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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
 
J Knot Theory Ramif. Author manuscript; available in PMC 2017 April 1.
Published in final edited form as:
J Knot Theory Ramif. 2016 April; 25(5): 1650024.
Published online 2016 March 1. doi:  10.1142/S0218216516500243
PMCID: PMC4918820
NIHMSID: NIHMS786302

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.

Keywords: Quandle, colorings, cocycle invariants, abelian extensions, composite knots

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, 2527]. 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 Krm(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) [mapsto] a * b satisfying the following conditions:

  1. For any a [set membership] X, a * a = a.
  2. For any b, c [set membership] X, there is a unique a [set membership] X such that a * b = c.
  3. For any a, b, c [set membership] X, we have (a * b) * c = (a * c) * (b * c).

A quandle homomorphism between two quandles X, Y is a map f : XY 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 [set membership] 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′ [set membership] X and y, y′ [set membership] Y.

Example 2.4 (see [21])

A generalized Alexander quandle is defined by a pair (G, f) where G is a group, f [set membership] 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 × XA for an abelian group A is called a quandle 2-cocycle [5] if it satisfies

ϕ(x,y)ϕ(x,z)+ϕ(x*y,z)ϕ(x*z,y*z)=0

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

(x,a)*(y,b)=(x*y,a+ϕ(x,y))

for x, y [set membership] X, a, b [set membership] 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 ZQ2(X,A). Two cocycles ϕ1 and ϕ2 are cohomologous if there is a function γ : XA such that

ϕ2(x,y)=γ(x)+ϕ1(x,y)+γ(x*y)

for any x, y [set membership] X. The set of equivalence classes is a group and it is denoted by HQ2(X,A). See [4] for more information on abelian extensions of quandles and [57] 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

β:X×XSym(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, βx,y(t)) for x, y [set membership] X and s, t [set membership] S (see [1] for details). This quandle is denoted by X ×β S. The map β satisfies the constant cocycle condition βx*y,zβx,y = βx*z,y*zβx,z for any x, y, z [set membership] X and the quandle condition βx,x = id for any x [set membership] X. Following [1], we also call these extensions non-abelian extensions.

Let X be a quandle. The right translation Ra : XX, by a [set membership] X, is defined by Ra(x) = x * a for x [set membership] X. Then Ra is a permutation of X by Axiom (2). The subgroup of Aut(X), the quandle automorphism group, generated by the permutations Ra, a [set membership] 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 [var phi] : X → Inn(X) defined by [var phi](a) = Ra 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*¯b=b1(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 [set membership] X.

A coloring of an oriented knot diagram by a quandle X is a map C from the set of arcs [mathematical script 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 SColX(K) and ColX(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 : S3S3 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 × XA. 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 (xτ, yτ) of colors assigned to a pair of nearby arcs is called the source colors. The third arc receives the color xτ * yτ.

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

When Zn is contained as a subgroup in Zm and in HQ2(X,m), and if a 2-cocycle ϕ : X × XZn is such that [ϕ] is a generator of the subgroup Zn in HQ2(X,m), then we say that ϕ is a generating 2-cocycle of the subgroup Zn.

Lemma 2.7

If the second homology group H2Q(X,) for X satisfies H2Q(X,)=n1n2nk, ni > 0 for all i, then we have

HQ2(X,n)n1n2nk,

where ni=gcd(ni,n).

Proof

It is known that HQ2(X,A) is isomorphic to Ham(H2Q(X,),A) by the universal coefficient theorem and from the fact that H1Q(X,) is torsion free [7].

The result follows from the standard facts

Hom(A1A2Ak,C)=Hom(A1,C)Hom(A2,C)Hom(Ak,C)

and Hom(Zn, Zm) [congruent with] Zgcd(n,m), for positive integers n and m.

The groups H2Q(X,) for some Rig quandles are found at [33]. Note that the groups given in [33] are rack homology H2R(X,), and the relationship is given by H2R(X,)H2Q(X,) [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 ColX(K) ≠ ColX(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, …, xn. At the end of the arc xi−1 we undercross the arc xk(i) and continue on arc xi. Let ε(i) be the sign of the crossing as in Fig. 1. Then the knot group is
    π1(𝕊3\K)x1,,xn|r1,,rn,
    where ri=xk(i)ε(i)xi1xk(i)ε(i)xi1 for all i.
  3. The map [partial differential] : π1(S3\K) → Z given by [partial differential](xi) = 1 for all i is a group homomorphism. By [3], Remark 3.13, the longitude lK can be written as a word w on all the generators x1, …, xn with [partial differential](w) = 0.
  4. Recall that a group G is residually finite if every non-trivial g [set membership] 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, lK ≠ 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(lK) ≠ 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|ColX(K1[equal or parallel]K2) = ColX(K1)ColX(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 [set membership] K and a small open neighborhood B of b in the 3-sphere S3 such that (B, KB) is a trivial ball-arc pair (so that KB 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 [set membership] 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

Col(X,x)(K,b)=ColX(K)/|X|

for any x [set membership] X.

Proof

First we show that Col(X,x)(K, b) = Col(X,y)(K, b) for any x, y [set membership] 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 [set membership] SCol(X,x)(K, b), h[equal or parallel](C) = h*C satisfies h[equal or parallel](C)(α) = y, hence h induces a bijective map h[equal or parallel] : SCol(X,x)(K, b) → SCol(X,y)(K, b). Then we have

ColX(K)=yXCol(X,y)(K,b)=|X|Col(X,x)(K,b)

for any x [set membership] 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[equal or parallel]K2, we assume that K1 and K2 are oriented, and the composite K1[equal or parallel]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|ColX(K1K2)=ColX(K1)ColX(K2).

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 [set membership] X. Let SCol(X,x)(Ki, bi), and Col(X,x)(Ki, bi), i = 1, 2, be the set and the number of colorings of Ki by X such that the arc that contains bi receives the color x. Let c1, c2 be points on a diagram K = K1[equal or parallel]K2 that result from taking a connected sum with respect to b1 and b2 by connecting 1-tangles, see Fig. 4.

For colorings Ci [set membership] Col(X,x)(Ki, bi), i = 1, 2, a coloring C = C1[equal or parallel]C2 of K is uniquely determined such that the colors of the arcs containing ci, i = 1, 2, coincide and is x. Conversely, any coloring C of K has the property that the color of the arcs containing ci, 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

xX[SCol(X,x)(K1,b1)×SCol(X,x)(K2,b2)]SColX(K).

By Lemma 4.3, we have Col(X,x)(Ki, bi) = ColX(Ki)/|X| for any x [set membership] X, hence the left side above has the cardinality

|X|(ColX(K1)/|X|)(ColX(K2)/|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|ColX(K1K2)=ColX(K1)ColX(K2)

for knots K1 and K2.

Corollary 4.8

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

ColX(RK)=ColX(Rrm(K)).

In particular, if X is a finite faithful quandle and K is reversible or positive-amphicheiral, respectively, then either ColX(R[equal or parallel]K) = ColX (R[equal or parallel]m(K)) or ColX(R[equal or parallel]K) = ColX(R[equal or parallel]r(K)).

Proof

By Lemma 4.7,

ColX(RK)=ColX(R)ColX(K)/|X|=ColX(R)ColX(rm(K))/|X|=ColX(Rrm(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, …, as], ai [set membership] Z, where ai represents the braid generator σk if ai = k > 0, and σk1 if k < 0. The sign of ai, sign(ai), is defined to be 1 (−1, respectively), if k > 0 (resp. k < 0). If [a1, …, as] ([b1, …, bt], respectively) is an m-braid (resp., n-braid) representative for a knot K (resp., K′), then

[a1,,as,b1+sign(b1)(m1),,bt+sign(bt)(m1)]

is an (m + n − 1)-braid representative for K[equal or parallel]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[equal or parallel]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[equal or parallel]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[equal or parallel]K and R[equal or parallel]rm(K) for knots R and K.

Proposition 5.1

Conjecture 3.1 implies that for any knot K such that Kf(K) for some f [set membership] G, there is a finite quandle X and a prime knot P (with braid index 2) such that ColX(P[equal or parallel]K) ≠ ColX(P[equal or parallel]f(K)).

Proof

First, we observe that for any knots K1 and K2 and f [set membership] G,

f(K1K2)=f(K1)f(K2),

and for any prime knot P and f [set membership] G, f(P) is prime. Let K = P1[equal or parallel](...)[equal or parallel]Pn be the prime factorization of K. Then

f(K)=f(P1)f(Pn)

is the prime factorization of f(K). Let P be a prime knot such that P is not in G(Pi) for i = 1, …, n and Prm(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[equal or parallel]KP[equal or parallel]f(K). The prime factorization of rm(P[equal or parallel]K) is

rm(P)rm(P1)rm(Pn)

and by the definition of P we again have by uniqueness of prime factorization that rm(P[equal or parallel]K) is not equal to P[equal or parallel]f(K). By the conjecture it follows that there is a finite quandle X such that ColX(P[equal or parallel]K) ≠ ColX(P[equal or parallel]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 Kf(K) for some f [set membership] G, there exists a prime knot P such that the fundamental quandles of P[equal or parallel]K and P[equal or parallel]f(K) are not isomorphicitalic.

Recall from Corollary 4.8 that if X is a finite faithful quandle, then we cannot distinguish R[equal or parallel]K from R[equal or parallel]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[equal or parallel]K and R[equal or parallel]m(K) for some closed 2-braids R by the condition

ColE(RK)ColE(Rrm(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 ϕZQ2(X,A) be a 2-cocycle with coefficient group A. Let Φϕ(K) = ∑g[set membership]A agg [set membership] Z[A] be the cocycle invariant of a knot K. We write Cgϕ(K)) = ag. In particular, Ceϕ(K)) [set membership] Z denotes the coefficient of the identity element e [set membership] 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 ColE(K) = Ceϕ(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 Φϕ(K) = ColX(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 [set membership] X. Let a0 [set membership] A and assign a color (x0, a0) [set membership] 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 [set membership] A is the contribution of the cocycle invariant d = [product]τ ϕ(xτ, yτ)ε(τ) [set membership] 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),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).

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),Q(10,1),Q(12,6),Q(12,5),Q(12,8),

Q(12,9),Q(12,7),Q(12,8),Q(12,8),Q(15,2),

Q(15,7),Q(16,4),Q(16,4),Q(16,5),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 ahh [set membership] Z[A], the element ā = ∑h ahh−1 [set membership] Z[A] is called the conjugate of a.

Lemma 5.8 (see [6])

Φϕ(K)=Φϕ(rm(K))¯.

Definition 5.9

The value of the quandle cocycle invariant Φϕ(K) of a knot K with respect to a 2-cocycle ϕ of a quandle X is called asymmetric if Φϕ(K)Φϕ(K)¯.

Corollary 5.10

If Φϕ(K) is asymmetric, then Krm(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|Φϕ(K1K2)=Φϕ(K1)Φϕ(K2).

The following corollary relates the condition

ColE(RK)ColE(Rrm(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 Φϕ(R) = ree + ruu for re, ru [set membership] N, the identity element e, and a non-identity element u [set membership] A, and that re = |X|, that is, any nontrivial coloring contribute u to the cocycle invariant. Suppose a knot K satisfies

Φϕ(K)=kee+kuu+ku1u1+V,

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

ColE(RK)ColE(Rrm(K)),

where E is the abelian extension of X by ϕ.

Proof

By Proposition 5.11,

Ce(Φϕ(RK))=(reke+ruku1)/|X|,

Ce(Φϕ(Rrm(K)))=(reke+ruku)/|X|.

By Lemma 5.4, kuku−1 if and only if

ColE(RK)=|A|(reke+ruku1)/|X||A|(reke+ruku)/|X|=ColE(Rrm(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

ColE(RK)ColE(Rrm(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[equal or parallel]K and R[equal or parallel]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 Φϕ(31) = 6 + 24u for Q(6, 2). With the same 2-cocycle, it is computed that Φϕ(K) = 6 + 24u−1. By Corollary 5.12, ColE(R[equal or parallel]K) ≠ ColE(R[equal or parallel]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 ColE(R[equal or parallel]K) ≠ ColE(R[equal or parallel]rm(K)), which then implies that Krm(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

ColE(RK)ColE(Rrm(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[equal or parallel]K and 31[equal or parallel]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 ColE(31[equal or parallel]K) ≠ ColE(31[equal or parallel]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 ColE(R[equal or parallel]K) ≠ ColE(R[equal or parallel]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 x2130 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 ColE(R[equal or parallel]K) ≠ ColE(R[equal or parallel]m(K)) with E [set membership] x2130 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 ColX(R[equal or parallel]K) ≠ ColX(R[equal or parallel]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).

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 ColX(R[equal or parallel]K) ≠ ColX(R[equal or parallel]rm(K)).

Table 2
Some chiral prime knots K distinguished from rm(K) by a quandle knot pair (X,R).

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 [set membership] A that is not the identity element e, there exists a knot Rυ such that Φϕ(Rυ) = ree + rυυ [set membership] Z[A]. Then

Cυ1(Φϕ(K))=1rυ|A|(|X|ColE(RυK)reColE(K)).

Proof

By Proposition 5.11, we have |Xϕ(Rυ[equal or parallel]K) = Φϕ(Rυϕ(K). By assumption Φϕ(Rυϕ(K) = (ree + rυυ)(∑u[set membership]A auu). The coefficient of the identity element in the left-hand side is reae + rυaυ−1. Hence we obtain |X|Ceϕ(Rυ[equal or parallel]K)) = reae + rυaυ−1. Let E be the abelian extension of X with respect to ϕ. Then by Lemma 5.4, we have

ColE(RυK))=Ce(Φϕ(RυK))|A|

and ColE(K) = ae|A|. By substitution and solving for aυ−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 = Zn (a notation usually used for the additive group of integers modulo n), we specify a multiplicative generator u, so that A = left angle bracketuright angle bracket 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:

Φϕ(31)=6+24u,  Φϕ(85)=30+24u2,  Φϕ(91)=6+24u3.

Proposition 6.1 implies that

Cu(Φϕ(K))=(1/(24·4))(6·ColE(91K)6·ColE(K)),

Cu2(Φϕ(K))=(1/(24·4))(6·ColE(85K)30·ColE(K)),

Cu3(Φϕ(K))=(1/(24·4)(6·ColE(31K)6·ColE(K)).

We also have

Ce(Φϕ(K))=(1/|A|)ColE(K)=(1/4)ColE(K)

from Lemma 5.4. Therefore we obtain

Φϕ(K)=116[4 ColE(K)+(ColE(91K)ColE(K))u+(ColE(85K)5 ColE(K))u2+(ColE(31K)ColE(K))u3].

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

Table 3
Some cocycle invariants for the quandle Q(6, 2).

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 ColE(R[equal or parallel]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 Φϕ(31) = 27 + 54u, where u is a multiplicative generator of A and it also implies that Φϕ(m(31)) = 27 + 54u2. Proposition 6.1 implies that

Cu(Φϕ(K))=(1/(54·3))(9·ColE(m(31)K)27·ColE(K)),=(1/18)(ColE(m(31)K)3·ColE(K)),

Cu2(Φϕ(K))=(1/18)(ColE(31K)9·ColE(K)).

Example 6.6

Let X = Q(12, 3). This quandle is not Alexander, not kei, not Latin, faithful, and HQ2(X,A)=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:

Φϕ(31)=12+60u,Φϕ(819)=12+60u2,Φϕ(52)=12+60u3,

Φϕ(m(929))=12+60u6,  Φϕ(51)=12+60u5,Φϕ(929)=12+60u6,

Φϕ(52)=12+60u7,Φϕ(m(819))=12+60u8,  Φϕ(86)=12+60u9.

One computes

Cu(Φϕ(K))=(1/(60·12))(12·ColE(86K)12·ColE(K))=(1/60)(ColE(86K))ColE(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 HQ2(X,4)=4. With a choice of a generating cocycle ϕ, up to eight crossings, all knots have the cocycle invariant of the form Φϕ(K) = a + bu2, a, b [set membership] Z. Thus we conjecture that this is the case for all knots. The trefoil has the invariant value Φϕ(31) = 12 + 96u2. Hence we obtain

Cu2(Φϕ(K))=(1/(96·4))(12·ColE(31K)12·ColE(K))=(1/32)(ColE(31K)ColE(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 HQ2(X,4)=4. With a generating 2-cocycle ϕ of Z4 the invariant values Φϕ(K) for K up to nine crossing knots are listed in Table 4. Thus, we conjecture that the invariant values are of the form

Φϕ(K)=a+bu2,

for a, b [set membership] Z and for all knots K. One computes

Cu2(Φϕ(K))=(1/(492·4))(12·ColE(935K)12·ColE(K))=(1/164)(ColE(935K)41·ColE(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).

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 HQ2(X,6)=6. With a generating cocycle ϕ of Z6, we obtain

Φϕ(31)=12+108u3,  Φϕ(85)=120+216u2,  Φϕ(815)=120+216u4.

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

Cu2(Φϕ(K))=(1/(108·6))(12·ColE(31K))12·ColE(K)),=(1/54)(ColE(31K))ColE(K)),

Cu3(Φϕ(K))=(1/(216·6))(12·ColE(815K))120·ColE(K)),=(1/108)(ColE(815K))12·ColE(K)),

Cu4(Φϕ(K))=(1/108)(ColE(85K))12·ColE(K)).

Remark 6.11

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

CLX1,,Xm,R1,,Rn(K)=[ColXi(RjK)]i=1,,m,j=1,,n.

It is, then, a natural question whether for any quandle 2-cocycle invariant Φϕ(K), there is a sequence of knots R1, …, Rn and quandles X1, …, Xm such that Φϕ(K) is derived from CLX1, …, 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 H2Q(X,)2 [33]. To get an abelian extension of X of order 30 we would have to have a non-trivial 2-cocycle X × XZ3 which would give an element of HQ2(X,3)=Hom(2,3)=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)) [congruent with] Z2, a similar argument implies that Q(36, 58) is not an abelian extension of Q(4, 1). We have H2(Q(12, 10)) [congruent with] Z6, and let f be a 2-cocycle that generates H2(Q(12, 10), Z6) [congruent with] 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)) [congruent with] 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 = [var phi](Y) [subset or is implied by] 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 [var phi] 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 HQ2(X,2)2×2. 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

ϕB:X×XB  and  ϕC:X×XC

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

Proof

Define ϕC:E(X,B,ϕB)×E(X,B,ϕB)C by

ϕC((x1,b1),(x2,b2))=ϕC(x1,x2).

Then ϕC is a 2-cocycle of E(X, B, ϕB) with coefficient C.

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

f((x1,(b1,c1))*(x2,(b2,c2)))=f((x1*x2,(b1,c1)+ϕ(x1,x2)))=f((x1*x2,(b1+ϕB(x1,x2),c1+ϕC(x1,x2)))=((x1*x2,b1+ϕB(x1,x2)),c1+ϕC(x1,x2)),

and

f((x1,(b1,c1)))*f((x2,(b2,c2))=((x1,b1),c1)*((x2,b2),c2))=((x1*x2,b1+ϕB(x1,x2)),c1+ϕC((x1,c1),(x2,c2)))=((x1*x2,b1+ϕB(x1,x2)),c1+ϕC(x1,x2)),

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 × XA be a 2-cocycle with abelian extension E(X, A, ϕ). Further, let pB and pC be the projections from A onto B and C respectively. Then pBϕ : X × XB is a 2-cocyle giving abelian extension E(X, B, pBϕ), and E(X, A, ϕ) is isomorphic to E(E(X, B, pBϕ), C, ϕ′), where ϕ′((x1, b1), (x2, b2)) = pCϕ(x1, x2) for (x1, b1), (x2, b2) [set membership] X × B.

Lemma 7.5 is generalized as follows.

Proposition 7.6

Let X be a finite quandle, and 0CιApBB0 be an exact sequence of finite abelian groups. Let ϕ : X × XA be a quandle 2-cocycle. Then E(X, A, ϕ) is an abelian extension of E(X, B, pBϕ) with coefficient group C.

Proof

Let s : BA be a section of the map pB, that is, pBs = idB. Then

pB(s(b1+b2)s(b1)s(b2))=0.

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

s(b1+b2)s(b1)s(b2)=ι(c)

for some c [set membership] C. Let η : B × BC be given by η(b1, b2) = c. Then pB(aspB(a)) = 0 and hence we can write aspB(a) = ι(pC(a)) where pC : AC. This yields

ιpC(a)+spB(a)=a

for all a [set membership] A.

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

ϕ((x1,b1),(x2,b2))=pCϕ(x1,x2)η(b1,pBϕ(x1,x2))

for (xi, bi) [set membership] E(X, B, pBϕ) = X × B, i = 1, 2. To show that ϕ′ is a 2-cocycle it suffices to show that E(E(X, B, pBϕ), C, ϕ′) is a quandle. For this it suffices to show that the mapping

f:E(E(X,B,pBϕ),C,ϕ)E(X,A,ϕ)

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) [set membership] X × A we see that

f((x,pB(a)),pC(a))=(x,spB(a)+ιpC(a))=(x,a).

Finally to show that f preservers the product we compute:

f(((x1,b1),c1)*((x2,b2),c2))=f(((x1,b1)*(x2,b2),c1+ϕ((x1,b1),(x2,b2))))=f(((x1*x2,b1+pBϕ(x1,x2)),c1+pCϕ(x1,x2)η(b1,ϕ(x1,x2))))=(x1*x2,s(b1+pBϕ(x1,x2))+ι(c1+pCϕ(x1,x2)η(b1,ϕ(x1,x2))))=(x1*x2,s(b1)+spBϕ(x1,x2)+ιη(b1,pBϕ(x1,x2))+ι(c1)+ιpCϕ(x1,x2)ιη(b1,pBϕ(x1,x2)))=(x1*x2,s(b1)+ι(c1)+ϕ(x1,x2)),

and

f(((x1,b1),c1))*f(((x2,b2),c2))=(x1,s(b1)+ι(c1))*(x2,s(b2)+ι(c2))=(x1*x2,s(b1)+ι(c1)+ϕ(x1,x2)),

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×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 EnX if E = E(X, Zn, ϕ) for some 2-cocycle ϕ such that E is connected. E2mE1dX if there is a short exact sequence 0 → ZmZnZd → 0 such that nHQ2(X,n) and E1, E2 are corresponding extensions as in Proposition 7.6. In this case E2nX where n = md. The notation 1X indicates that HQ2(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.

1Q(8,1)2Q(4,1)

(Kei)1Q(24,1)2Q(12,1)2Q(6,1)

1Q(24,2)2Q(12,2)2Q(6,2)

(Kei)1Q(27,1)3Q(9,2)=Q(3,1)×Q(3,1)

1Q(27,6)3Q(9,3)=3[t]/(t2+1)

1Q(27,14)3Q(9,6)=3[t]/(t2+2t+1)

1Q(24,8)=Q(3,1)×Q(8,1)2Q(12,4)=Q(3,1)×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.

?Q(120,)6Q(20,3)2Q(10,1)

?Q(120,)5Q(24,7)2Q(12,3)

?Q(120,)2Q(60,)5Q(12,3)

?Q(48,)2Q(24,4)2Q(12,5)

?Q(48,)2Q(24,3)2Q(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 (...)Qn(...)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 [set membership] A for any x, y [set membership] X, in additive notation.

Proof

One computes, for any x, y [set membership] X and a, b [set membership] A,

[(x,a)*(y,b)]*(y,b)=(x*y,a+ϕ(x,y))*(y,b)=((x*y)*y,a+ϕ(x,y)+ϕ(x*y,y)).

For any x, y [set membership] X and a, b [set membership] 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 [set membership] X.

Remark 7.9

Let X = Q(12, 7). Then HQ2(X,4)2×4. By computer calculation, there is a particular generating 2-cocycle of the Z2-factor, Q(24,15)2Q(12,7). Notice that HQ2(Q(24,15),4)4. We also note that there are epimorphisms Q(24, 14) → Q(12, 7) and Q(24, 18) → Q(12, 7), whereHQ2(Q(24,14),4)4 and HQ2(Q(24,14),2)2. Hence there is a quandle of order 48 corresponding to the Z4-factor of HQ2(X,4), that has epimorphic image Q(24, 14) or Q(24, 18).

Remark 7.10

Let X = Q(12, 8). Then HQ2(X,2)(2)3. There are three epimorphisms from Rig quandles of order less than 36:

Q(24,5)X,  Q(24,16)X,  Q(24,17)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 HQ2(X,4)4×4. There are two extensions in Rig quandles of order less than 36:

Q(24,6)Q(12,9),  Q(24,19)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 HQ2(X,6)6. There is one extension among Rig quandles of order less than 36, Q(24, 20) → Q(12, 10) and we have HQ2(Q(24,20),3)3×3. 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 [mathematical script N]F of 315 non-faithful connected generalized Alexander quandles with respect to pairs (G, f) for non-abelian groups G and f [set membership] 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 [set membership] 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 [mathematical script N]F 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 [var phi] (Y) = X [subset or is implied by] 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 × XZ2 by ϕ(x, y) = 0 if βx,y = id and ϕ(x, y) = 1 if βx,y is the transposition. Then βx,y(t) = t + ϕ(x, y) for t [set membership] 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 [mathematical script N]F.

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 [mathematical script N]F. 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 [mathematical script N]F.

We observe the following generalization of Lemma 8.1. Let Y be a finite connected quandle, and let [var phi](Y) = X [subset or is implied by] 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 Hβ of Sym(S) generated byx,y | x, y [set membership] 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 Hβ is cyclic of order k and k is a prime power, it is generated by a k-cycle σ. We can identify S with Zk in such a way that σ = (1, 2, …, k). Then σ(t) = 1+t for any t [set membership] Zk. Hence for any x, y [set membership] X, βx,y = σi for some i [set membership] Zk, so that βx,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 Φϕ is of the form Φϕ(K) = a + bu2, where a, b [set membership] 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 Φϕ is of the form Φϕ(K) = a + bu2, where a, b [set membership] 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 Φϕ(K) = a + bu + cu2 + du3 + eu4 + fu5, where a, b, c, d, e, f [set membership] 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 (...)Qn(...)Q1?

Question 9.4

Is any abelian extension of a finite kei a kei?

Acknowledgments

MS was partially supported by the National Institutes of Health under Award Number R01GM109459. The content of this paper is solely the responsibility of the authors and does not necessarily represent the official views of NIH. LV was partially supported by Conicet, ICTP, UBACyT 20020110300037 and PICT-2014-1376. We thank Santiago Laplagne for the computer where some calculations were performed.

Appendix

Cocycle Invariants for Q(6, 2)

In this appendix we list the cocycle invariant Φϕ(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 bd then by Lemma 4.3 each of the corresponding knots K satisfes Krm(K) (see Tables A.1A.3).

Table A.1

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

Cocycle invariantKnot
541062, 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
1981099
6 +24u31, 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 + 72u12n0546
150 + 24u11n126, 12n0440

Table A.2

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

Cocycle invariantKnot
6 +48u2940, 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 + 96u212a0701, 12a0987
30 + 24u285, 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 + 168u212n0604
54 + 144u212n0508
54 + 48u + 48u212a0742, 12n0380
78 + 48u + 24u212a0574, 12n0571, 12n0574
102 + 96u212n0518
126 + 72u212a0647, 12n0605
150 + 48u212a1288, 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 [set membership] Z with d ≠ 0.

Cocycle invariantKnot
6 + 24u361, 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+ 72u3948, 11a293, 12a0895
6 + 144u2+ 24u31098
6 + 24u + 72u312n0666
6 +24u + 48u2+ 48u311n164, 12n0402
6 +24u + 96u2+ 24u311n167
6 + 48u + 48u3818, 12a1260, 12n0403
6 +48u + 48u2+ 24u312n0565
6 + 72u + 24u3947, 12n0549
30 + 120u2+ 24u312n0737
30 + 24u + 72u2+ 24u312a0576, 12n0570
54 + 72u311a314
54 + 48u2+ 48u311a332, 12n0386
54 + 24u + 48u2+ 24u312a0297, 12n0379
54 + 48u + 24u3946, 11a291, 12n0567
78 + 24u2+ 48u312a1283
78 + 24u + 24u2+ 24u312n0883
78 + 48u + 72u2+ 48u311a44, 11a47, 11a57
11a231, 11a263, 11n71
11n72, 11n73, 11n74
11n75, 11n76, 11n77
11n78, 11n81
12a0167, 12a0692, 12a0801
102 + 48u312n0806
102 + 24u + 24u311a277, 12a1225

Contributor Information

W. Edwin Clark, Department of Mathematics and Statistics, University of South Florida, Tampa, Florida, USA.

Masahico Saito, Department of Mathematics and Statistics, University of South Florida, Tampa, Florida, USA.

Leandro Vendramin, Departamento de Mathemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires Buenos Aires, Argentina.

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