Quandle colorings of knots and applications

Abstract

We present a set of 26 finite quandles that distinguish (up to reversal and mirror image) by number of colorings, all of the 2977 prime oriented knots with up to 12 crossings. We also show that 1058 of these knots can be distinguished from their mirror images by the number of colorings by quandles from a certain set of 23 finite quandles. We study the colorings of these 2977 knots by all of the 431 connected quandles of order at most 35 found by Vendramin. Among other things, we collect information about quandles that have the same number of colorings for all of the 2977 knots. For example, we prove that if Q is a simple quandle of prime power order then Q and the dual quandle Q* of Q have the same number of colorings for all knots and conjecture that this holds for all Alexander quandles Q. We study a knot invariant based on a quandle homomorphism f : Q₁ → Q₀. We also apply the quandle colorings we have computed to obtain some new results for the bridge index, the Nakanishi index, the tunnel number, and the unknotting number. In an appendix we discuss various properties of the quandles in Vendramin’s list. Links to the data computed and various programs in C, GAP and Maple are provided.

1. Introduction

Sets with certain self-distributive operations called quandles have been studied since the 1940s in various areas with different names. They have been studied, for example, as algebraic systems for symmetries [43], as quasi-groups [17], and in relation to modules [19, 36]. Typical examples of quandles arise from conjugacy classes of groups and from modules over the integral Laurent polynomial ring ℤ[t, t⁻¹], called Alexander quandles. The fundamental quandle of a knot was defined in a manner similar to the fundamental group [25, 31] 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 [6, 16], and investigated in [28, 33, 37–39]. Extensions of quandles by cocycles have been studied [1, 5, 13], and invariants derived thereof are applied to various properties of knots and knotted surfaces (see [7] and references therein).

Tables of small quandles have been made previously (e.g. [7, 14, 36]). Computations using GAP [44] by Vendramin [47] significantly expanded the list for connected quandles. He found all connected quandles of order up to 35. There are 431 of them. These quandles may be found in the GAP package RIG [46]. We refer to these quandles as RIG quandles, and use the notation C[n, i] for the ith quandle of order n in his list. As a matrix C[n, i] is the transpose of the quandle matrix SmallQuandle(n, i) in [46].

In this paper, we investigate to what extent the number of quandle colorings of a knot by a finite quandle can distinguish the oriented knots with at most 12 crossings from the knot table at KnotInfo [8].

In Sec. 2, we provide definitions and conventions. The coloring of knots by quandles and the relationship to symmetry of knots are discussed in Secs. 3 and 4. It turns out that RIG quandles do not suffice to distinguish all of the prime, oriented knots with at most 12 crossings. To distinguish all of these knots we generated several thousand conjugation quandles and generalized Alexander quandles. Eventually we found a set of 26 quandles that distinguish, up to orientation and mirror image, all knots with up to 12 crossings. These computations extend the results by Dionisio and Lopes [11] for 10 Alexander quandles and 249 prime knots with at most 10 crossings.

We write m(K) for the mirror image of knot K and r(K) for K with the orientation reversed. The 2977 knots given at KnotInfo [8] are representatives up to mirrors and reverses of the prime, oriented knots with at most 12 crossings (see [9]). It is known [29, 31] that quandle colorings do not distinguish K from rm(K) for any knot K. Thus it is of interest which finite quandles, if any, can distinguish K from m(K). This is only possible for a knot which is chiral or negative amphicheiral. There are 1366 chiral or negative amphicheiral knots among the 2977 knots with up to 12 crossings at KnotInfo [8]. We have found a set of 23 quandles that distinguish K from m(K) for 1058 knots out of these 1366 knots. It remains an open question whether the remaining 308 of these knots can be so distinguished.

Many sets of RIG quandles share the same number of colorings for all knots in the KnotInfo table. We explore this phenomenon in Sec. 5. For example, we conjecture that a connected Alexander quandle and its dual quandle always give the same number of colorings for any knot, and we prove this for a special class of Alexander quandles. In Sec. 6, we introduce an invariant of knots based on quandle homomorphisms and mention a relation to the quandle cocycle invariant. The computational results are applied to other knot invariants in Sec. 7, such as the bridge index, the Nakanishi index, the tunnel number, and the unknotting number. Various properties of quandles are further discussed in Appendix A, and a summary of computational results is given as to these properties for RIG quandles. Descriptions of quandles used for distinguishing knots and their mirror images can be found in Appendix B. Files containing the quandle matrices, programs and the results of computations can be found at [21, 22]. Appendix C contains the list of 12 crossing knots discussed in Sec. 7. Appendix D contains some information on programs used for colorings.

2. Preliminaries

We briefly review some definitions and examples of quandles. More details can be found, for example, in [1, 7, 16].

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*_Xb) = 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.

Typical examples of quandles include the following.

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

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

• A generalized Alexander quandle is defined by (e.g., [4]) a pair (G, f) where G is a group and 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 quandle.

• A Galkin quandle is defined as follows. Let A be an abelian group, also regarded naturally as a ℤ-module. Let μ : ℤ₃ → ℤ, τ : ℤ₃ → A be functions. These functions μ and τ need not be homomorphisms. Define a binary operation on ℤ₃ × A by

  $$(x,a)*(y,b)=(2y-x,-a+\mu (x-y)b+\tau (x-y)),x,y\in {\mathbb{Z}}_{3}a,b\in A.$$

  Then for any abelian group A, the above operation * defines a quandle structure on ℤ₃ × A if μ(0) = 2, μ(1) = μ(2) = −1, and τ(0) = 0. Galkin gave this definition in [17, p. 950], for A = ℤ_p, and this definition was given in [10].
• A function ϕ : X × X → A for an abelian group A is called a quandle 2-cocycle [6] if it satisfies

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

  and ϕ(x, x) = 0 for any x, y, z ∈ X. For a quandle 2-cocycle ϕ, X × A becomes a quandle by (x, a) * (y, b) = (x * y, a + ϕ(x, y)) for x, y ∈ X, a, b ∈ A, and it is called an abelian extension of X by A, see [5].

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. Similarly the left translation ℒ_a is defined by ℒ_a (x) = a*x. Then ℛ_a is a permutation of X by axiom (2). The subgroup of Sym(X) 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. 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, *). We also denote the dual of X by X*. If X* is isomorphic to X, then X is called self-dual.

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′. A coloring of an oriented knot diagram by a quandle X is a map 𝒞 : 𝒜 → X 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 [7, 12], 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 a diagram, and hence is a knot invariant. In Fig. 2, a coloring of a figure-eight knot is indicated. In the figure, the elements x, y, z of a quandle are assigned at the top arcs. The coloring conditions are applied to color other arcs as depicted. Since the bottom arcs are connected to the top arcs, we need the conditions

$$x=z\overline{*}(x*y),$$

$$y=(x*y)\overline{*}(y*(z\overline{*}(x*y))),$$

$$z=y*(z\overline{*}(x*y)),$$

to obtain a coloring. Each crossing corresponds to a standard generator σ_i of the 3-strand braid group or its inverse, and the closure of ${\sigma }_1{\sigma }_2^{-1}{\sigma }_1{\sigma }_2^{-1}$ represents the figure-eight knot. The dotted lines indicate the closure, see [42] for more details on braids. We give the orientation downward for braids, see Fig. 2. For the rest of this paper all knots will be prime and oriented.

Fig. 1.

The coloring rule at crossings.

Fig. 2.

Figure-eight knot as the closure of the braid ${\sigma }_1{\sigma }_2^{-1}{\sigma }_1{\sigma }_2^{-1}$.

The fundamental quandle is defined in a manner similar to the fundamental group [25, 31]. 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, [15]), by assigning generators to arcs of a knot diagram, and relations corresponding to crossings. Then, the set of colorings of a knot diagram K by a quandle X is in one-to-one correspondence with the set of quandle homomorphisms from the fundamental quandle of K to X.

If Q is a quandle and K is a knot, we denote by ${\text{Col}}_Q^N(K)$ the number of non-trivial colorings of K by Q. The number of colorings including trivial colorings is ${\text{Col}}_Q(K)={\text{Col}}_Q^N(K)+|Q|$, where |Q| is the order of Q. It is well known that for each quandle Q, ColQ(K) is an invariant of knots (see, for example, [41]). We say that the quandle Q distinguishes knots K and K′ if Col_Q(K) ≠ Col_Q(K′).

Let m : 𝕊³ → 𝕊³ be an orientation-reversing homeomorphism. For a knot K, m(K) is the mirror image of K. Let r(K) denote the same 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.

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 [8] 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 it has all three symmetries, that is,

  $$K=r(K)=m(K)=rm(K).$$

The symmetry type of each knot on at most 12 crossings is given at [8]. Thus each of the 2977 knots K given there represents as many as four knots K, m(K), r(K) and rm(K).

It is known [29, 31] that the fundamental quandles of K and K′ are isomorphic if and only if K = K′ or K = rm(K′).

3. Distinguishing the Knots K, m(K), r(K), rm(K)

For the problem of distinguishing knots by quandle coloring it suffices to consider only connected quandles by the following lemma (see [40, Sec. 5.2]).

Lemma 3.1

The image of a quandle homomorphism of the fundamental quandle of any knot in another quandle is connected. In particular, given two knots K₁ and K₂, a quandle Q of smallest order satisfying Col_Q(K₁) ≠ Col_Q(K₂) is connected.

Lemma 3.2

The following equations hold for all quandles Q and all knots K:

1. Col_Q(K) = Col_Q(rm(K)),

2. Col_Q(r(K))) = Col_Q(m(K))),

3. Col_Q(m(K)) = Col_Q* (K),

where Q* in (3) denotes the dual quandle of Q.

Proof

For (1) and (2) see [26]. Item (3) follows easily from the definitions.

Corollary 3.3

For all quandles Q and for knots K of any symmetry type except chiral or negative amphicheiral,

$${\text{Col}}_Q(rm(K))={\text{Col}}_Q(m(K))={\text{Col}}_Q(r(K))={\text{Col}}_Q(K).$$

Conjecture 3.4

If 𝒦 is any finite set of knots closed under the action of 𝒢, there exists a finite sequence of finite quandles 𝒮 = (Q₁, Q₂, …, Q_n) such that the invariant

$$C(K)≔({\text{Col}}_{Q_1}(K),{\text{Col}}_{Q_2}(K),\dots ,{\text{Col}}_{Q_n}(K))$$

satisfies, for all K, K′ ∈ 𝒦,

$$C(K\prime )=C(K)\mathit{\text{if and only if}}K\prime =K\mathit{\text{or}}K\prime =mr(K).$$

Our computations verify the truth of the following weaker conjecture where 𝒦 is the set of all knots with at most 12 crossings. We also verify the truth of Conjecture 3.4 for smaller sets of knots (see Sec. 4).

Conjecture 3.5

If 𝒦 is any finite set of knots closed under the action of 𝒢, there exists a finite sequence of finite quandles 𝒮 = (Q₁, Q₂, …, Q_n) such that the invariant

$$C(K)≔({\text{Col}}_{Q_1}(K),{\text{Col}}_{Q_2}(K),\dots ,{\text{Col}}_{Q_n}(K))$$

satisfies, for all K, K′ ∈ 𝒦,

$$C(K\prime )=C(K)\mathit{\text{implies}}𝒢(K\prime )=𝒢(K).$$

It is clear that if Conjecture 3.4 is true then the following conjecture is true.

Conjecture 3.6

There exists a sequence of finite quandles

$$𝒮=(Q_1,Q_2,\dots ,Q_n,\dots )$$

such that the invariant

$$C(K)≔({\text{Col}}_{Q_1}(K),{\text{Col}}_{Q_2}(K),\dots ,{\text{Col}}_{Q_n}(K),\dots )$$

satisfies, for all knots K, K′,

$$C(K\prime )=C(K)\mathit{\text{if and only if}}K\prime =K\mathit{\text{or}}K\prime =mr(K).$$

This conjecture is similar to a conjecture made by Fenn and Rourke [15, p. 385] for racks.

Let 𝒦 be a finite set of knots closed under the action of the group 𝒢. Let ℛ = {k₁, k₂, …, k_n} be a set of representatives of the orbits of 𝒢 acting on 𝒦. Let 𝒮 = (Q₁, Q₂, …, Q_m) be a list of finite quandles and for any knot K let

$$C(K)≔({\text{Col}}_{Q_1}(K),{\text{Col}}_{Q_2}(K),\dots ,{\text{Col}}_{Q_m}(K)).$$

To simplify the following proof, for knots K and K′ we write

$$K~K\prime \text{if and only if}C(K)=C(K\prime ).$$

Thus K ≁ K′ if and only if there exist Q_i ∈ 𝒮 such that Col_Qi(K) ≠ Col_Qi(K′). The analysis of our computations is based on the following.

Proposition 3.7

Staying with the notation above, suppose for all pairs of knots k, k′ ∈ ℛ the list 𝒮 satisfies the following conditions:

• (A)if k ≠ k′ then k ≁ k′,
• (B)if k ≠ k′ then k ≁ m(k′), and
• (C)if k ≠ k′ then m(k) ≁ m(k′).

Then if K, K′ ∈ 𝒦 and K ~ K′ we have 𝒢(K) = 𝒢(K′). If in addition 𝒮 satisfies the following condition:

• (D)m(k) ≁ k when k is chiral or negative amphicheiral, then if K, K′ ∈ 𝒦 and K ~ K′ we have K′ = K or K′ = rm(K).

Proof

Assume first that conditions (A)–(C) hold and that K, K′ ∈ 𝒦 and K ~ K′. We must prove that 𝒢(K) = 𝒢(K′). Let k, k′ be elements of ℛ such that

$$K\in 𝒢(k)\text{and}K\prime \in 𝒢(k\prime ).$$

Note that

$$𝒢(k)=\{k,rm(k)\}\cup \{r(k),m(k)\}\text{and}𝒢(k\prime )=\{k\prime ,rm(k\prime )\}\cup (r(k\prime ),m(k\prime )\}.$$

By Lemma 3.2(1) and (2) we have k ~ rm(k) and r(k) ~ m(k), and similarly for k replaced by k′. We consider four cases and show that each case leads to 𝒢(k) = 𝒢(k′).

• Case (i)

  K ∈ {k, rm(k)} and K′ ∈ {k′, rm(k′). Then we have k ~ K ~ K′ ~ k′. This implies by (A) that k = k′, and hence 𝒢(k) = 𝒢(k′).

• Case (ii)

  K ∈ {k, rm(k)} and K′ ∈ {m(k′), r(k′)}. Then k ~ K ~ K′ ~ m(k′) which implies by (B) that k = k′, hence 𝒢(k) = 𝒢(k′).

• Case (iii)

  K ∈ {m(k), r(k)} and K′ ∈ {k′, rm(k′)}. This is similar to Case (ii).

• Case (iv)

  K ∈ {m(k), r(k)} and K′ ∈ {m(k′), r(k′)}. Then m(k) ~ K ~ K′ ~ m(k′), and by (C) we have k = k′, hence 𝒢(k) = 𝒢(k′).

This proves the first part of the proposition. Now assume that conditions (A)–(D) hold and K ~ K′. Let k and k′ be as above. By what we just proved we have 𝒢(k) = 𝒢(k′) and so we have K, K′ ∈ 𝒢(k). If K = K′ there is nothing to prove so we can assume that one of the following four cases holds and that K ≠ K′.

• Case (1)

  k is reversible or positive amphicheiral. Hence 𝒢(k) = {k, rm(k)}. In this case we may take K = k and K′ = rm(k) so K′ = rm(K).

• Case (2)

  k is negative amphicheiral. Hence 𝒢(k) = {k, m(k)}. In this case by (D) we have that k ≁ m(k) so K ≠ K′, K ~ K′ and K, K′ ∈ 𝒢(k) is impossible.

• Case (3)

  k is chiral. Hence 𝒢(k) = {k, r(k), m(k), rm(k)}. Of the four elements of 𝒢(k) there are six possible pairs that might be {K, K′}. Note that the pair {k, m(k)} is ruled out by (D). Since we always have k ~ rm(k) and r(k) ~ m(k), the pairs {k, r(k)}, {rm(k), m(k)}, {rm(k), r(k)} are also ruled out. This leaves the only possibilities to be {k, rm(k)} and {r(k), rm(k)}. If r(k) ~ rm(k) then m(k) ~ r(k) ~ rm(k) ~ k which is a contradiction to (D), so we are left with {k, rm(k)}as the only possibility for {K, K′}, as desired.

• Case (4)

  k is fully amphicheiral. Hence 𝒢(k) = {k}. This is impossible since there is only a single knot in 𝒢(k).

This completes the proof.

4. Computational Support for Conjectures 3.4 and 3.5

Let 𝒬₂₆ = (Q₁, …, Q₂₆) be the list of 26 quandles in the first table of Appendix B. Matrices for these 26 quandles may be found at [22], where they are denoted by Q[i], i = 1, …, 26. The list 𝒬₂₆ is closed under taking dual quandles. This is to simplify computing the number of colorings of the mirror image of a knot. Included in the file is the list dual satisfying dual[i] = j if and only if $Q_i^{*}$ is isomorphic to Q_j.

If i ≠ 15 each quandle Q_i is faithful and hence isomorphic to a conjugation quandle on some conjugacy class in its inner automorphism group Inn(Q_i). The quandle Q₁₅ is not faithful and is not a conjugation quandle (Vendramin, personal communication). We note that some non-faithful quandles are conjugation quandles, so this is not trivial. However, Q₁₅ is a generalized Alexander quandle on GAP’s SmallGroup(32, 50) = (C2 × Q8) : C2. The quandles in 𝒬₂₆ of order less than 36 are RIG quandles and the remaining 11 were found by searching for conjugation quandles on conjugacy classes of finite groups using GAP. In retrospect all of these 26 quandles could have been found by searching for conjugation quandles and generalized Alexander quandles.

We denote the knots in KnotInfo up to 12 crossings by

$${ℛ}_{12}=\{K[i]:i=1,\dots ,2977\}.$$

In our notation, the knot K[i] represents the knot with name rank = i + 1 in [8]. In particular, K[1] is the trefoil, which has name rank 2 in [8].

The 26 × 2977 matrix M found at [22] has the property that

$$M_{i,j}={\text{Col}}_{Q_i}^N(K[j]).$$

See Appendix D for comments on the computation of ${\text{Col}}_Q^N(K)$.

Since ℛ₁₂ is a set of representatives of the set 𝒦₁₂ of (isotopy classes of prime, oriented) knots with at most 12 crossings, including all symmetry types, to verify that Conjecture 3.4 holds when 𝒦 = 𝒦₁₂ and the list of quandles 𝒮 = 𝒬₂₆, it suffices, by Proposition 3.7, to verify that the matrix M and the list dual have the following properties for i, j ∈ {1, 2, …, 2977} and k ∈ {1, 2, …, 26}.

1. For all i ≠ j there exists k such that M_k,i ≠ M_k,j.

2. For all i ≠ j there exists k such that M_k,i ≠ M_s,j, where s = dual[k].

3. For all i ≠ j there exists k such that M_s,i ≠ M_s,j, where s = dual[k].

These conditions are easy to verify directly from the matrix M and the list dual. This shows that the set 𝒬₂₆ distinguishes the knots in 𝒦₁₂ up to mirror images and orientation.

Remark 4.1

Note that the unknot is not included among the knots K[i], i = 1, …, 2977. Since the unknot has only trivial colorings, to show that we can distinguish it from all of the 2977 knots it suffices to note that for every knot K[j] there is a quandle Q_i such that ${\text{Col}}_{Q_i}^N(K[j])\ne 0$, that is, no column of the matrix M contains only 0s. This is confirmed from the matrix M.

As pointed out above the only knots we can hope to distinguish from their mirror images (equivalently from their reversals) are chiral knots and negative amphicheiral knots. Among 2977 knots in KnotInfo [8] up to 12 crossings, 1319 knots are chiral and 47 are negative amphicheiral. Denote this set of 1366 knots by 𝒦^c. We were not able to find quandles to distinguish all of these 1366 knots from their mirror images. Thus we cannot verify Conjecture 3.5 when 𝒦 = 𝒦₁₂. However, we were able to find quandles that distinguish 1058 of the knots in 𝒦^c from their mirror images. The quandles that distinguish these 1058 knots are the quandles T₁, T₂, …, T₂₃ in the second table of Appendix B. Matrices for the quandles can be found in [22]. A list of 1058 pairs (i, n) where K[n] is one of 1058 knots in 𝒦^c and T_i is a quandle that distinguished knot K[n] from m(K[n]) can be found in [22].

With the exception of quandles T₂ and T₃ the quandles T_i are conjugation quandles on conjugation classes in the group Inn(T_i). T₂ and T₃ are generalized Alexander quandles, both on the group PSL(2, 8). We note that Q₁₇ = T₂₂ and Q₂₄ = T₂₃. Otherwise the quandles in the second table of Appendix B are different from those in the first table of that appendix.

In the first and second tables of Appendix B, the structure descriptions are given by GAP’s Structure Description function. In particular, A × B is the direct product, N : H is a semidirect product and A·B is a non-split extension. The group ID, SmallGroup(n, i), is the ith group of order n in the GAP Small Groups library for n at most 2000. If a group has order n > 2000 we write SmallGroup(n, 0). Nevertheless for all of the inner automorphism groups we have computed, GAP does give a Structure Description.

5. Similarity of Quandles Over Knot Colorings

For two quandles Q₁ and Q₂, and a set 𝒦 of knots, we write Q₁ ≈ _𝒦 Q₂ if Col_Q1 (K) = Col_Q2 (K) for all K ∈ 𝒦. This equivalence relation was considered in [10]. We omit the subscript 𝒦 when it is the set of all knots. The following is immediate from the definitions and a property of the number of colorings.

Lemma 5.1

If $Q_1\approx Q_1^{\prime }$ and $Q_2\approx Q_2^{\prime }$ then $(Q_1\times Q_2)\approx (Q_1^{\prime }\times Q_2^{\prime })$.

Conjecture 5.2

For every connected Alexander quandle Q, Q ≈ Q*.

We prove the above conjecture for simple Alexander quandles of prime power order. (Following [1] we say that a quandle Q is simple if it is not trivial and whenever f : Q → Q′ is an epimorphism then f is an isomorphism or |Q′| = 1. This implies that |Q| > 2 since the quandles of orders 1 and 2 are trivial.) The following proposition gives alternative characterizations of these quandles.

Proposition 5.3

The following statements are equivalent for a quandle Q and a prime p.

1. Q is a simple quandle with p^k elements.

2. Q is isomorphic to an Alexander quandle on ℤ_p[t, t⁻¹]/(h(t)) where h(t) ∈ ℤ_p[t] is irreducible, of degree k and different from t and t−1 and the quandle operation is defined by x * y = t̄x + (1 − t̄)y where t̄ is the coset t + (h(t)).

3. Q is isomorphic to an Alexander quandle on the vector space (𝔽_p)^k where T is the companion matrix of a monic irreducible polynomial h(t) ∈ ℤ_p[t] of degree k, different from t and t − 1 and the quandle operation is given by x * y = Tx + (1 − T)y.

4. Q is isomorphic to a quandle defined on the finite field 𝔽_q where q = p^k, a ∈ 𝔽_q, a ≠ 1, a generates 𝔽_q, and the quandle operation is given by x*y = ax+(1 − a)y for x, y ∈ 𝔽_q.

Proof

The equivalence of (1), (3) and (4) is proved in [1, Sec. 3]. The equivalence of (2) and (4) is proved in [2].

Proposition 5.4

If Q is a simple quandle of prime power order, then Q ≈ Q*.

Proof

By Proposition 5.3, the quandle Q is equal to (𝔽_q, *) where 𝔽_q = ℤ_p[t, t⁻¹]/(h(t)), q = p^k, h(t) ∈ ℤ_p[t] is irreducible of degree k. The quandle operation is defined by x*y = t̄x + (1 − t̄)y where t̄ is the coset t + (h(t)). Let S ∈ M_n(𝔽_q) denote the reduction modulo (p, h(t)) of a Seifert matrix of K, and let N = t̄S−S^T, which is denoted by N(1, p, h(t)) in [2]. Although p is restricted to odd primes in [2], we claim that their results hold for p = 2 also. The authors of [2] agree (personal communication).

Let $\phi :{𝔽}_q^n\to {𝔽}_q^n$ be the linear map that N represents. Then in [2] it was shown that the set of colorings SCol_Q(K) is in one-to-one correspondence with the module 𝔽_q ⊕ ker(φ).

The dual quandle Q* is equal to (𝔽_q, ⁎¯) where a ⁎¯ b = t̄⁻¹a + (1 − t̄⁻¹)b. Let N* = t̄⁻¹ S − S^T, and ${\phi }^{*}:{𝔽}_q^n\to {𝔽}_q^n$ be the corresponding linear map. Then the number of colorings by Q* is the cardinality of 𝔽_q ⊕ ker(φ*). It remains to show that the dimensions of ker(φ) and ker(φ*) are the same. One computes

$$(-{\overline{t}}^{-1})N^T=-{\overline{t}}^{-1}(\overline{t}{S}^T-S)=-S^T+{\overline{t}}^{-1}S=N^{*}.$$

Since rank(N^T) = rank(N), we have rank(N) = rank(N*) and hence the dimension of the kernel of φ* is equal to the dimension of the kernel of φ, so the number of colorings is the same for Q and Q*.

This proof is similar to the proof of the fact that the Alexander polynomial is symmetric, Δ(t⁻¹) = Δ(t) modulo a power of t.

From Proposition 5.4 and Lemma 5.1 we obtain the following corollary.

Corollary 5.5

If a quandle Q is isomorphic to a product of simple quandles of prime power order then Q ≈ Q*.

There are 273 connected Alexander quandles of order at most 35. Only 45 of these quandles are not a product of simple quandles of prime power order.

Remark 5.6

The following is a list of all RIG quandles that satisfy Proposition 5.3 except those of prime power order.

• C[4, 1], C[8, 2], C[8, 3], C[9, 3], C[9, 7], C[9, 8], C[16, 3],

• C[16, 8], C[16, 9], C[25, 3], C[25, 11], C[25, 12], C[25, 13], C[25, 19],

• C[25, 20], C[25, 31], C[25, 32], C[25, 33], C[25, 34], C[27, 31], C[27, 32],

• C[27, 33], C[27, 34], C[27, 62], C[27, 63], C[27, 64], C[27, 65], C[32, 10],

• C[32, 11], C[32, 12], C[32, 13], C[32, 14], C[32, 15].

Let 𝒦 be the set of all 2977 knots in the table in KnotInfo [8] up to 12 crossings. We observe the following for ≈_𝒦 and for the 431 RIG quandles.

• There are 151 classes of ≈_𝒦 consisting of more than one quandle.

• Among these classes, 145 classes consist of two quandles.

• Among the 151 classes, six classes have 3 or more element. Let 𝒬₍₆₎ denote this family of six classes.

• Among 145 classes of pairs, all but 35 classes consist of a pair of dual Alexander quandles. Let 𝒬₍₃₅₎ denote this family of 35 classes.

• Among 35 classes of pairs in 𝒬₍₃₅₎, one class is a pair of self-dual Alexander quandles, some are non-Alexander dual quandles, and some are non-Alexander self-dual quandles.

We now explain to the extent possible the six non-trivial classes 𝒬₍₆₎ of ≈_𝒦 that have more than two elements. Among these six classes, the first three classes listed below form equivalence classes under ≈ by Lemma 5.1 and Proposition 5.4. In all the other cases unless otherwise stated we do not know whether they form equivalence classes under ≈.

• {C[25, 6], C[25, 7], C[25, 8]}: This situation is explained as follows:

  • C[25, 6] = C[5, 2] × C[5, 2], C[25, 7] = [5, 3] × C[5, 3],
  • C[25, 6] = C[5, 2] × C[5, 3],

  where C[5, 2] = ℤ₅[t]/(t + 2) is dual to C[5, 3] = ℤ₅[t]/(t + 3).
• {C[35, 8], C[35, 9], C[35, 10], C[35, 11]}: This situation is explained as follows:

  • C[35, 8] = C[5, 2] × C[7, 4], C[35, 9] = C[5, 2] × C[7, 5],
  • C[35, 10] = C[5, 3] × C[7, 4], C[35, 11] = C[5, 3] × C[7, 5],

  where C[7, 4] = ℤ₇[t]/(t + 2) is dual to C[7, 5] = ℤ₇[t]/(t + 4).
• {C[35, 12], C[35, 13], C[35, 14], C[35, 15]}: This situation is explained as follows:

  • C[35, 12] = C[5, 3] × C[7, 3], C[35, 13] = C[5, 3] × C[7, 2],
  • C[35, 14] = C[5, 2] × C[7, 3], C[35, 15] = C[5, 2] × C[7, 2],

  where C[7, 2] = ℤ₇[t]/(t + 3) is dual to C[7, 3] = ℤ₇[t]/(t + 5).
• {C[30, 7], C[30, 8], C[30, 9], C[30, 10]}: This situations is explained as follows:

  • C[35, 12] = C[5, 3] × C[7, 3], C[35, 13] = C[5, 3] × C[7, 2],
  • C[35, 14] = C[5, 2] × C[7, 3], C[35, 15] = C[5, 2] × C[7, 2],

  where C[5, 2] = ℤ₅[t]/(t + 2) is dual to C[5, 3] = ℤ₅[t]/(t + 3). As mentioned before it is conjectured [10] that C[6, 1] ≈ C[6, 2]. So we conjecture that this forms a class for ≈.

• {C[24, 5], C [24, 6], C[24, 16], C[24, 17]}: These quandles are abelian extensions of C[12, 8], C[12, 9], C[12, 8], C[12, 8], respectively, and none is Alexander. The quandles C[12, 8], C[12, 9] have C[6, 1], C[6, 2] as subquandles, respectively, and have C[6, 1] as an epimorphic image. These facts alone, however, do not seem to imply that they constitute a class for ≈. We conjecture that this forms a class for ≈.

• {C[24, 29], C[24, 30], C[24, 31]}: These three quandles are self-dual Galkin quandles. This class was reported in [10].

The remaining classes in 𝒬₍₃₅₎ consist of pairs, that are not dual Alexander quandles, and we categorize them as follows. We conjecture that they form equivalence classes under ≈.

There is one class with a pair of Alexander self-dual quandles: C[27, 17] and C[27, 22], both of which are Alexander quandles on the abelian group ℤ₃ × ℤ₉.

Each of the following classes is a pair of non-Alexander quandles that are duals of each other:

• {C[27, 9], C[27, 11]}, {C[27, 35], C[27, 36]}, {C[27, 37], C[27, 38]},

• {C[27, 41], C[27, 42]}, {C[27, 43], C[27, 46]}, {C[27, 44], C[27, 45]},

• {C[27, 56], C[27, 57]}, {C[27, 58], C[27, 59]}, {C[28, 11], C[28, 12]},

• {C[30, 17], C[30, 18]}, {C[32, 7], C[32, 8]}.

The remaining classes are pairs of self-dual non-Alexander quandles. In [10], the relation ≈ was studied for a family called Galkin quandles. Galkin quandles are known to be self-dual.

The following are two element ≈_𝒦-equivalence classes of (non-Alexander) Galkin quandles that were reported in [10]:

• {C[6, 1], C[6, 2]}, {C[12, 5], C[12, 6]}, {C[12, 8], C[12, 9]},

• {C[18, 1], C[18, 4]}, {C[18, 5], C[18, 8]}, {C[24, 27], C[24, 28]},

• {C[24, 38], C[24, 39]}, {C[30, 12], C[30, 14]}, {C[30, 13], C[30, 15]}.

The following classes are pairs of self-dual non-Alexander, non-Galkin quandles:

• {C[18, 2], C[18, 3]}, {C[18, 6], C[18, 10]}, {C[18, 7], C[18, 9]},

• {C[20, 5], C[20, 6]}, {C[20, 9], C[20, 10]}, {C[24, 3], C[24, 4]},

• {C[24, 15], C[24, 18]}, {C[24, 22], C[24, 23]}, {C[24, 34], C[24, 36]},

• {C[24, 35], C[24, 37]}, {C[24, 41], C[24, 42]}, {C[30, 2], C[30, 6]},

• {C[30, 23], C[30, 24]}, {C[32, 5], C[32, 6]}.

6. Quandle Homomorphisms and Knot Colorings

We write SCol_Q(K) to denote the set of all colorings of the knot K by the quandle Q.

Let f : Q₁ → Q₀ be a quandle homomorphism, and K be a knot. Note that for any coloring y ∈ SCol_Q1 (K) regarded as a quandle homomorphism Q(K) → Q₁, the composition x = f ◦ y : Q(K) → Q₀ is a coloring in SCol_Q0 (K). In this case we say that x lifts to y. This correspondence defines a map f_# : SCol_Q1 (K) → SCol_Q0 (K).

For x ∈ SCol_Q0 (K), denote by $f_{\#}^{-1}(x)\subset \mathrm{S}{\text{Col}}_{Q_1}(K)$ the set of colorings y ∈ SCol_Q1 (K) such that f_#(y) = x. (This set may be empty.) A pair of colorings x ∈ SCol_Q0 (K) and y ∈ SCol_Q1 (K) such that x = f ◦ y for a knot diagram determines another such a pair after each Reidemeister move, and hence for each x ∈ SCol_Q0 (K), the cardinality $|f_{\#}^{-1}(x)|$ does not depend on the choice of diagrams. Thus we obtain the following knot invariant.

Definition 6.1

For a given quandle homomorphism f : Q₁ → Q₀, we denote by Col_f (K) the multiset $\{|f_{\#}^{-1}(x)||x\in \mathrm{S}{\text{Col}}_{Q_0}(K)\}$, that does not depend on the choice of a diagram of a knot.

We use the notation [a, k] for k copies of the element a. For example, {[0, 3], [2, 4]} represents {0, 0, 0, 2, 2, 2, 2}. We observe the following.

• If Col_f (K) = {[h₁, k₁], …, [h_n, k_n]}, then Col_Q0 (K) = k₁ + ⋯ + k_n and Col_Q1 (K) = h₁k₁ + ⋯ + h_nk_n.

• Suppose Q₁ is connected. If ${\text{Col}}_{Q_0}^N(K)=0$ (i.e. K is only trivially colored by Q₀) and f is an epimorphism, then Col_f (K) = {[h, k]}, where k = |Q₀| and h = Col_Q1 (K)/|Q₀|.

• Suppose Q₁ is connected. If ${\text{Col}}_{Q_1}^N(K)=0$, then Col_f (K) = {[0, ℓ], [h, k]}, where $\ell ={\text{Col}}_{Q_0}^N(K)$, k = |Q₀|, and h = |Q₁|/|Q₀|.

• If f : Q₁ → Q₀ has image $Q_0^{\prime }$, let $f\prime :Q_1\to Q_0^{\prime }$ be the same map with the codomain $Q_0^{\prime }$, then Col_f (K) = Col_f′ (K) ∪ {[0, h]} where h is the number of colorings by Q₀ such that some colors are not from $Q_0^{\prime }$.

• If $f:Q_1=Q_0\times Q_0^{\prime }\to Q_0$ is the projection from a product quandle, then Col_f (K) = {[h, k]}, where k = Col_Q0 (K) and $h={\text{Col}}_{Q_0^{\prime }}(K)$.

Example 6.2

We computed the invariant Col_f (K) for i = 1, 2 and for 2977 knots up to 12 crossings, where f is the unique (up to isomorphisms of C[6, i] and C[3, 1]) epimorphism from C[6, i] to C[3, 1]. The values of the invariants for i = 1, 2 are the same for each K up to 12 crossings.

For 2977 knots, the types of multisets are

• {[2, 3]}, {[2, 3], [4, 6]}, {[2, 3], [4, 24]}, {[2, 3], [8, 6]}, {[2, 3], [8, 24]},

• {[2, 3], [4, 6], [8, 18]}, {[2, 3], [4, 12], [8, 12]}, {[2, 3], [4, 18], [8, 6]},

• {[2, 3], [8, 18], [16, 6]}, {[2, 3], [4, 36], [8, 24], [16, 18]}.

Example 6.3

There is an epimorphism f : C[8, 1] → C[4, 1] such that there are 12 non-trivial colorings of the trefoil K = 3₁ by C[4, 1], none of which lifts to a coloring by C[8, 1]. Hence for this f : C[8, 1] → C[4, 1] and for K = 3₁, the invariant Col_f:Q1→Q0 (K) contains 12 copies of 0s. There are four and eight trivial colorings of K by C[4, 1] and C[8, 1], respectively, and every trivial coloring by C[4, 1] has two lifts. Therefore we have Col_f (K) = {[0, 12], [2, 4]}.

We note that C[8, 1] is not Alexander but is an abelian extension of C[4, 1] = ℤ₂[t]/(t² + t + 1). Hence this computation is equivalent to the quandle cocycle invariant. All possible patterns over all of the first 1000 knots are

• {[2, 4]}, {[2, 16]}, {[2, 64]}, {[0, 12], [2, 4]},

• {[0, 24], [2, 40]}, {[0, 48], [2, 16]}.

Example 6.4

For f : C[9, 1] = R₉ → C[3, 1] = R₃, the patterns over all of the first 1000 knots are

• {[3, 3]}, {[9, 9]}, {[27, 27]}, {[0, 6], [9, 3]}, {[0, 18], [27, 9]}, {[0, 24], [27, 3]}.

Example 6.5

There is an epimorphism f : C[18, 5] → C[6, 2]. The quandle C[18, 5] is not Alexander, not a kei, not Latin, self-dual, and faithful. It is not an abelian extension of C[6, 2]. It has three subquandles, C[3, 1], a trivial quandle of two elements, a 6-element non-connected quandle, and does not have C[6, 1] nor C[6, 2] as subquandles. The RIG quandles onto which it has epimorphisms are C[3, 1], C[6, 2], and C[9, 6].

For this f : C[18, 5] → C[6, 2] based on the first 100 knots, the patterns of the invariant are

• {[3, 6]}, {[9, 30]}, {[9, 54]}, {[27, 102]}, {[0, 96], [27, 6]}, {[0, 120], [27, 6]}.

Remark 6.6

For all examples of epimorphisms f : Q₁ → Q₀ we computed, the invariant Col_f (K) is determined by (Col_Q1 (K), Col_Q0 (K)). Specifically, for these epimorphisms and all pairs of knots K and K′ with at most 12 crossings, if (Col_Q1 (K), Col_Q0 (K)) = (Col_Q1 (K′), Col_Q0 (K′)) then Col_f (K) = Col_f (K′). This holds for epimorphisms C[6, 1] → C[3, 1], C[8, 1] → C[4, 1], and C[9, 1] → C[3, 1]. It also holds for C[18, 5] → C[6, 2], for many knots (not necessarily all 2977 knots) we computed.

Furthermore, for C[6, 1] → C[3, 1], the invariant is determined by Col_{C[6, 1]}(K) alone for many knots, while for C[8, 1] → C[4, 1] and C[9, 1] → C[3, 1], both Col_Q1 (K) and Col_Q0 (K) are needed to determine the invariant.

Remark 6.7

From the considerations in the preceding remark, it is of interest whether the number of colorings of a quandle determines that of another. There are many such pairs among RIG quandles over all 2977 knots. We note that many of the pairs do not have epimorphisms between them, and more studies on this phenomenon may be desirable.

Remark 6.8

The quandle 2-cocycle invariant was defined in [6] as follows. Let ϕ : X × X → A be a quandle 2-cocycle, for a finite quandle X and an abelian group A. For a knot diagram K and a coloring x ∈ SCol_X(K), the weight at a crossing τ as depicted in Fig. 1 is defined by B_ϕ(τ, x) = εϕ(x, y), where ε = ±1 is the sign of τ, defined as +1 if the under-arc points right in the figure, otherwise −1. The 2-cocycle invariant Φ_ϕ(K) is defined as the multiset {∑_τ B_ϕ(τ, x) | x ∈ SCol_X(K)}.

The cocycle invariant and the invariant Col_f (K), which we defined for an epimorphism f : Q₁ → Q₀ for quandles Q_i, i = 0, 1, can be naturally combined as follows. Let A be an abelian group, and ϕ_i, i = 0, 1, be 2-cocycles of Q_i. Then define Φ_f;ϕ1,ϕ0(K) to be a multiset

$$\left\{\right(\sum _{\tau }{B}_{{\varphi }_0}(\tau ,x),\{\sum _{\tau }{B}_{{\varphi }_1}(\tau ,y)|y\in f_{\#}^{-1}(x)\left\}\right)|x\in \mathrm{S}{\text{Col}}_{Q_0}(K)\}.$$

7. Applications to Other Knot Invariants

In this section we give applications of the number of quandle colorings to a few invariants for the 2977 knots in [8]. In particular, we determine the tunnel number for some of the knots with 11 and 12 crossings in [8]. For definitions of knot invariants, we refer to [8, 42]. Recall that Col_X(K) denotes the number of colorings of a knot K by a finite quandle X. Let Lq_X(K) denote ⌈log_|X|(Col_X(K))⌉, where |X| denotes the order of X.

We computed Lq_Q(K) for the 2977 knots over a set 𝒬ℒ of 439 quandles consisting of all 431 RIG quandles and the 8 quandles Q_i, i = 16, 18, 19, 20, 22, 23, 25 from the 26 quandles Q_i in the first table of Appendix B. The additional eight quandles are not Alexander. Let MLq(K) be the maximum of Lq_Q(K) over all of the 439 quandles in 𝒬ℒ. Among 2977 knots with 12 crossings or less, there are 1473 knots K with MLq(K) = 2, 1441 knots with MLq = 3, and 63 (15 11-crossing and 48 12-crossing) knots with MLq = 4. Let MLq^F (K) denote the maximum of Lq_X(K) over all Alexander RIG quandles of the form Λ_p/(h(t)) where h(t) is irreducible in ℤ_p[t] for prime p and different from t and t − 1. Note that MLq and MLq^F are defined over all quandles in the particular set 𝒬ℒ. The computational results on these are listed in [21].

Bridge index

Let Bg(K) denote the bridge index of a knot K. In [41], a lower bound of the bridge index is given in terms of the number of quandle colorings: Let X be a finite quandle. For an n-bridge presentation of a knot, an assignment of colors at the n maxima determines the colors of the remaining arcs of the diagram, if it extends to the entire diagram and gives a well-defined coloring. Hence we have |Col_X(K)| ≤ |X|ⁿ, where |X| denotes the order of X. Thus we obtain Lq_X(K) ≤ Bg(K), and therefore, MLq(K) ≤ Bg(K).

It is proved in [3] that a Montesinos knot K with r rational tangle summands has Bg(K) = r. In [34], Montesinos knots were used for determining the bridge indices of knots with 11 crossings. In particular, the set of 4-bridge 11 crossing knots identified in [34] agreed with the list of 11 crossing knots with MLq = 4. These are:

• 11a_43, 11a_44, 11a_47, 11a_57, 11a_231, 11a_263, 11n_71, 11n_72,

• 11n_73, 11n_74, 11n_75, 11n_76, 11n_77, 11n_78, 11n_81.

At the time of writing, KnotInfo [8] did not have information on the bridge index for 12 crossing knots. The set of 12 crossing Montesinos knots with four rational tangle summands were computed by Jablan using LinKnot [24], and the list was provided to us by Musick. There are 48 knots in their table. Our computation showed that there are 48 knots with 12 crossings with MLq = 4, and the list can be found in [21]. These two lists coincide.

Nakanishi index

We use the notations Λ = ℤ[t, t⁻¹] and Λ_p = ℤ_p[t, t⁻¹]. The Nakanishi index NI(K) of a knot K is the minimum size of square presentation matrices of the Alexander module (H₁(Ỹ) as a Λ-module, where Ỹ is the infinite cyclic covering of the complement Y) of K [35]. The Nakanishi index for 11 and 12 crossing knots was blank in [8] when this project was started (June, 2012). We examine how much of the Nakanishi index can be determined for these knots using quandle colorings.

Lemma 7.1

For any knot K and Alexander quandle Q of the form Λ_p/(h(t)) where h(t) is irreducible in ℤ_p[t] for prime p and different from t and t − 1, it holds that

$$\mathrm{L}{\mathrm{q}}_Q(K)\le \mathrm{N}\mathrm{I}(K)+1\le \mathrm{B}\mathrm{g}(K).$$

Proof

As in the proof of Proposition 5.4, Λ_p/(h(t)) is a finite field 𝔽_q of order q denoted by 𝔽(p, h(t)) in [2], where q = p^k for a prime p and k = deg(h(t)).

It is shown in [2] that if A is a presentation matrix of the Alexander module, then the same matrix reduced modulo p, A^(p), is a presentation matrix of H₁(Ỹ;ℤ_p) as a Λ_p-module. Let $\overline{A^{(p)}}$ be the matrix A^(p) with entries reduced modulo h(t).

Let $\psi :{\mathrm{\Lambda }}_p^n\to {\mathrm{\Lambda }}_p^n$ denote the map corresponding to A^(p), and $\overline{\psi }:{𝔽}_p^n\to {𝔽}_p^n$ denote the map on the vector space ${𝔽}_p^n$ corresponding to $\overline{A^{(p)}}$. Then in [2] it was shown that the set of colorings SCol_Q(K) is in one-to-one correspondence with the vector space 𝔽_q ⊕ ker(ψ̅).

If NI(K) = n, then there is an n × n presentation matrix A for the Alexander module. Then ker(ψ̅) has dimension at most n. Hence

$${\text{Col}}_Q(K)=|{𝔽}_q\oplus \text{ker}(\overline{\psi })|=q^{1+\text{dim ker}(\overline{\psi })}\le q^{1+n},$$

so that we obtain Lq_Q(K) ≤ 1 + n. It is well known that NI(K) + 1 ≤ Bg(K).

Corollary 7.2

For any knot K, MLq^F (K) ≤ NI(K) + 1 ≤ Bg(K).

For 10 crossing knots, in KnotInfo [8] as of October 2013, the following knots are listed as having NI = 2: 10_k for

• k = 61, 63, 65, 69, 74, 75, 98, 99, 101, 103, 122, 123, 140, 142, 144, 155, 157, 160.

The list of 10 crossing knots such that Corollary 7.2 determines that NI = 2 consists of 10_ℓ for

• ℓ = 61, 63, 65, 74, 75, 98, 99, 103, 115, 122, 123, 140, 142, 144, 155, 157, 163.

Thus Corollary 7.2 determines knots with NI = 2 for all but k = 69, 101, and 160. We point out that Corollary 7.2 implies NI = 2 for k = 115 and 163, these values of NI are posted in [8] incorrectly as having NI = 1. This error was corrected in [27] (the knot 10₁₆₃ is denoted as 10₁₆₄ in [27]).

The 11 and 12 crossing knots with MLq^F (K) = 3, and therefore, NI(K) ≥ 2 by Corollary 7.2, are listed in [21]. Among these, knots with Bg(K) = 3 are determined to have NI(K) = 2 by Corollary 7.2. In particular, 11 crossing knots in this list (46 knots) have bridge index 3 by [34], and therefore, have NI = 2.

Tunnel number

Let τ(K) denote the tunnel number of a knot K. At the writing of this paper (October 2013), KnotInfo [8] did not contain the tunnel number for knots with 11 or more crossings.

Let Q be a quandle of the form Λ_p/(h(t)) for a prime p and an irreducible polynomial h(t) as before. It was shown in [23] (see also [2]) that Lq_Q(K) ≤ τ(K)+1 for any knot K. It is well known that τ(K) + 1 ≤ Bg(K) holds for any knot K. Hence we obtain the following lemma.

Lemma 7.3

If Lq_Q(K) = Bg(K) for some quandle Q of the form Λ_p/(h(t)), where h(t) is irreducible in ℤ_p[t] for prime p and different from t and t − 1, then τ(K) = Bg(K) − 1.

As we mentioned earlier the bridge index was determined for all 11 crossing knots in [34]. For RIG Alexander quandles Q of the form Λ_p/(h(t)), the following 11 crossing knots satisfy the condition Lq_Q(K) = Bg(K) = 3, and therefore they have tunnel number 2: 11a_k and 11n_ℓ where

• k = 87, 97, 107, 123, 132, 133, 135, 143, 155, 157, 165, 173, 181, 196, 239, 249, 277, 291, 293, 297, 314, 317, 321, 322, 329, 332, 340, 347, 352, 354, 366.

• ℓ = 49, 83, 90, 91, 126, 133, 148, 157, 162, 164, 165, 167, 175, 184, 185.

These are 46 knots among 552 total of 11 crossing knots.

Unknotting number

It is known [35] that the unknotting number u(K) of a knot K is bounded below by NI(K): NI(K) ≤ u(K). We use this fact to examine the unknotting number of knots in the KnotInfo table.

We consider the knots with MLq^F (K) = 3. The knots in the table with MLq^F (K) = 3 are posted at [21], together with the first simple prime power Alexander quandle X that gives Lq_X(K) = 3. A file containing knots with MLq_F (K) = 3 and their unknotting numbers is also found at [22]. In the list, the notation [2, 3] means u(K) = 2 or 3, which we also denote by u(K) = [2, 3] below for short-hand. By Corollary 7.2 and the inequality NI(K) ≤ u(K), for these knots with MLq^F (K) = 3 we obtain 2 = MLq^F (K) − 1 ≤ NI(K) ≤ u(K). Hence we obtain the following information on the Nakanishi index and the unknotting number for these knots with MLq^F (K) = 3 and with crossings 11 or 12:

• (U1)
  If u(K) = 2, then it is determined that NI(K) = 2. The knots that satisfy this condition are listed below.

  • 11a_ : 87, 97, 107, 132, 133, 135, 143, 155, 157, 165, 173, 181, 196, 239, 249, 277, 293, 297, 314, 317, 321, 322, 332, 347, 352.
  • 11n_ : 90, 164, 175, 185.
  • 12a_ : 216, 253, 408, 444, 466, 679, 701, 987, 1183, 1206.
  • 12n_ : 273, 332, 403, 436, 508, 510, 526, 549, 565, 570, 592, 604, 617, 643, 666, 839, 887.
• (U2)
  If u(K) = [1, 2], then it is determined that NI(K) = 2 and u(K) = 2. The knots that satisfy this condition are listed below.

  • 11a_ : None.
  • 11n_ : 49, 83, 91, 157, 162, 165, 167.
  • 12a_ : There are 63 knots in this category. See Appendix C for the list.
  • 12n_ : There are 74 knots in this category. See Appendix C for the list.
• (U3)
  If u(K) = 3, then we obtain the information that NI(K) = [2, 3]. The knots that satisfy this condition are listed below.

  • 11a_ : 123.
  • 11n_ : 126, 133, 148, 183.
  • 12a_ : 295, 311, 327, 386, 433, 561, 563, 569, 576, 615, 664, 683, 725, 780, 907, 921, 1194.
  • 12n_ : 147, 276, 387, 402, 494, 496, 581, 626, 654, 660.
• (U4)
  If u(K) = [2, 3], then NI(K) = [2, 3] and no new information is obtained for u(K). The knots that satisfy this condition are listed below.

  • 11a_ : 329.
  • 11n_ : None.
  • 12a_ : 297, 970, 1286.
  • 12n_ : 294, 509, 881.
• (U5)
  If u(K) = [1, 2, 3], then NI(K) = [2, 3] and the possibility for the unknotting number is narrowed to u(K) = [2, 3]. The knots that satisfy this condition are listed below.

  • 11a_ : None.
  • 11n_ : None.
  • 12a_ : 244, 291, 376, 381, 481, 493, 494, 634, 886, 1124, 1142, 1202, 1205, 1269, 1288.
  • 12n_ : 270, 601, 602, 630, 701, 844, 873.
• (U6)
  If u(K) = [2, 3, 4]; [3, 4]; 4, respectively, then NI(K) = [2, 3, 4] and no new information is obtained for u(K). The knots that satisfy these conditions are listed below.

  • 11a_ : 354, 366; 291, 340; None, respectively.
  • 11n_ : None; None; None.
  • 12a_ : 1097, 1164; 973; 574, 647.
  • 12n_ : None; 600, 764, 806; 386, 518.

We observed no other cases in the list. In particular, in our list of knots with MLq^F = 3, there is no knot K with u(K) containing 5, nor u(K) = [1, 2, 3, 4]. All the other positive integer intervals containing less than 5 are listed above.

Remark 7.4

Among 12 crossing knots K with MLq(K) = 4 (recall that there are 48 of them), the following seven knots have MLq^F (K) = 4, and for each of the seven knots, the only RIG quandle Q of type F(p, h(t)) such that Lq_Q(K) = 4 is Q = C[3, 1]:

• 12a_ : 554, 750.

• 12n_ : 553, 554, 555, 556, and 642.

These knots have the Nakanishi index at least 3 from Corollary 7.2. The bridge indices of these knots were determined to be 4. Hence by Corollary 7.2, we obtain that NI(K) = 3 for these knots.

The unknotting number for all of these knots except 12n_642 is listed in [8] as [1, 2, 3], and it is posted as [2, 3, 4] for 12n_642. Since NI = 3, the unknotting number of these knots except 12n_642 is determined to be 3, and it is [3, 4] for 12n_642.

Appendix A. Enumeration of Quandles and Properties of RIG Quandles

Sequences enumerating quandles of order n with various properties may be found in the Online Encyclopedia of Integer Sequences [45]. We list the OEIS identification numbers of some of these sequences below. We also list below some properties and characterizations of RIG quandles that may be found in files at [22].

• http://oeis.org/A193024. The number of isomorphism classes of Alexander quandles of order n.

• http://oeis.org/A193067. The number of isomorphism classes of connected Alexander quandles of order n.

• http://oeis.org/A225744. The number of isomorphism classes of connected Generalized Alexander quandles of order n.

• http://oeis.org/A196111. The number of isomorphism classes of simple quandles of order n. A quandle is simple if it has more than one element, and if it has no homomorphic images other than itself or the singleton quandle. However, in [1], it is further assumed that simple quandles are not trivial. This also rules out the quandles of order 2.

• http://oeis.org/A177886. The number of isomorphism classes of Latin quandles of order n. A Latin quandle is a quandle such that for each a ∈ X, the left translation ℒ_a is a bijection. That is, the multiplication table of the quandle is a Latin square.

• https://oeis.org/A226172. The number of isomorphism classes of faithful connected quandles of order n. A quandle is faithful if the mapping a ↦ ℛ_a from X to Inn(X) is an injection.

• https://oeis.org/A226173. The number of isomorphism classes of connected keis (involutory quandles) of order n. A quandle X is involutory, or a kei, if the right translations are involutions: ${ℛ}_a^2=\mathrm{i}\mathrm{d}$ for all a ∈ X.

• https://oeis.org/A226174. The number of isomorphism classes of self-dual connected quandles of order n.

We also obtained the following information about RIG quandles. This is available at [22]:

• Subquandles of RIG quandles that are RIG quandles and trivial quandles. We also identified the orders of subquandles that are neither connected nor trivial.

• The connected RIG Alexander quandles are presented as ℤ[t]/J where J = (f(t)) or J = (f(t), g(t)) except for one Alexander RIG quandle, namely C[27, 17], which is not a cyclic ℤ[t, t⁻¹]-module.

• We identified RIG quandles that are products of other RIG quandles.

• We identified dual quandles of RIG quandles. In particular, if the dual is itself, then it is self-dual.

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

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

  and ϕ(x, x) = 0 for any x, y, z ∈ X. For a quandle 2-cocycle ϕ, X × A becomes a quandle by (x, a) * (y, b) = (x * y, a + ϕ(x, y)) for x, y ∈ X, a, b ∈ A, and it is called an abelian extension of X by A. See [5], for example, for more details of extensions. A partial list of RIG quandles that are abelian extensions of RIG quandles is found in [22].

• For each RIG quandle C[n, i] we identified a set Gen[n, i] which is a smallest set of generators for C[n, i]. We also produced the set of all [n, i] such that any two elements of C[n, i] will generate C[n, i]. Note that if [n, i] is in this set then there are no non-trivial subquandles of C[n, i].

Remark A.1

It is known that connected quandles of prime order p and those of order p² are Alexander quandles [20]. By examining characterizations and properties of RIG quandles, listed above, one might wonder if these results might generalize. For example, RIG quandles of order 2ⁿ for some n, and those with order pq for primes p, q > 3 are Alexander. However, inspections of higher-order conjugation quandles reveal that there are quandles with order 64 and 55 that are not Alexander. The former is a size 64 conjugacy class of GAP’s SmallGroup(192, 1023). Two of the latter are a conjugacy class of PSL(2, 11) and a conjugacy class of SmallGroup(1210, 7).

Remark A.2

As in [18, 46], for computational purposes we represent quandles by matrices. The entries of the matrix A of a quandle of order n are integers 1, …, n and the quandle operation is given by i * j = A_i,j. We note that the quandles in the RIG package are left distributive whereas for knot colorings we prefer right distributive quandles. Hence we have reversed the order of the product in the RIG quandle by taking the transposes of the matrices representing the RIG quandles. In the RIG package the ith quandle of order n is denoted by SmallQuandle(n, i), for which we use the notation C[n, i].

Appendix B. Tables of Quandles Used to Distinguish Knots and Their Mirrors

List of 26 quandles that distinguish 2977 knots up to orientation and mirror image. The quandle matrices can be found at [22].

Quandle	Order	GAP ID for Inn(Q)	Structure Description(Inn(Q))
Q1	12	SmallGroup(60, 5)	A5
Q2	13	SmallGroup(52, 3)	C13 : C4
Q3	13	SmallGroup(52, 3)	C13 : C4
Q4	13	SmallGroup(156, 7)	(C13 : C4) : C3
Q5	13	SmallGroup(156, 7)	(C13 : C4) : C3
Q6	15	SmallGroup(60, 5)	A5
Q7	17	SmallGroup(136, 12)	C17 : C8
Q8	17	SmallGroup(136, 12)	C17 : C8
Q9	20	SmallGroup(120, 34)	S5
Q10	24	SmallGroup(168, 42)	PSL(3, 2)
Q11	25	SmallGroup(75, 2)	(C5 × C5) : C3
Q12	27	SmallGroup(702, 47)	((C3 × C3 × C3) : C13) : C2
Q13	27	SmallGroup(702, 47)	((C3 × C3 × C3) : C13) : C2
Q14	30	SmallGroup(120, 34)	S5
Q15	32	SmallGroup(160, 199)	((C2 × Q8) : C2) : C5
Q16	40	SmallGroup(320, 1635)	((C2 × C2 × C2 × C2) : C5) : C4
Q17	40	SmallGroup(320, 1635)	((C2 × C2 × C2 × C2) : C5) : C4
Q18	42	SmallGroup(168, 42)	PSL(3, 2)
Q19	48	SmallGroup(288, 1025)	(A4 × A4) : C2
Q20	60	SmallGroup(660, 13)	PSL(2, 11)
Q21	72	SmallGroup(576, 8652)	(A4 × A4) : C4
Q22	72	SmallGroup(504, 156)	PSL(2, 8)
Q23	84	SmallGroup(1512, 779)	PSL(2, 8) : C3
Q24	84	SmallGroup(1512, 779)	PSL(2, 8) : C3
Q25	90	SmallGroup(720, 765)	A6 · C2
Q26	182	SmallGroup(1092, 25)	PSL(2, 13)

List of 23 quandles that distinguish 1058 knots from their mirror images. The quandle matrices can be found at [22].

Quandle	Order	GAP ID for Inn(Q)/Structure Description(Inn(Q))
T1	351	SmallGroup(2106, 0)	(((C3 × C3 × C3) : C13) : C3) : C2
T2	504	SmallGroup(3024, 0)	C2 × (PSL(2, 8) : C3)
T3	504	SmallGroup(4536, 0)	PSL(2, 8) : C9
T4	18	SmallGroup (216, 90)	(((C2 × C2) : C9) : C3) : C2
T5	27	SmallGroup(216, 86)	((C3 × C3) : C3) : C8
T6	27	SmallGroup(486, 41)	((C3.((C3 × C3) : C3) = (C3 × C3) · (C3 × C3)) : C3) : C2
T7	28	SmallGroup(168, 43)	((C2 × C2 × C2) : C7) : C3
T8	720	SmallGroup(7920, 0)	M11
T9	112	SmallGroup(1344, 816)	(((C2 × C2 × C2) · (C2 × C2 × C2)) : C7) : C3
T10	112	SmallGroup(1344, 816)	(((C2 × C2 × C2) · (C2 × C2 × C2)) : C7) : C3
T11	117	SmallGroup(1053, 51)	((C3 × C3 × C3) : C13) : C3
T12	162	SmallGroup(1296, 2890)	(C3 · (((C3 × C3) : Q8) : C3) = (((C3 × C3) : C3) : Q8).C3) : C2
T13	162	SmallGroup(1296, 2891)	((((C3 × C3) : C3) : Q8) : C3) : C2
T14	192	SmallGroup(1344, 814)	(C2 × C2 × C2) · PSL(3, 2)
T15	125	SmallGroup(1500, 36)	(((C5 × C5) : C5) : C4) : C3
T16	135	SmallGroup(1620, 421)	((C3 × C3 × C3 × C3) : C5) : C4
T17	135	SmallGroup(1620, 421)	((C3× C3 × C3 × C3) : C5) : C4
T18	168	SmallGroup(1512, 779)	PSL(2, 8) : C3
T19	64	SmallGroup(448, 179)	((C2 × C2 × C2) · (C2 × C2 × C2)) : C7
T20	64	SmallGroup(768, 1083508)	((((C8 × C4) : C2) : C2) : C2) : C3
T21	64	SmallGroup(768, 1083509)	((((C8 × C4) : C2) : C2) : C2) : C3
T22	40	SmallGroup(320, 1635)	((C2 × C2 × C2 × C2) : C5) : C4
T23	84	SmallGroup(1512, 779)	PSL(2, 8) : C3

Appendix C. Knots with Unknotting Number 2

In this section we list knots with 12 crossings such that their unknotting number is listed as 1 or 2 in KnotInfo [8], and our computation that MLq^F = 3 determines that their unknotting number is in fact 2.

The list of alternating knots is 12a_k for

• k = 100, 177, 215, 218, 245, 248, 249, 265, 270, 279, 298, 312, 332, 347, 348, 396, 413, 427, 429, 435, 448, 465, 475, 503, 594, 703, 712, 742, 769, 787, 806, 808, 810, 868, 873, 895, 904, 905, 906, 941, 949, 960, 975, 990, 1019, 1022, 1026, 1053, 1079, 1092, 1093, 1102, 1105, 1123, 1152, 1167, 1181, 1225, 1229, 1251, 1260, 1280, 1283.

The list of non-alternating knots is 12n_ℓ for

• ℓ = 144, 145, 257, 268, 269, 274, 297, 333, 334, 355, 356, 357, 379, 380, 388, 389, 393, 394, 397, 414, 420, 440, 442, 460, 462, 480, 495, 498, 505, 533, 546, 567, 571, 582, 583, 598, 605, 611, 622, 636, 637, 651, 652, 669, 706, 714, 717, 737, 742, 745, 746, 752, 756, 757, 760, 779, 781, 798, 813, 817, 837, 838, 840, 843, 846, 847, 869, 874, 876, 877, 878, 879, 883.

Appendix D. Remarks on the Computation of Col_Q(K)

We used several techniques to compute Col_Q(K) for quandles Q and knots K. We take for each knot the braid representation given at KnotInfo [8]. We take the braids to be oriented from top to bottom. This induces an orientation of the knots. The two parameters that are most important for computing the number of colorings are the braid index, b(K) (the number of strands) and the order |Q| of the quandle. A straightforward computation requires |Q|^b(K) steps. Using the fact that our quandles are connected, i.e. the group Inn(Q) acts transitively on Q we may assume that one of the arcs has a fixed color. This allowed us to reduce the number of steps to |Q|^b(K)−1, which reduces the time by a factor of 1/|Q|. In the case of quandles of large order this makes a noticeable difference in program running times (hours vs. months).

We initially used Mace4 [32] to find the number of colorings. Later on, we wrote a program in C to verify that the results we obtained using Mace4 were correct. The C program uses the same braid representations as those used for the Mace4 computations.

Let K be a knot with a braid index b(K). Let Q be a quandle of order n. Let a_i, i = 1, …, b, represent the element of a given quandle assigned to the top of the ith strand of a braid, where b = b(K) is the braid index. To count the number of knot colorings, we find the colorings at the bottom strands of the braid, and check assignments such that the top and bottom colors are the same for each strand; cf. Fig. 2. The following pseudocode demonstrates that our algorithm runs in O(n^b−1) time, n = |Q|.

/* fix strand a1 to be 0. This allows us to get away *

* with coloring b−1 strands instead of b strands. */

a1 = 0;

for (a2=0; a2 < n ; a2++) {

for (a3 …) {

for (a4 …) {

for (a5 …) {

… up to the braid index ’b ’

/* constant time quandle coloring of knot strands */

}

}

}

}

We implemented a multithreaded version of the program using POSIX threads to further improve performance. Combined with fixing a color of one strand, we found that multithreading reduces the computation time considerably on multicore processors. The unexpectedly large speedup of multithreading over trivial parallelization may be due to operating system characteristics and prevention of CPU cache swapping to the slower RAM when the operating system is switching tasks.

The computations were performed on a GNU/Linux computer equipped with 24GB of RAM and an Intel Core i7 processor with four cores. The kernel was compiled with a task switching latency of 100 Hz.

In the following table, the first column indicates 26 quandles in the first table of Appendix B, the second column gives the order of the quandle, the third column is the computation time for all 2977 knots with at most 12 crossings using a single thread, the fourth column is time using multiple threads, and the last column is the ratio of column 3 and column 4. A log plot of timings of three different coloring programs is given in Fig. D.1.

Quandle	Order	Serial seconds	Multithread seconds	Speedup factor
Q1	12	1	4	0.25 × (slower)
Q2	13	2	4	0.5 × (slower)
Q3	13	2	4	0.5 × (slower)
Q4	13	2	4	0.5 × (slower)
Q5	13	2	4	0.5 × (slower)
Q6	15	3	4	0.75 × (slower)
Q7	17	6	4	1.5 × (faster)
Q8	17	6	4	1.5 × (faster)
Q9	20	12	5	2.4 × (faster)
Q10	24	31	5	6.2 × (faster)
Q11	25	38	5	7.6 × (faster)
Q12	27	60	5	12 × (faster)
Q13	27	60	5	12 × (faster)
Q14	30	105	7	15 × (faster)
Q15	32	125	7	17.86 × (faster)
Q16	40	455	9	50.56 × (faster)
Q17	40	455	9	50.56 × (faster)
Q18	42	542	10	54.2 × (faster)
Q19	48	1215	14	86.79 × (faster)
Q20	60	(hours)	28	?
Q21	72	(days)	56	?
Q22	72	(days)	56	?
Q23	84	(days)	112	?
Q24	84	(days)	112	?
Q25	90	(weeks)	155	?
Q26	182	(months)	5602	(a lot faster)

Fig. D.1.

Log plot of timings.

Multithreading was used to perform computations in parallel for the outermost loop of our knot coloring program. We set the number of threads to be the order of the quandle. None of the child threads communicate with each other. The parent thread waits for all of the child threads to return the number of colorings from its search. The parent sums each of the results from the child threads. The number of colorings of the knot is the sum of the counts from the child threads. The multithreaded version of the program is required to compute the colorings by the quandle of order 182.