Notation and Basic Facts in Knot Theory

Home , Alexander duality, Hyperbolic link, Knot group

Appendix A Notation and Basic Facts in Knot Theory

In this appendix, our aim is to provide a quick review of basic terminology and some facts in knot theory, which we use or need in this book. This appendix lists them item by item (without proof); so we do no attempt to give a full rigorous treatments; Instead, we work somewhat intuitively. The reader with interest in the details could consult the textbooks [BZ, R, Lic, KawBook]. • First, we ﬁx notation on the circle

S1 := {(x, y) ∈ R2 | x2 + y2 = 1}, and consider a finite disjoint union S1. A link is a C∞-embedding of L :S1 → S3 in the 3-sphere. We denote often by #L the number of the disjoint union, and denote the image Im(L) by only L for short. If #L is 1, L is usually called a knot, and is written K instead. This book discusses embeddings together with orientation. For an oriented link L, we denote by −L the link with its orientation reversed, and by L∗ the mirror image of L. • (Notations of link components). Given a link L :S1 → S3, let us fix an open tubular neighborhood νL ⊂ S3. Throughout this book, we denote the complement S3 \ νL by S3 \ L for short. Since we mainly discuss isotopy classes of S3 \ L,we may ignore the choice of νL. 2 • For example, for integers s, t ∈ Z , the torus link Ts,t (of type (s, t)) is defined by 3 2 s t 2 2 S Ts,t := (z,w)∈ C z + w = 0, |z| +|w| = 1 .

This Ts,t is a knot if and only if s and t are relatively prime. • In addition, as a trivial example, the unknot (or the trivial knot) is the embedding K : S1 → S3; (x, y) → (x, y, z,w).If K is the unknot, the complementary space S3 K is homeomorphic to S1 × D2. • Two links L and L are called isotopic, if there is a smooth family of diffeomor- 3 3 phisms ht : S → S such that ht (L) is a link, and h1(L) = L and h0 = idS3 .

Fig. A.1 The trefoil knot, ﬁgure eight knot, and the Tm,n-torus link with labeled arcs

An invariant (of links) is a map I from the set of links to some set S such that, for any two links L and L which are isotopic, I (L) = I (L ) holds in S (Fig. A.1). 3 • The fundamental group π1(S K ) is sometimes called the knot group. • We will mention meridians and longitude. Given a link L, ﬁx a tubular neighborhood νL ⊂ S3,as before. Then νL \ L is homotopic to #L S1 × S1, which is regarded as a S1-bundle over Im(L). Thus, we have a section s : Im(L) =#L S1 → νL \ L, called a framing. The -th component of Im(s) is called the (-th) longitude (of L). Furthermore, a representative circle of the -th component of the S1-ﬁber is called the (-th ) meridian. We denote them by l and m, respectively; see Fig. 2.4 as an example. Furthermore, with a choice of basepoint in the complementary space S3 L,we often regard the loops m and l, up to homotopy, as elements in the fundamental 3 group π1(S L). The abelian subgroup, P, generated by m and l is usually called the (-th) peripheral subgroup. Since we deal with links up to isotopy, we should give a remark: That is, in contrast to many choices of basepoints and the embedding 3 L, the peripheral subgroups P1,...,P#L ⊂ π1(S L) considered up to conjugacy depend on only the isotopy class of L.

Example A.1 Fix four integers (a, b, s, t) ∈ Z4 with as + bt = 1. Then, it is known [BZ, Lic] that the knot group of the (s, t)-torus knot has the presentation

3 ∼ s t π1(S Ts,t ) = x, y, | x = y .

Then, the meridian m and the longitude l are represented by xb ya and m−st xs , respec- s tively, and this π1 has the center generated by x . It is known [Sim] that a knot group 3 π1(S K ) contains a non-trivial center, if and only if K is a torus knot.

• A(-th) Seifert surface of an oriented link L is an oriented surface embedded in 3 S L such that the boundary is the longitude l compatible with orientation. For any link and , we can construct a (-th) Seifert surface of L (via “Seifert algorithm”), although we can choose many such surfaces. • The integral homology of S3 L is as follows (which is immediately shown by the Alexander duality). Appendix A: Notation and Basic Facts in Knot Theory 101 ⎧ ⎪ Z, = , ⎨⎪ if i 0 Z#L , = , ( 3 ; Z) =∼ if i 1 Hi S L ⎪ Z#L−1, = , (A.1) ⎩⎪ if i 2 0, otherwise.

3 ∼ #L Here, generators of the ﬁrst H1(S L) = Z are represented by the meridians m. We will brieﬂy review geometrical properties of knots. First, we observe that the 3 isotopy types of knots are almost determined by the fundamental group π1(S K ) and a meridian-longitude pair. Precisely,

Theorem A.2 (Waldhausen [Wal]) Let K and K be two knots in S3. Then, K is isotopic to either K or −K ∗ if and only if there is a group isomorphism Φ : 3 3 π1(S K ) −→ π1(S K ) such that Φ(m) = m and Φ(l) = l .

3 Furthermore, as is known, knot group π1(S K ) has interesting properties:

Theorem A.3 (The loop theorem and the sphere theorem [Pa]) Let K be a knot. 3 1. The longitude l is not trivial in π1(S K ), if and only if K is not the unknot, 3 3 2. S K is an Eilenberg-MacLane space K (π, 1) with π = π1(S K ). That is, 3 the homotopy group πn(S K ) for any n ≥ 2 is zero.

Furthermore, knot groups have a special property in the category of groups:

Theorem A.4 (Gonzéalez-Acuña [Gon], Johnson [Joh]) Let G be a group, and ﬁx z0 ∈ G. Then, the followings are equivalent: 3 1. There are a knot K with meridian m and a group epimorphism f : π1(S K ) → G such that f (m) = z0, −1 2. The group G is ﬁnitely generated, and is generated by the set {g z0g}g∈G .

Proposition A.5 (Ryder [Ry]) Let Q K be the knot quandle of a knot K ; see Sect.2.3 ∼ 3 for the detailed deﬁnition: The map κ : Q K → As(Q K ) = π1(S K ) is injective, if and only if K is trivial or prime.

Next, we review basic notions appearing in Theorem 7.20 (which are also impor- tant for the geometrization of knots). • A knot is said to be prime if, for any decomposition as a connected sum, one of the factors is unknotted. If not so, the knot is called composite. Here, see [Lic, BZ] for the deﬁnition of connected sum among (oriented) knots. • A knot K is cable,ifK is not any torus knot, and if there is a solid torus V embedded in S3 such that V contains K as the (p, q)-torus knot for some p, q ∈ Z. More generally, a knot K is satellite, if it is embedded in a small solid torus neighborhood of some knot K0, not the unknot, and K is not isotopic to K0 nor is contained in a ball inside the solid torus. Here, as is common in 3-dimensional topology, we should suppose that the unknot is not a torus knot, and torus knots are not cable knots. 102 Appendix A: Notation and Basic Facts in Knot Theory

Theorem A.6 (JSJ-decomposition of knots (see [Bud, HWO] for the details)) Let 3 L ⊂ S be a prime knot. Then, there exists a sequence of open sets V1 ⊂ V2 ⊂···⊂ 3 Vn ⊂ S satisfying the following properties: 3 1. The set Vi is an open solid torus in S , and V1 contains the knot L. 2. The difference Vi+1 − Vi for any i ∈ Z≥0 is one of a hyperbolic link, or an 2 (ni , mi )-torus knot for some (ni , mi ) ∈ Z , or a composite knot in the solid torus.

As is known, the decomposition is unique in some sense. Here, we should notice that L is cable iff Vj − Vj−1 for some j is an (n j , m j )-torus knot in the solid torus. Changing to 2-dimensional descriptions, we next focus on a diagrammatic approach to links: Precisely, we will review the Reidemeister Theorem A.7, which enables us to diagrammatically deal with links. To state, we deﬁne a link diagram to be a smooth immersion D :S1 → R2 whose double points are transverse, and in general position (so no triple intersections, or higher, occur) where the diagram contains the information if the arcs of the crossings are over or under. Here, an arc of D is a path as a subset of Im(D) running from a over-crossing to the next crossing. We call such a double point of D a crossing (in D). Two diagrams D and D are 2 2 called isotopic, if there is a smooth family of diffeomorphisms ht : R → R such that each ht (D) is a diagram, and h1(D) = D and h0 = idR2 . Furthermore, when a link diagram D is obtained as the image of a link L via the projection R3 → R2, D is also called a diagram of the link L. It is well known that every link (with a little perturbation) ensures some link diagram, and conversely that every link diagram is obtained from some link L. Moreover,

Theorem A.7 (Reidemeister theorem) Let L and L be two links. Choose two diagrams D and D of L and L , respectively. Then, L and L are isotopic in R3 if and only if D and D are related by a sequence of isotopies in R2 and the R-I, R-II, R-III moves. Here the R-I, R-II, R-III moves are the local relation as illustrated in Fig.3.3, and called the Reidemeister moves (of type I, II, III), respectively. In each diagram, the diagrams are equivalent outside of the pictures, and the change only occurs in the region pictured.

In this theorem, we can replace the 2-plane R2 by the 2-sphere S2. Thus, in this book, we often work with diagrams on the 2-sphere. 3 The following diagrammatic discussion is useful to describe the group π1(S L).

Theorem A.8 (Wirtinger presentation) Let L be a link, and D be a diagram of L. 3 The fundamental group π1(S L) has the following ﬁnite presentation: (α ) −1 −1 ( τ ) . eα is an arc of D eγτ eβτ eατ eβτ for all crossing as in Fig. 3.1

Here, the indeterminate eα can be represented by a meridian around the arc α. Appendix A: Notation and Basic Facts in Knot Theory 103

3 Example A.9 It happens often that the presentation of π1(S L) is complicated.

Even concerning the ﬁgure eight knot K41 , the Wirtinger presentation implies

π ( 3 ) =∼ , | −1 = −1 −1 −1 . 1 S K41 g h h gh g h ghg hg

A.1 Cyclic Branched Coverings, and Hyperbolic Links

This section reviews cyclic branched coverings, and hyperbolic links. Given a link L ⊂ S3 and r ∈ N, we will construct the cyclic branched cover- 3 ing space. By the abelianization (A.1), we have q : π1(S L) → Z/r deﬁned by (m ) = r → 3 q i 1. Then we have the associated cover EL S L of r-sheets. By con- r struction, the boundary of EL is a union of #L tori; hence, we can canonically attach r r #L solid tori to EL along the boundary, such that every meridian in EL vanishes by the attachment. The resulting closed 3-manifold is called the r-fold cyclic covering 3 r of S branched along a link L. In this book, we denote it by CL . multiplication 3 Ab #L In addition, the covering space associated with π1(S L) −→ Z −−−−−−−→ Z 3 ∞ is called the inﬁnite cyclic cover of S L. This book denotes it by EL . Starting from a link diagram D of L, we describe presentations of the fundamental π ( r ) π ( r ) η ,...,η ∈ Z groups 1 EL , 1 CL .Let 0 n be the arcs of this D. For an index s ,we formally take a copy ηi,s of the arc ηi . Proposition A.10 ([Kab, Sect. 3] or [R, Sect. 10], shown by Reidemeister-Schreier ∈ N π ( r ) method) For r and a link L, the fundamental group 1 EL can be presented by

generators: ηi,s (0 ≤ i ≤ n, s ∈ Z),

η = η−1 η γ relations: γ,s β,s−1 α,s−1 β,s for each crossings such as Fig. 3.1, and η0,0 = η0,1 =···=η0,r−2 = 1, and ηk,s+r = ηk,s.

Moreover, if r =∞, the above presentation is isomorphic to π1 of the inﬁnite cyclic ∞ cover EL , and we can deﬁne the inclusion

ι : π ( ∞)→ π ( 3 ); ι(η ) = ηs−1η η−s . 1 EL 1 S L i,s 0 i 0

< ∞ π ( r ) Moreover, if r , then the fundamental group 1 CL is isomorphic to this presented group with the additional relation η0,r−1 = 1. Next, we briefly review the Alexander modules. For this, we restrict ourselves = ∞ =∞ to the knot case L K and the infinite cyclic cover EL , that is, r . The first ( ∞; Z) integral homology H1 EK is called the Alexander module of K . 3 There is another description. Consider the abelianization π1(S K ) → Z = t . Then, we can consider the local coefficients over the Laurent polynomial Z[t±1], 3 ±1 and the associated homology H1(S K ; Z[t ]). Then, as is known as Shapiro’s 104 Appendix A: Notation and Basic Facts in Knot Theory

( ∞; Z) =∼ lemma (see [Bro, Wei1]), we can canonically obtain an isomorphism C∗ EK 3 ±1 C∗(S K ; Z[t ]) of cellular complexes; Thus,

( ∞; Z) =∼ ( 3 ; Z[ ±1]). H∗ EK H∗ S K t (A.2)

The Alexander module (and Alexander polynomial) has a long history, and is well- studied in many ways; see [Lic, BZ, Kau, R, Tro] with examples. Changing the subject, we will end this subsection by explaining roughly hyperbolic links. To explain this, we consider the upper half space with metric:

dx2 + dy2 + dz2 H3 := {(x, y, z) ∈ R3 | z > 0 }, dg2 = . z2

Then, a link L ⊂ S3 is hyperbolic,ifS3 L has a (complete) metric such that every local coordinate is isometric to H3. Consider the universal covering space of a hyperbolic link L. Since the space with negative curvature is simply connected and complete, it must be isometric to (H3, dg2). Hence,

Proposition A.11 Any hyperbolic link complement is an Eilenberg-MacLane space.

In addition, we mention the well-known fact:

3 Proposition A.12 The orientation preserving isometry group Isom+(H , g) is isomorphic to P SL2(C).

3 Hence, we have the covering transformation ρL : π1(S L) → PSL2(C), deﬁned up to conjugacy. This ρL is commonly called the holonomy representation of L. Furthermore, as is known, the image of each peripheral subgroup is contained in the unipotent subgroup of PSL2(C), up to conjugacy. That is, ρL is a parabolic PSL2-representation. Next, we will explain the geometrization theorem of links. This theorem says that most links admit hyperbolic structures, except essential torus case. Precisely,

Theorem A.13 (Geometrization theorem of links) Let L be a link. (1) Let L be a prime knot (#L = 1). Then, S3 L has a hyperbolic structure if and only if L is neither the torus knot nor a satellite knot. (2) Let #L ≥ 2, and L be indecomposable (i.e., it cannot be separated into two parts which can be isotopic to disjoint 3-balls). Suppose that no component is a torus knot and L is not a satellite link (Here, L is satellite, if one or more components of L is satellite to a component of the unlink). Then, S3 L has a hyperbolic structure.

The hyperbolic structure provides interesting properties and invariants of a knot complement when it exists. Here is a property that we use in this book: Appendix A: Notation and Basic Facts in Knot Theory 105

Proposition A.14 (Algebraic atroidality; see, e.g., [AFW] and references therein.) 3 Let L be a hyperbolic link, and let G be π1(S L). Let K be the peripheral subgroup of G with respect to the -th component. Then, for any (i, j) and any g ∈ G with g ∈/ Ki , the intersection satisﬁes the −1 condition g Ki g ∩ K j ={1G }. Appendix B Automorphism Groups from Quandles

Abstract As seen in Theorems 2.23 and 2.24, it is signiﬁcant to determine the associated group As(X) and the 2-nd homology H2(Inn(X)). As a study of quandles from group theory, we analyse the groups in detail. In Sect.B.1, we give a method for describing the inner automorphism group Inn(X), and give some examples (see also Sect. B.3). Furthermore, Sect.B.2 discusses quotients of quandles, and gives the deﬁnitions of presentation of quandles. As an analogy of abelianization in group theory, we give the Alexanderization in Sect. B.4. In this chapter, we list examples of automorphism groups and quotient groups. We suppose notation in Chap.2.

B.1 Calculations of Inner Automorphism Groups

Recall the central extension As(X) → Inn(X) in (2.6). Thus, in order to study As(X), we shall develop a method of describing the inner automorphism group Inn(X).After that, we give several examples (where some known groups are recovered). To begin, here is the key statement to determine Inn(X). Theorem B.1 ([N8] cf. Deﬁnition 3.10 of augmented quandles) Let a group G act on a quandle X. Let a map κ : X → G satisfy the following two conditions: 1. The identity x y = x · κ(y) ∈ X holds for any x, y ∈ X. 2. The image κ(X) ⊂ G generates the group G, and the action X G is effective. Then, there is a group isomorphism Inn(X) =∼ G, and the action X G agrees with the natural action of Inn(X). Proof. Identify the action X G with a group homomorphism F : G → Bij(X, X). It follows from (1.) that F(κ(X)) ⊂ Inn(X) and F(κ(X)) generates Inn(X); thus, F extends to an epimorphism F : κ(X) →Inn(X), where κ(X) is the subgroup of G generated by κ(X). Then, the former assumption of (2.) ensures κ(X) =G, and the second implies the bijectivity of F, i.e., Inn(X) =∼ G. Moreover, the agreement of the two actions follows by construction. © The Author(s) 2017 107 T. Nosaka, Quandles and Topological Pairs, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-981-10-6793-8 108 Appendix B: Automorphism Groups from Quandles

Thanks to this theorem, we can Inn(X) of many quandles X. For example,

× Lemma B.2 For r ∈ F , recall from Example 2.10 the symplectic quandle Xr = 2n F {0} over a ﬁeld F with r. Further, X be the conjugacy quandle of the form ( ) r∈F×/(F×)2 Xr . Then, Inn X is isomorphic to the symplectic group

Sp(2n; F) ={ h ∈ GL(2n; F) | thΩh = Ω }. −E Here, by t A we mean the transpose of a matrix A, and we let Ω = 0 n . En 0

Proof. For any y ∈ X,themap(• y) : X → X is a restriction of a linear map F 2n → F 2n. It thus yields a map κ : X → GL(2n; F); y →• y, which factors through Sp(2n; F) and satisﬁes the ﬁrst condition in Theorem B.1. Furthermore, the second condition is exactly the classical statement of the Cartan-Dieudoné theorem (possibly, in the case Char(F) = 2; see [O’M]), which says that Sp(2n; F) is generated by transvections. Therefore Inn(X) =∼ Sp(2n; F) as desired.

Example B.3 We now explain that Theorem B.1 is inspired by the Cartan embeddings in symmetric space theory. Recall from Example2.8 that every symmetric space X admits a quandle structure on X. Consider the group Inn(X) ⊂ Diff(X) generated by the symmetries •y with compact-open topology. As is well known, Inn(X) has a Lie group structure, and the map X → Inn(X) that sends y to sy is commonly called the Cartan embedding. As seen in textbooks on symmetric spaces, Theorem B.1 has been used to determine Inn(X) concretely.

Next, we observe that As(X) is compatible with the case where Inn(X) is perfect. ⊕O(X) Proposition B.4 Take the abelianization ⊕i εi : As(X) → Z in Lemma 2.27. ( ) gr( ( ); Z) = If Inn X is perfect, i.e., H1 Inn X 0, then there is an isomorphism

∼ ⊕O(X) As(X) = Ker(⊕i∈O(X)εi ) × Z . (B.1)

Further, this Ker(⊕i∈O(X)εi ) is a centrally extended group of Inn(X) and is perfect. gr( ( )) = ( ) =∼ ( ) × Z In particular,if X is connected and H2 Inn X 0, then As X Inn X . gr( ( )) = (ψ )→ Proof. Since H1 Inn X 0 by assumption, the composite Ker X As . ( ) −→proj gr( ( )) = Z⊕O(X) X H1 As X obtained from (2.6) is surjective. Then, we can ⊕O(X) ⊕O(X) choose a section s : Z → Ker(ψX ) of the composite, since Z is free. Hence, by the equality (2.5) and the inclusion Ker(ψX ) ⊂ As(X), the semi-direct ∼ ⊕O(X) product As(X) = Ker(⊕i∈O(X)εi ) Z is trivial, leading to (B.1) as desired. Furthermore the kernel Ker(⊕i∈O(X)εi ) is a central extension of Inn(X) by con- ∼ ⊕O(X) struction, and is perfect by the Künneth theorem and As(X)ab = Z , which completes the proof.

Remark B.5 Here recall the basic fact (see [Wei1, Sect. 6.9]) that, for a perfect group G, there is uniquely a central extension G → G such that G˜ is perfect and Appendix B: Automorphism Groups from Quandles 109

˜ ˜ H2(G; Z) = 0. Such a G is called the universal central extension of G. However, we remark that the kernel Ker(⊕i∈O(X)εi ) is not always the universal central extension of the perfect group Inn(X); see Theorem B.8 with g = 3 as a counterexample (cf. gr(M ; Z) =∼ Z ⊕ Z/ the fact H2 3 2 shown by Sakasai [Sak]). However, in the Sp-case, we can recover the universal central extensions: Proposition B.6 Let X be the quandle in Lemma B.2. Then As(X) =∼ ZO(X) × Sp(2n; F), where Sp(2n; F) is the universal central extension of Sp(2g; F). Remark B.7 The central kernel of Sp(2g; F) → Sp(2g; F),orH2(Sp(2g; F); Z), is called the Milnor-Witt K2-group of F;ThisK2-group has a long history includ- ing number theory, the metaplectic group, A1-homotopy theory, and stability problem (see [Lam, Mor, Wei2]). For example, as is known, the inclusion SL (F) → ∼ 2 Sp(2g; F) induces the stable isomorphism H2(SL(2; F); Z) = H2(Sp(2g; F); Z).

Proof. (Outline) Consider the augmentation ∪r κr : X → Sp(2g; F), and the map K : → ZO(X) × ( ; ) × MW( ) ( .κ , ) X SL 2g F K2 F which sends xr to 1r r 0 . Then, the author [N5] showed that, if g = 1, this map yields a group homomorphism K: As(X) → ZO(X) × SL(2; F). Hence, by Proposition B.1 and the universality of central extensions, this Kis an isomorphism. Finally, for any g > 1, the stable isomorphism with functriality implies that the map K with g > 1 yields the required isomorphism. We end this section by introducing two examples without proof. The first is on the spherical quandle (see Example 2.9 for the definition): ( ) = n Exercise 21 Let F be a field with Char F 2, and SF be the spherical quandle. > ( n ) If n is odd and 1, Inn SF is isomorphic to the orthogonal group

t O(n + 1; F) ={A ∈ Mat((n + 1) × (n + 1); F) | A A = In+1}.

( n ) ( + ; ) := ( + ; ) ∩ ( + If n is even, Inn SF is isomorphic to SO n 1 F O n 1 F SL n 1; F). Furthermore, look over the deﬁnition of the spin group Pin(n; F), and construct ( 2n) → ( + ; ) → ( + ; ) a sequence of central extensions As SF Spin 2n 1 F SO 2n 1 F .

The second example is with respect to the mapping class group Mg, which is deﬁned to be the group of isotopy-classes of orientation preserving homeomorphisms of g. ≥ M 2 (M ; Z) =∼ Z Then, if g 3, it is known (see [FM]) that g is perfect and Hgr g .So we can set up the universal central extension,

proj. 0 −→ Z −→ Tg −→ Mg −→ 0 (central extension). (B.2)

≥ D ns D Theorem B.8 ([Ger],seealso[N6]) Let g 3. Let g and g be the Dehn quandle as in Example 2.11. Then, there are three group isomorphisms

(D ns) =∼ Z × T , (D ) =∼ Z[g/2]+2 × T , As g g As g g 110 Appendix B: Automorphism Groups from Quandles

∼ ⊕∞ As(Conj(Mg)) = Z × Tg. (B.3)

(D ns) =∼ (D ) =∼ ( (M )) =∼ M . In addition, Inn g Inn g Inn Conj g g In general, the adjoint relation via hom-sets makes symmetry the most apparent (see [Mac]), and, it is sometimes sensible to explicitly describe a gap between the two categoriesm. When discussing symmetry and adjoint relations, the statements about quandles and group (Theorem B.8) are astute observations.

B.2 Quotients and Presentations on Quandles

In this section, we study quotients and presentations on quandles. Here, we use the augmented quandles (i.e., the triple of a set X, a group G, and a map κ satisfying some conditions; see Definition 3.10). We need a preparation. Given an augmented quandle (Q, G) and a group homomorphism f : G → H, we will define another augmented quandle (Q ×G H, H) as follows. Note that the product Q × H is made into a right H-set with action (q, h) · k := (q, hk). Define a G-congruence on Q × H by

(q, h) ∼ (p, k) iff kh−1 = f (g) and q = p · g for some g ∈ G.

Then, we deﬁne Q ×G H to be the set of congruent classes, and introduce a map −1 κ : Q ×G H → H; (q, h) −→ h f (κ(q)) h.

Then, we can easily see that this κ is well-defined, and the pair (Q ×G H, H) is an augmented quandle. Using this construction, we define quotients of quandles as follows. Definition B.9 ([Joy, Sect. 10]) Let X be a quandle, and N be a normal subgroup of G = As(X). Take the augmented quandle (X, G = As(X)) from Example 3.12. Then, the quotient of X subject to N is the quandle on X ×G G/N, where the quandle operation is given by [p][q]:=[p · κ (q)] with p, q ∈ X. We write the quotient quandle by X/N. Then we have a canonical quandle epimorphism X → X/N. Conversely, starting from a quandle epimorphism with some conditions, we can recover a quotient of the quandle: Lemma B.10 Let f : X → Y be a quandle epimorphism, which induces a bijection between the orbit sets O(X) O(Y ), e.g., when X is connected. Then Y is isomorphic to the quandle from the augmented data (X ×As(X) As(Y ), As(Y )). This lemma is an analogy of the homomorphism theorem in group theory; in fact, the latter group As(Y ) is isomorphic to As(X)/Ker(As( f )), Appendix B: Automorphism Groups from Quandles 111

Next, we study presentations of quandles, similar to those of group presentations. free For this, we brieﬂy discuss the free quandle Q I associated with an index set I , i.e., the conjugacy classes of I in the free group FI (see Example 2.16). Via the : free → natural inclusion to the free group I Q I FI we have an augmented quandle ( free, , ) Q I FI I . Furthermore, by deﬁnitions, the connected components of FI is ( free) exactly I , and the associated group As Q I is FI . Further, we can easily see Proposition B.11 (cf. free group) Let I be an index set, and X a quandle. Then, α : → : free → any map I X uniquely gives rise to a quandle homomorphism f Q I X such that f (xi ) = α(i).

Hence, for any normal subgroup N of FI , i.e., a group presentation, we can free/ consider the quotient quandle Q I N. Conversely, we see that free/ Proposition B.12 Every quandle X is some quotient quandle of the form Q I N. = : free → ( ) = Proof. Let I X. Proposition B.11 admits a map f Q X X with f ey y ∈ = ( free) → ( ) for any y X, and the associated group epimorphism FI As Q X As X . ⊂ free/ = free × ( ) Letting N FI be the kernel, Q I N Q X FI As X is isomorphic to X by construction.

In a categorical way, Proposition B.12 implies the adjointness (see [Mac, Chap. IV])

( , U ( )) =∼ ( free, ), HomSet I X HomQnd Q I X where U is the forgetful functor to the category of sets. In this way, we explicitly reach at an concept of the quandle presentation:

Definition B.13 Let X be a quandle. A quandle presentation of X isapairofan =∼ free/ index set I and a normal subgroup N of FI such that X Q I N. This X is finitely presentable if we can choose an index set I and groups N FI =∼ free/ / such that X Q I N and the quotient FI N is a finite presentation as a group.

Example B.14 (Link quandle [Joy]) To present the link quandle Q L , let us fix a diagram D of a link L, and denote the arcs by α1,...,αn, and denote {1,...,n} by −1 −1 I . Consider the normal subgroup generated by ατ βτ γτ βτ where the indices run over all the crossings τ depicted in Fig. 3.1. The Wirtinger presentation (Theorem 3 A.8) implies that the quotient group FI /N is isomorphic to π1(S L). In addition, free/ Proposition B.15 The link quandle Q L is isomorphic to the quotient one Q I N. : ( ) → / Proof. By Theorem 2.31, the homomorphism f As Q L FI N that eH x sends −1m =#L ( \ ) to x x is a group isomorphism. Recall Q L =1 H G as a set, by definition. × / = Hence, we have the identification Q L As(Q L ) FI N Q L by Lemma B.10. α−1α−1α α As seen in [Joy, Sect. 10], it is common to replace the relation k j i j by ‘αk = αi α j ’. Then, we can describe many quandle presentations of link quandles. For example, the knot quandle of the trefoil knot 31 is reduced to 112 Appendix B: Automorphism Groups from Quandles

=∼ , , | = , = , = Q31 a b c a b c b c a c a b

=∼ a, b | (a b) a = b,(b a) b = a .

Example B.16 (Coxeter quandle) As another example, we examine the Coxeter quandle (see Example 2.15 for the deﬁnition). For n ∈ N, consider the index set I ={s1, s2,...,sn}. Further, consider a Coxeter graph, that is, a map T : I × I → N satisfying T (s, s) = 1 and T (s, t) = T (t, s) if s = t. Then, the Coxeter group is deﬁned by the group presentation

W := s ∈ I | (st)m(s,t) = 1 ((s, t) ∈ I × I ) .

In a parallel fashion, the associated Coxeter quandle has the quandle presentation

m(s,t) 2 XT = s ∈ I | (s t) t = s, s (t s) = t ((s, t) ∈ I ) .

For example, the dihedral quandle of order 2m + 1 is presented by

s, t | (s t) t = s, s m (t s) = t .

Exercise 22 Show that Inn(XT ) is isomorphic to W/ZW subject to the center ZW .

B.3 Examples; Alexander and Core Quandles

This section focuses on the Alexander quandles, and determines the associated groups. After that, we also brieﬂy consider the core quandles. We often consider an Alexander quandle X to be a Z[T ±1]-module or an abelian group (see Example 2.6 for the deﬁnition). First, we discuss the connectivity:

Lemma B.17 (LN) Its connected components are bijective to X/(1 − T )X. In particular, the quandle X is connected if and only if 1 − T is invertible in X.

Proof. Let X¯ be the quotient module X/(1 − T )X, and take the projection π : X → X¯ . The quotient quandle on X¯ is trivial; hence, it sufﬁces to show that, for any a¯ ∈ X¯ , every two elements x, y in π −1(a¯) are connected. Indeed, since x − y ∈ Ker(π) = (1 − T )X,wehavex − y = (1 − T )z for some z ∈ X, which gives y = x(x + z).

We next determine the inner automorphism groups Inn(X).

Proposition B.18 Let X be an Alexander quandle of type m, and let X denote the Z[T ±1]-module (1 − T )X. Then the group Inn(X) is isomorphic to the semi-direct product X Z/mZ.HereZ/mZ acts on X by the multiplication of T . In particular, if X is connected, we have an isomorphism Inn(X) =∼ X Z/mZ. Appendix B: Automorphism Groups from Quandles 113

Proof. We consider the action X X Z/mZ by the formula y · (x, n) = (x + T n(y − x)) where y ∈ X and (x, n) ∈ X Z/mZ. Furthermore, deﬁne a map κ : X → X Z/mZ by κ(y) := (y − yT, 1). Then they satisfy the assumption in Theorem B.1. Hence we have the isomorphism Inn(X) =∼ X Z/mZ.

Next, to discuss automorphism groups Aut(X), we focus on quandle homomorphisms between Alexander quandles in the following lemma.

Lemma B.19 Let X and Y be Alexander quandles. For x ∈ X, denote by cx the constant map Y →{x}⊂X. Assume that Y is connected. Then the map

Φ : X ⊕ HomZ[T ±1]-mod(Y, X) −→ HomQnd(Y, X) deﬁned by setting Φ(x, f ) := (cx + f ) is a bijection.

Proof. Given a quandle homomorphism g : Y → X, we can check that the map ±1 g − cg(0) : Y → X is a Z[T ]-module homomorphism, from the fact (Lemma B.17) that 1 − T is invertible in Y . Hence, the map Φ has the inverse mapping.

In addition, we determine the automorphism group of X, i.e., Aut(X) := { f : X → X | f is a quandle isomorphism.}. Denote by AutZ[T ±1]-mod(X) the group consisting of Z[T ±1]-module automorphisms of X. Consider the action of the group AutZ[T ±1]-mod(X) on X by setting x · f := f (x). Regarding X as an abelian group, we can deﬁne the semi-direct product X AutZ[T ±1]-mod(X). Proposition B.20 Let X be a connected Alexander quandle, and let Y = X. Then, the restriction on Aut(X) of the map Φ gives a group isomorphism ∼ resΦ : X AutZ[T ±1]-mod(X) = Aut(X).

Proof. It sufﬁces to check that this Φ is a group homomorphism. Indeed, compute: Φ(x, f ) · Φ(y, g) (z) = Φ x, f (cy + g(z)) = cx + c f (y) + g ◦ f (z) = Φ x, c f (y) + g ◦ f (z) = Φ (x, f ) · (y, g) (z) ∈ X, where x, y, z ∈ X and f, g ∈ AutZ[T ±1]-mod(X).

Remark B.21 In general, the automorphism groups Aut(X) and Inn(X) are different. For example, if X is the Alexander quandle on the direct product (Z/p)n with T =−1 andaprimep > 2, then it follows from Propositions B.18 and B.20 that Aut(X) is the n n afﬁne group (Z/p) GL(n, Fp), and that Inn(X) is the dihedral group (Z/p) Z/2.

Finally we focus on a simple presentation of the associated group As(X), due to Clauwens [Cla2]. Set up a homomorphism μX : X ⊗Z X → X ⊗Z X deﬁned by 114 Appendix B: Automorphism Groups from Quandles

μX (x ⊗ y) = x ⊗ y − Ty⊗ x.

We set the direct product Z × X × Coker(μX ), and equip it with a group operation

(n, a,κ)· (m, b,ν)= (n + m, T ma + b,κ+ ν +[T ma ⊗ b]). (B.4)

Theorem B.22 ([Cla2]) Let X be a connected Alexander quandle. Then a homomorphism As(X) → Z × X × Coker(μX ) which sends the generator ex to (1, x, 0) is a group isomorphism. Proof. (Sketch) The point is to concretely construct the inverse mapping; See [Man, Theorem 2.1.1] (the proof is simpler than the origin [Cla2]). Changing the subject, we will work with the core quandle and show Proposi- tion B.23. Here is some terminology: given a group G, we equip X = G with the quandle operation g h := hg−1h.TakeZ/2 ={±1} and the wreath product (G × G) Z/2, and consider the epimorphism (G × G) Z/2 → G/[G, G] which sends (g, h,σ)to [gh]. Then, the kernel is of the form

G1 := { (g, h,σ) ∈ (G × G) Z/2 | gh ∈[G, G]}. (B.5)

−1 σ With respect to x ∈ X and (g, h,σ)∈ G1, we deﬁne x · (g, h,σ):= h (x )g, which ensures an action of G1 on X. Furthermore, consider a subgroup of the form 2 −1 σ G2 := (z, z,σ)∈ (G × G) Z/2 z ∈[G, G], k zk = z for any k ∈ G , which is contained in the center of G1.

Proposition B.23 Let X be the core quandle on G. The quotient group G1/G2 is isomorphic to Inn(X).

Proof. The quotient action subject to G2 is effective. Furthermore, consider the map −1 κ : X → G1/G2 which sends g to [(g, g , −1)]. Then, G1/G2 is generated by the image (κ). Indeed, we can easily verify that any element (g, h,σ) in G1 with , ∈ = −1 −1 ··· −1 −1 gi hi G and gh g1h1g1 h1 gm hm gm hm is decomposed as

σ −1 (g, h,σ)= κ(1G ) · κ(gh )· κ( ) · κ( ) · κ( −1) · κ( ) ··· κ( ) · κ( ) · κ( −1) · κ( ) . g1h1 1G g1 h1 gm hm 1G gm hm ∼ These justify the conditions in Theorem B.1; hence, Inn(X) = G1/G2 as desired.

As seen in G2, this proposition implies the difﬁculty to determine Inn(X),in general. Thus, it also seems hard to determine As(X). Actually, even if a core quandle X is connected, Proposition B.23 deduces that the kernel Ker(ψ) has the complexity gr( ) gr( ( )) characterized by the second homology H2 G and H2 Inn X . For example, if X is the product of h-copies of the cyclic group Z/m, i.e., X is the Alexander quandle of the form (Z/m)h[T ]/(T + 1), then TheoremB.22 implies such a complexity. Appendix B: Automorphism Groups from Quandles 115

B.4 Alexanderizations of Quandles

As an analogy of abelianization of groups, we introduce “Alexanderization” with respect to quandles ([Joy, Sect. 17]). For this, consider the following condition on a quandle X:

(ab)(cd) = (ac)(bd) for any a, b, c, d ∈ X. (B.6)

Example B.24 Every Alexander quandle X satisﬁes this identity.

To form the quotient of X modulo the relation (B.6), take the augmented quandle (X, G = As(X)) in Example 3.12, where κ(x) = ex . Then the identity (B.6)is expressed equivalently to the statement that the expression (B.7) be equal to 1.

κ(b)κ(c)−1κ(d)κ(b)−1κ(c)κ(d)−1. (B.7)

Let N be the normal subgroup of As(X) generated by the elements of the form (B.7). Then the quotient X/N satisﬁes the identity (B.6).

Deﬁnition B.25 Let X be a quandle, and let N be the normal subgroup of As(X) above. The Alexanderization of X, denoted by Al(X), is the quotient quandle X/N.

We will show (Theorem B.26) that the Alexanderization of any connected quandle X is some Alexander quandle. To this end, let As(X) be the commutator subgroup ±1 n of As(X), which is acted on by Z = T (Precisely, Z ={e } ∈Z acts on As(X) x0 n by g → e−n gen ; see Lemma 2.27). Further consider the commutator subgroup, x0 x0 As(X) ,ofAs(X) . Then, the quotient As(X) /As(X) is an abelian group with action by Z. Hence, considering it to be a Z[T ±1]-module, we can regard any quotient of As(X) /As(X) as an Alexander quandle.

Theorem B.26 Let X be a connected quandle. Fix x0 ∈ X. Then, the Alexanderiza- tion of X is isomorphic to an Alexander quandle on the following abelian group:

As(X) /As(X) / (Stab(x0) ∩ As(X) / Stab(x0) ∩ As(X) .

Remark B.27 This theorem is false in non-connected cases. Actually, there is a non- connected quandle even of order 3 which is not Alexander, but satisﬁes (B.6).

Proof. We ﬁx notation G = As(X) and H = Stab(x0) in the proof. First, we discuss the form of X. By Theorem 2.23, we may assume that X is the quandle of the form (G, H, x0). Note that the action restricted on X of G is transitive (Why?); Hence X is also of the form (G , G ∩ H, T ) in Example 2.17. Next, we further compute the Alexanderization Al(X) as

Al(X) = X/G = X ×G (G/G ) = X ×G (G /G ) = (G /G )/(H ∩ G /H ∩ G ), 116 Appendix B: Automorphism Groups from Quandles as sets, where the ﬁrst equality is due to N = G by Lemma B.28 below, and the third and forth are obtained from the transitive action X As(X) . Hence, the quotient X/N is equal to the quandle arising from the triple (G /G , G ∩ H/G ∩ H, T ). Finally, it is enough to observe the quandle structures of X/N. By deﬁnition, the operation of the quandle X = (G, H, z0) is given by

= −1 = ( −1 )( −1 −1 ) , , ∈ . Hx Hy Hz0xy z0 y H z0 xz0 z0 y z0 y for x y G

Thus, in the quotient (G /G , H ∩ G /H ∩ G , T ) as a Z[T ±1]-module, this operation is rewritten in

xy = Tx − Ty+ y = Tx + (1 − T )y, as Alexander operation. In summary, we have the required quandle isomorphism.

Lemma B.28 ([Joy, Lemma 17.1]) Let X be a connected quandle. The double commutator subgroup As(X) is equal to the above normal subgroup N.

Proof. G ⊂ N: Fix arbitrary a, b ∈ G . To prove aba−1b−1 ∈ N, it sufﬁces to show that a commutes with b in the quotient G /N. First, notice that if x, y ∈ Im(κ), then xy−1 = y−1z where z = yxy−1 ∈ Im(κ). Iterating this process, we can see, by the connectivity of X and Lemma 2.27, that there are some ai ∈ κ(X) such that = −1 −1 ··· −1. a a1a2 a3a4 an−1an We can rewrite b similarly. In addition, we should notice the equality, for any w, x, y, z ∈ κ(X),

xy−1wz−1 ≡ wy−1xz−1 ≡ wz−1xy−1 ∈ G /N.

Iterating this process, we conclude that a commutes with b. Hence, G ⊂ N. N ⊂ G : for any a, b, c ∈ X, we denote κ(a), κ(b), κ(c) ∈ As(X) by A, B, C. The connectivity permits x, y ∈ As(X) such that B = x−1 Ax and C = y−1 Ay. Hence,

N AB−1CA−1 BC−1 = (Ax−1 A−1x)(y−1 AyA−1)(x−1 Ax A−1)(Ay−1 A−1 y).

Noting that each term in the right side lies in As(X) ,wehaveN ⊂ G as desired.

As a corollary, we will calculate the Alexanderization of the knot quandle of a knot 3 3 ∼ K . To describe this, recalling the abelianization π1(S K ) → H1(S K ) = Z, ∞ → 3 consider the associated inﬁnite covering EK S K . Then, the ﬁrst homology ( ∞; Z) H1 EK is called the Alexander module of K . By the covering transformation ∞ Z= ±n Z[ ±1] EK T , the Alexander module has a T -module structure. Corollary B.29 ([Joy, Theorem 17.3]) The Alexanderization of the knot quandle ( ∞; Z) Q K is isomorphic to the Alexander quandle on the Alexander module H1 EK . Appendix B: Automorphism Groups from Quandles 117

3 Proof. Let π denote π1(S K ) for short. Suppose that K is not the unknot. Theo- rem 2.31 says that Q K is connected and is the quandle from the triple (π, H, m), and ( ) =∼ π π =∼ π ( ∞) that As Q K . Because of the topological fact 1 EK [see Proposition ( ∞; Z) π /π A.10 in detail], the Alexander module H1 EK is isomorphic to . Further- more, we can easily see that this isomorphism is a Z[T ±1]-module isomorphism. Thereby, Theorem B.26 implies that it is enough for the proof to show the identity ∩ π = ∩ π l ∈ π l ∞ H H . Notice that , because the lift of in EK is also bounded by the lift of a Seifert surface. Since H is, by deﬁnition, Z2 generated by a meridian- longitude pair (m, l), we have H ∩ π = H ∩ π = (l) as desired. Appendix C Small Quandles, and Some Quandle Homology

C.1 Classiﬁcation of Small Quandles

We explain some lists which classify small quandles. In the ﬁnite group theory, it had been a long-problem to classify simple groups. Similarly, we here introduce simple quandles.

Deﬁnition C.1 A quandle X is said to be simple, if the order of X is larger than 2 and any quandle epimorphism X → Y is either an isomorphism or a constant map.

Since quandle is something like homogenous sets, the number of quandles of small order is larger than that of ﬁnite groups. However, thanks to the help of computers (see [JMcC2]), here is the numbers of quandle isomorphism classes of quandles of order ≤10 (Here, the notation is according to Chap. 1):

|X| Quandles Connected q’dl Simple q’dl Non-Alexander conn. q’dl Latin q’dl 2 1 0 0 0 0 3 3 1 1 0 1 4 7 1 1 0 1 5 22 3 3 0 3 6 73 2 0 2 2 7 298 5 5 0 5 8 1581 3 0 1 2 9 11079 8 2 0 8 10 Unknown 1 1 1 1

Furthermore, the binary operations of many quandles of order ≤ 6 are concretely listed in Appendix in the book [CKSb]. Next, we restrict ourselves to connected cases. Let a(n) be the number of the isomorphism classes of connected quandles of order n.From[Ven], here is a list of a(n) with n < 48:

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 a(n) 1 0 1 1 3 2 5 3 8 1 9 10 11 0 7 9 15 12 17 10 9 0 21 42 34 0

n 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 a(n) 65 13 27 24 29 17 11 0 15 73 35 0 13 33 39 26 41 9 45 0 45

Furthermore, we state two theorems to classify connected quandles of prime power order. For this, we mention an interesting property:

Proposition C.2 ([HSV]) For any connected quandle X of prime power order, the group Inn(X) is solvable.

Proof. For a group G with a ∈ G, consider the set aG := {g−1ag | g ∈ G }. Kazarin proved [Kaz] that if a ∈ G is such that |aG | is a prime power, then the subgroup generated by aG is solvable. Let G = Inn(X), and a := (• x) for some x ∈ X. The connectivity implies that |X| is divisible by |aG | and Inn(X) is generated by aG . Hence, Inn(X) is solvable.

Next, connected quandles of order p and of order p2 are completely classiﬁed:

Theorem C.3 (p-case [EGS],p2-case [Gra]) Let X be a connected quandle. Let 2 p ∈ Z≥0 be odd prime. Assume the order |X|=por|X|=p . Then, the quandle X is isomorphic to an Alexander quandle.

Proof. Let G be Inn(X). By the proof of Theorem B.26, the quandle X is represented by (G , Stab(x0) ∩ G , T ). We should notice G G by Proposition C.2. First, we discuss the case |X|=p. Consider the quotient quandle on G /G . Since G G , G must be zero by |X|=p. Hence, as in Theorem B.26, the Alexander- ization of X is itself. That is, X is an Alexander quandle. The case of |X|=p2 is similarly done. Since this is a little technical, we only outline it. For this, consider the subquandle X = (G , Stab(x0) ∩ G , T ). Then, we can verify that the subquandle is connected. By the preceding case, G is of order p or 0. Similarly, the quotient quandle on G /G is of order pk with k ≥ 0. Hence, |G |=p2. In particular, G is abelian; hence, the Alexanderization of X is itself.

However, when |X|=pk with k > 2, such a classiﬁcation is hopeless by the reason of complexity of ﬁnite groups. Next, we mention connected quandle of order 2p.

Theorem C.4 ([JMcC1]) For any prime p > 5, there is no connected quandle of order 2p.

We do not give the proof, because it is not so easy, and it is based on a group property of S2p; see also [HSV, Sect. 10] as another proof. Incidentally, any connected quandle of order 2p < 11 forms a subquandle of the conjugacy quandle Conj(Sp). Finally, concerning |X|=3p, we mention latin quandles. Appendix C: Small Quandles, and Some Quandle Homology 121

Deﬁnition C.5 A quandle X is said to be latin (or strongly connected), if for any x ∈ X the map X → X; y → x y is bijective.

We notice that if X is latin, it is connected. Indeed, any x, z ∈ X admit uniquely y ∈ X such that x y = z. Here is a condition for latin quandles:

Lemma C.6 ([HSV]) Let X be a connected quandle with x0 ∈ X, and G = Inn(X). Denote (• x0) ∈ Inn(X) by g0. Then, X is latin if and only if for every α ∈ G − ( ) [ ,α]= −1α−1 α ( ) Stab x0 the commutator g0 g0 g0 is not contained in Stab x0 . This lemma can be proven by direct computation, although the proof is a little long. This lemma gives many examples of latin quandles. First, any connected Alexan- der quandle X is latin. Furthermore, if the pair Stab(x) ⊂ G is malnormal, the quandle X = (G, Stab(x), x) is latin. In addition, the knot quandle Q K is latin iff the knot K is prime, and neither a torus knot nor a cable knot (cf. Theorem 7.20). The paper [HSV] discussed relations between connected quandles of 3p and latinness.

C.2 Known Results on Quandle Homology

This section lists known results on the quandle homology and the second homotopy group of the rack space (For the proof see references therein). Since Sects. 6.3–6.5 discussed the second homology, we mainly focus on homology of degree > 2. First, we give a list of the symplectic and orthogonal quandles over Fq ,

Theorem C.7 ([N7]) Let q = pd be odd: q = 3, 32, 33, 5, 7. n F 1. Let X be the symplectic quandle Spq over the ﬁnite ﬁeld q . Then, 0, n > 1, H Q(X) =∼ 3 Z/(q2 − 1) ⊕ (Z/p)d(d+1)/2, n = 1, Z ⊕ Z/(q2 − 1), n > 1, π (BX) =∼ 2 Z ⊕ Z/(q2 − 1) ⊕ (Z/p)d , n = 1.

n F ≥ 2. Let X be the spherical quandle Sq over q , where n 2. Then , > , Q( ) =∼ 0 n 2 H2 X [1/2] Z/(q − δq ), n = 2, , > , Q( ) =∼ 0 n 2 H3 X [1/2] 2 Z/(q − 1) ⊕ Z/(q − δq ), n = 2, Z ⊕ Z/( 2 − ), > , π ( ) ∼ q 1 n 2 2 BX =[1/2] 2 Z ⊕ Z/(q − 1) ⊕ Z/(q − δq ), n = 2. 122 Appendix C: Small Quandles, and Some Quandle Homology

Table C.1 Notational remarks: Z[T ]/(p, T − ω) means the Alexander quandle with ω = 0, −1; The other quandles are conjugacy ones, where S4 and A4 mean the symmetric group and the extended alternating group of order 24, respectively | | Q ( ) Q ( ) π ( ) Connected quandle X X Type H2 X H3 X Tor 2 BX Z[T ]/(3, T + 1) 3 2 0 Z/3 Z/3 Z[T ]/(2, T 2 + T + 1) 4 3 Z/2 Z/2 ⊕ Z/2 ⊕ Z/8 Z/4 Z[T ]/(5, T + 1) 5 2 0 Z/5 Z/5 Z[T ]/(5, T − ω) 5 4 0 0 0 { −1( ) } ⊂ S Z/ Z/ ⊕ Z/ g 12 g g∈S4 4 6 2 2 6 12 Z/2 { −1( ) } ⊂ Z/ Z/ ⊕ Z/ ⊕ Z/ g 1234 g g∈S4 6 4 4 4 4 24 S4 Z/24 Z[T ]/(7, T + 1) 7 2 0 Z/7 Z/7 Z[T ]/(7, T − ω) 7 3 0 0 0 − X ={g 1(12)g} 8 3 0 Z/8 Z/8 g∈A4

Here δq =±1 is according to q ≡±1 (mod 4).

Next, we draw up a list of connected quandles of order ≤ 8 : Theorem C.8 ([N7]) Let X be one of connected quandles of order ≤ 8. Then, the second and third quandle homology groups and π2(BX) are listed in TableC.1.

Theorem C.9 ([N4]) Let X be the dihedral quandle of order 2m + 1. Then, there (π ( )) =∼ Q( ) =∼ Z/ + . Q( ) =∼ . are isomorphisms Tor 2 X H3 X 2m 1 , and H2 X 0 D ns As to the nonseparating Dehn quandle g , here is a result in a stable range: ≥ π ( D ns) Theorem C.10 ([N6, N7]) If g 7, the group 2 B g is isomorphic to either Z/24 ⊕ Z or Z/48 ⊕ Z. ≥ Q(D ns; Z) =∼ Z/ . Moreover, if g 5, the second homology is determined as H2 g 2 We now discuss link quandles for knots and hyperbolic links.

Theorem C.11 ([E1], and [N14]) For the knot quandle, Q K , of a non-trivial knot,

Q( ; Z) =∼ Q( ; Z) =∼ Z,π( ) =∼ Z ⊕ Z. H2 Q K H3 Q K 2 BQK

Furthermore, let L be a hyperbolic link, and Q L be the link quandle. Then,

Q( ; Z) =∼ Z#L , Q( ; Z) =∼ Z#L ,π( ) =∼ Z#L ⊕ Z#L . H2 Q L H3 Q L 2 BQL

Q() Π () We discuss the groups H3 X and 2 X of centrally extended quandles X. Appendix C: Small Quandles, and Some Quandle Homology 123

Theorem C.12 ([N6, N8]) Let X be a connected quandle of type tX . Let p : X → X be the universal covering mentioned in Example 8.9. Then, the second quandle homology of X is isomorphic to the kernel of the : () → ( ) Q() =∼ ( ) pushforward p∗ As X As X . Namely, H2 X Ker p∗ . Further, if the type < ∞ Q() tX , then H2 X is annihilated by tX . gr( ( )) [ / ] Furthermore, if H3 As X is ﬁnitely generated, then there are 1 tX -isomorphisms

Q() =∼ Π () =∼ gr( ( )). H3 X [1/tX ] 2 X [1/tX ] H3 As X

The following is a list of known facts. • In [Moc2] (see Theorem C.13), the third quandle cohomology was computed for Alexander quandles of the form Fq [T ]/(T − ω), where ω ∈ Fq {0, 1}. Based on Mochizuki method [Moc2], with respect to such quandles on Fq ,S.Abe [Abe] gave many quandle 4-cocycles. • R( ; Z) (On the torsion subgroup of Hn X ) In 2003, it was shown [LN] that the torsion R( ) | |n subgroup TorHn X is annihilated by X if X is latin. • The author showed [N2] that, for connected Alexander quandle X with |X| < ∞, R( ) | | Tor Hn X is annihilated by X . More generally, it is shown [PY1] (see also R( ) | | [PY2]) that, if X is latin and of ﬁnite order, then TorHn X is annihilated by X . • Niebrzydowski and Przytycki [NP2] discussed some (cohomology-like) operation R on the quandle homology H∗ (X; Z) of some quandles. • Inspired by [NP2], when X = Z[T ]/(T + 1, p) and p ∈ Z is odd prime, Clauwens R( ; Z) [Cla1] completely determined the rack homology Hn X . Furthermore, we mention other versions of the quandle complex. For example, the paper [CEGS, Sect. 2] introduces “quandle algebras” from quandles, and deﬁne the associated complex. In another way, for some quandles, Carter–Ishii–Saito–Tanaka [CIST] considered the complex like a mixture between simplicial and cube complexes; this complex has applications for handlebody-links; see also [IIJO]. Lebed [Leb] introduced a “qualgebra”, which is roughly a quandle with group-like operation, and constructed the cocycle invariants of handlebody-links. Concerning applications to virtual knots, the concept of biquandle and the homology has been studied in many ways; see [EN] and references therein. Furthermore, Przytycki considered a family of distributive operations and discussed a homology theory compatible with the family and Yang-Baxter equations; see [Pr] and references therein.

C.3 Some Cocycles of Alexander Quandles

This subsection focuses on the Alexander quandle over a ﬁnite ﬁeld Fq with ω ∈ d Fq {0, 1}, and reviews Mochizuki’s 3-cocycle [Moc2]. Precisely, q = p , and the quandle is the set Q = Fq with the operation (x, y) → ω(x − y) + y. Hereafter, we use notation X = x − y, Y = y − z, Z = z, and regard polyno- 3 mials in the ring Fq [X, Y, Z] as functions from Q to Fq , and as being in the quan- 124 Appendix C: Small Quandles, and Some Quandle Homology

3 ( ; F ) F dle complex CQ Q q in Sect. 8.4. Take the following three polynomials over q ([Moc2, Sect. 2.2]): χ(X, Y ) = (−1)i−1i −1 X p−i Y i = (X + Y )p − X p − Y p /p, 1≤i≤p−1 a b a −1 b E0(p · a, b) = χ(ωX, Y )−χ(X, Y ) Z , E1(a, p · b) = X χ(Y, Z)−χ(ω Y, Z) .

+ Deﬁne the following set Iq,ω consisting of the polynomials under some conditions:

+ · + := { ( · , ) | ω p q1 q2 = , < } Iq,ω E0 p q1 q2 1 q1 q2 q1+p·q2 ∪{E1(q1, p · q2) | ω = 1, q1 ≤ q2}

q1 q2 q3 q1+q2+q3 ∪{X Y Z | ω = 1, q1 < q2 < q3}. (C.1)

Here the symbols qi range over powers of p with qi < q. Furthermore, we review polynomials denoted by Γ(q1, q2, q3, q4). For this, we 4 deﬁne a set Qq,ω ⊂ Z consisting of quadruples (q1, q2, q3, q4) such that

q1+q3 q2+q4 • q2 ≤ q3, q1 < q3, q2 < q4, and ω = ω = 1. Here, if p = 2, we omit q2 = q3. + • One of the following holds: Case 1 ωq1 q2 = 1.

q1+q2 Case2 ω = 1, andq3 > q4. q1+q2 Case3 (p = 2), ω = 1, andq3 = q4. q1+q2 q1 q2 Case4 (p = 2), ω = 1, q2 ≤ q1 < q3 < q4,ω = ω . q1+q2 q1 q2 Case5 (p = 2), ω = 1, q2 < q1 < q3 < q4,ω = ω .

Furthermore, for (q1, q2, q3, q4) ∈ Qq,ω in each case, we deﬁne the associated polynomial Γ(q1, q2, q3, q4) as follows:

q1 q2+q3 q4 Case1 Γ(q1, q2, q3, q4) := X Y Z . q1 q2+q3 q4 q2 q1+q4 q3 Case2 Γ(q1, q2, q3, q4) := X Y Z − X Y Z − + + + −(ωq2 − 1) 1(1 − ωq1 q2 )(X q1 Y q2 Z q3 q4 − X q1 q2 Y q4 Z q3 ). q1 q3+q4 q2 Case3 Γ(q1, q2, q3, q4) := X Y Z . q3 q1+q2 q4 Case4 and Case5 Γ(q1, q2, q3, q4) := X Y Z .

Then we state the main theorem in [Moc2]: Theorem C.13 ([Moc2]) The following set composed of quandle 3-cocycles gives a 3 ( ; F ) < basis of the third cohomology HQ X q .Hereqi means a power of p with qi q.

+ + ∪{Γ( , , , ) | ( , , , ) ∈ Q }∪{ q1 q2 | ωq1 q2 = , < }. Iq,ω q1 q2 q3 q4 q1 q2 q3 q4 q,ω X Y 1 q1 q2

Q + ( ) ωq1 q2 = < Moreover, H2 X is zero if and only if 1 for any q1 q2: In particular, if + + 3 ( ; F ) { q1 q2 q3 |ωq1 q2 q3 = , < < } so, HQ X q is generated only by X Y Z 1 q1 q2 q3 . Appendix C: Small Quandles, and Some Quandle Homology 125

Remark C.14 Unfortunately the original statement and his proof of this theorem contained slight errors, which had however been corrected by Mandemaker [Man].

As seen in Theorem 8.17, the polynomial Γ is most interesting among them, and agrees with diagrammatic computations of the cocycle invariants. Finally, we will mention Alexander quandles of prime order:

Theorem C.15 ([N2]) Let ω ∈ Z/pbeω = 0, 1. Let X be the Alexander quandle Z[T ]/(p, T − ω) , and let d be the type, i.e., d is the minimal satisfying ωd = 1. Q( ; Z) =∼ (Z/ )bn ≥ Then, the integral quandle homology groups are Hn X p for n 1, where bn is determined by the recursion bn = bn−2d + bn−2d+1 + bn−2d+2, b1 = b2 =···=b2d−2 = 0, and b2d−1 = b2d = 1.

Furthermore, the author described explicit presentations of all the cocycles of the quandle cohomology group; see [N2] for details. References

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A D Alexanderization, 115 Dehn quandle, 8 ( ∞; Z) Alexander module H1 EK , 117 Dehn twist, 8 ( ∞; Z) Alexander module H1 EL , 103 Dihedral quandle, 6 Alexander quandle, 6 Dijkgraaf-Witten invariant DWκ (M), 67 Algebraic atroidality, 105 Arcs in link diagram, 102 Associated group As(X), 10 E Augmented quandle, 23 Eilenberg-MacLane space, K (π, 1)-space, 61, 71, 76, 101

B Bloch group, 81, 85 F Braided set, 31 Finitely generated, 16 Fox coloring, 19 Fox derivative, 81 free C Free quandle Q I , 9 Cable knot, 101 Fundamental 3-class [S ], 50 Chern-Simons 3-class, 80 Fundamental homotopy class, 65 Chern-Simons invariant, 87 Classifying map, 62 Cohomology pairing of knots, 44 G Coinvariant, 73 Geometrization theorem for links, 104 Coloring set, ColX (D), 19 G-family of Alexander quandles, 28 Composite knot, 101 G-invariant group cocycle, 95 Conﬁguration complex, 72 Group homology of G, Hn(G; M), 71 Conjugacy quandle Conj(G), 9 Guitar map, 32 Connected, 10 Connected components O(X), 10 Core quandle, 8 H Coxeter quandle, 9, 56, 112 Hochschild group homology, 76 Crossed module, 23 Holonomy representation (from hyperbolic- Crossing of a link diagram, 102 ity), 104 Cup product (on the group cochain), 75 Homogenous complex (as a chain group), 72 H Cup product on the rack complex, 51 Hurewicz homomorphism Y,y0 , 49 Cusp shape, 56 Hyperbolic link, 104 © The Author(s) 2017 135 T. Nosaka, Quandles and Topological Pairs, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-981-10-6793-8 136 Index

I Quandle homomorphism, 5 Inﬁnite cyclic cover of a link, 103 Quandle space BXQ , 35 Inner automorphism group Inn(X), 12 Quasi-isomorphic, 78 Quasi-isomorphism (between chain groups), 77 J Quotients of quandles, 110 JSJ-decomposition, 102

R K Rack, 17 Kei, 17 R R Rack chain group (C∗ (X, Y ; A), ∂∗ ), 46 Knot group, 100 Rack homology, 46 ⊂ 3 Knot, K S , 99 Rack homotopy invariant, 34 Knot quandle, 14 Rack space B(X, Y ), 48 Kronecker product, 75 Reidemeister move, 102

L S Latin quandle, 121 Satellite knot, 101 Link diagram, 102 Seifert surface, 100 Link invariant, 100 Seifert matrix, 82 ⊂ 3 Link, L S , 14, 99 Shadow coloring, 36 Link quandle QL , 14 Shadow coloring set SCol , (D), 36 l X y0 ( -th) longitude (of L), , 100 Simple quandle, 119 Small knot, 26 M Spherical quandle, 7 Malnormal, 77 Subquandle, 6 Symmetric space, 7, 108 (-th ) meridian (of L), m, 100 MW( ) Symplectic quandle, 8 Milnor-Witt K2-group of F, K2 F , 109 Mochizuki cocycle, 38 T N Torus link Ts,t , 99 Non-abelian 2-cocycle, 40 Transvection, 8 Non-abelian quandle cocycle invariant, 40 Trivial coloring, 19 Twisted cohomology pairings, 43, 96 Type (quandle), 5 P Parabolic quandle X F,r , 26 Peripheral subgroup, 100 U Positive and negative crossings in a diagram, 109 36 Universal quandle covering, 90 Postnikov tower, 62 Unknot, the trivial knot, 99 Presentations of quandles, 111 Prime knot, 101 W Primitive X-set, 36 Wirtinger presentation, 102

Q X Quandle, 5 X-coloring, 19 Quandle 2-cocycle, 37 X-set, 36 Quandle cocycle invariant, 38 Quandle covering, 55 Quandle extension, 55 Z Q Z Quandle homology Hn (X, Y ; A), 46 -equivariant part of DW invariant, 67