be the 2-cocycle choosen in Example 6.7. Then for each knot K the cocycle invariant Φ_ϕ is of the form Φ_ϕ(K) = a + bu², where a, b ∈ ℤ.

Conjecture 9.3

Let X = Q(12, 6) and ϕ be the 2-cocycle choosen in Example 6.8. Then for each knot K the cocycle invariant Φ_ϕ is of the form Φ_ϕ(K) = a + bu², where a, b ∈ ℤ.

Conjecture 9.4

Let X = Q(12, 10) and ϕ be the 2-cocycle choosen in Example 6.10. For each knot K write Φ_ϕ(K) = a + bu + cu² + du³ + eu⁴ + fu⁵, where a, b, c, d, e, f ∈ ℤ. Then b = f = 0 for all K.

In Sec. 6, we posed the following questions.

Question 9.2

What is a condition on cocycles for abelian, or non-abelian extensions to be connected?

Question 9.3

Is there an infinite sequence of abelian extensions of connected quandles ⋯→ Q_n → ⋯ → Q₁?

Question 9.4

Is any abelian extension of a finite kei a kei?

Appendix

Cocycle Invariants for Q(6, 2)

In this appendix we list the cocycle invariant Φ_ϕ(K) for the quandle X = Q(6, 2) and the 2-cocycle over ℤ₄ discussed in Example 6.2. The list is for all knots in [14] that have braid index 4 or less, and 12 crossings or less. Knots with only trivial colorings (the invariant value 6) are not listed. These values are computed using the formula described in Example 6.2 and programs similar to those in [11]. Note that if the 2-cocycle invariant below has the form a + bu + cu² + du³ where b ≠ d then by Lemma 4.3 each of the corresponding knots K satisfes K ≠ rm(K) (see Tables A.1–A.3).

Table A.1.

Some cocycle invariants for the quandle Q(6, 2) of the form a + bu for some a, b ∈ ℤ.

Cocycle invariant	Knot
54	1062, 1065, 10140, 10143, 10165 11a108, 11a109, 11a139, 11a157 11n85, 11n106, 11n118, 11n119 12a0290, 12a0375, 12a0390, 12a0571 12a0668, 12a0672, 12a0941, 12a0949 12a1184, 12a1191, 12a1207, 12a1215 12n0425, 12n0426, 12n0533, 12n0807 12n0811, 12n0812, 12n0831, 12n0868
198	1099
6 +24u	31, 811, 94, 910, 917, 934 105, 109, 1040, 10103, 10106 10136, 10146, 10158, 10159, 10163 11a73, 11a99, 11a146, 11a171, 11a175 11a176, 11a184, 11a196, 11a216, 11a239 11a248, 11a306, 11a346, 11a353, 11n13 11n14, 11n86, 11n98, 11n109 11n125, 11n137, 11n138, 11n158 12a0234, 12a0346, 12a0409, 12a0411 12a0422, 12a0509, 12a0519, 12a0523 12a0567, 12a0588, 12a0617, 12a0626, 12a0718 12a0723, 12a0878, 12a0894, 12a0904, 12a0907 12a0916, 12a0923, 12a0944 12a0986, 12a1002, 12a1025, 12a1029 12a1060, 12a1079, 12a1115, 12a1120 12a1136, 12a1170, 12a1177, 12a1180 12a1197, 12a1201, 12a1214, 12a1226 12a1247, 12a1248, 12a1262, 12a1270 12a1272, 12a1276, 12n0147, 12n0329 12n0369, 12n0377, 12n0409, 12n0413 12n0419, 12n0439, 12n0493, 12n0502 12n0543, 12n0597, 12n0653, 12n0655 12n0657, 12n0660, 12n0667, 12n0668 12n0752, 12n0767, 12n0782, 12n0803 12n0825, 12n0866, 12n0284
54 + 72u	12n0546
150 + 24u	11n126, 12n0440

Table A.2.

Some cocycle invariants for the quandle Q(6, 2) of the form a + bu + cu² for some a, b, c ∈ ℤ with c ≠ 0.

Cocycle invariant	Knot
6 +48u2	940, 1061, 1064, 1066 10139, 10141, 10142, 10144, 10164 11a106, 11a194, 11a223, 11a232 11a244, 11a338, 11a340, 11n87 11n104, 11n105, 11n107, 11n145 11n146, 11n173, 11n183, 11n184, 11n185 12a0428, 12a0670, 12a0737, 12a0739, 12a0855 12a0864, 12a0970, 12a1111, 12a1147, 12a1212 12a1219, 12a1221, 12n0483, 12n0484, 12n0536 12n0627, 12n0779
6 +48u + 96u2	12a0701, 12a0987
30 + 24u2	85, 810, 815, 819, 820, 821, 916, 924, 928 1076, 1077, 1082, 1084, 1085, 1087 11a71, 11a72, 11a245, 11a261 11a264, 11a305, 11a351 11n38, 11n121 12a0577, 12a0578, 12a0852 12a0861, 12a0930, 12a0979 12a0981, 12a0982, 12a0999 12a1000, 12a1059, 12a1061 12a1100, 12a1187, 12a1252 12a1253, 12a1261, 12a1284 12a1285, 12n0084, 12n0106 12n0107, 12n0290, 12n0291 12n0572, 12n0573, 12n0575 12n0576, 12n0577, 12n0578 12n0638, 12n0674, 12n0675 12n0700, 12n0753, 12n0833 12n0845, 12n0850
30 + 168u2	12n0604
54 + 144u2	12n0508
54 + 48u + 48u2	12a0742, 12n0380
78 + 48u + 24u2	12a0574, 12n0571, 12n0574
102 + 96u2	12n0518
126 + 72u2	12a0647, 12n0605
150 + 48u2	12a1288, 12n0888

Table A.3.

Some cocycle invariants for the quandle Q(6, 2) of the form a + bu + cu² + du³ for some a, b, c, d ∈ ℤ with d ≠ 0.

Cocycle invariant	Knot
6 + 24u3	61, 74, 77, 91, 96, 911, 923, 929, 938 1014, 1019, 1021, 1032 10108, 10112, 10113, 10114 10122, 10145, 10147, 10160 11a179, 11a203, 11a236, 11a274 11a286, 11a300, 11a318, 11a335 11a355, 11a365, 11n65, 11n66, 11n92 11n94, 11n95, 11n99, 11n122, 11n136 11n143, 11n148, 11n149, 11n153, 11n176 11n182, 12a0236, 12a0321, 12a0496 12a0580, 12a0762, 12a0805 12a0806, 12a0807, 12a0809 12a0876, 12a0909, 12a0952 12a0972, 12a1036, 12a1091 12a1101, 12a1129, 12a1157 12a1196, 12a1200, 12a1210 12a1216, 12a1224, 12a1237 12a1239, 12a1255, 12n0330 12n0368, 12n0375, 12n0412 12n0438, 12n0441, 12n0443 12n0464, 12n0500, 12n0603 12n0640, 12n0641, 12n0717 12n0738, 12n0740, 12n0750 12n0751, 12n0754, 12n0769 12n0770, 12n0781, 12n0791 12n0823, 12n0832, 12n0836 12n0865, 12n0874, 12n0875 12n0882
6 +48u2+ 72u3	948, 11a293, 12a0895
6 + 144u2+ 24u3	1098
6 + 24u + 72u3	12n0666
6 +24u + 48u2+ 48u3	11n164, 12n0402
6 +24u + 96u2+ 24u3	11n167
6 + 48u + 48u3	818, 12a1260, 12n0403
6 +48u + 48u2+ 24u3	12n0565
6 + 72u + 24u3	947, 12n0549
30 + 120u2+ 24u3	12n0737
30 + 24u + 72u2+ 24u3	12a0576, 12n0570
54 + 72u3	11a314
54 + 48u2+ 48u3	11a332, 12n0386
54 + 24u + 48u2+ 24u3	12a0297, 12n0379
54 + 48u + 24u3	946, 11a291, 12n0567
78 + 24u2+ 48u3	12a1283
78 + 24u + 24u2+ 24u3	12n0883
78 + 48u + 72u2+ 48u3	11a44, 11a47, 11a57 11a231, 11a263, 11n71 11n72, 11n73, 11n74 11n75, 11n76, 11n77 11n78, 11n81 12a0167, 12a0692, 12a0801
102 + 48u3	12n0806
102 + 24u + 24u3	11a277, 12a1225