LLinking numbers, quandles and groups
Lorenzo TraldiLafayette CollegeEaston, PA 18042, [email protected]
Abstract
We introduce a quandle invariant of classical and virtual links, denoted Q tc ( L ). This quandle has the property that Q tc ( L ) ∼ = Q tc ( L (cid:48) ) if andonly if the components of L and L (cid:48) can be indexed in such a way that L = K ∪ · · · ∪ K µ , L (cid:48) = K (cid:48) ∪ · · · ∪ K (cid:48) µ and for each index i , there is amultiplier (cid:15) i ∈ {− , } that connects virtual linking numbers over K i in L to virtual linking numbers over K (cid:48) i in L (cid:48) : (cid:96) j/i ( K i , K j ) = (cid:15) i (cid:96) j/i ( K (cid:48) i , K (cid:48) j )for all j (cid:54) = i . We also extend to virtual links a classical theorem of Chen,which relates linking numbers to the nilpotent quotient G ( L ) /G ( L ) . Keywords : link group; linking number; quandle.Mathematics Subject Classification 2020: 57K10
Linking numbers are among the oldest, and simplest, invariants of classical knottheory. They were mentioned by Gauss in a diary almost 200 years ago; see [13]for a discussion. For our purposes, the most convenient way to describe linkingnumbers is to calculate them from link diagrams.We use standard conventions regarding classical and virtual link diagrams;for a thorough discussion we refer to the book of Manturov and Ilyutko [9]. A link diagram is a subset D of R obtained from the union of a finite number µ of piecewise smooth closed curves. The curves must be in general position, i.e.,there are only finitely many (self-)intersections among them, and all of these(self-)intersections are transverse double points (crossings). Each crossing iseither classical or virtual . The diagram is obtained from the union of the curvesby (a) at each classical crossing, removing a short piece of the underpassingsegment on each side of the crossing, and (b) at each virtual crossing, drawinga small circle around the crossing. The result of removing the short pieces nearclassical crossings is to cut the original curves into arcs; the set of arcs in D is denoted A ( D ). These arcs are called the long arcs of D , to contrast withshorter arcs obtained by cutting also at overpasses, or at virtual crossings. Wesay a diagram D represents a link of µ components, L = K ∪ · · · ∪ K µ .1 a r X i v : . [ m a t h . G T ] F e b wo link diagrams are equivalent or of the same type if they are relatedthrough a finite sequence of four kinds of moves: detour moves affecting onlyvirtual crossings, and classical Reidemeister moves affecting only classical cross-ings. Any of these moves provides a natural way to identify the closed curvesunderlying one diagram with the closed curves underlying the other diagram,so it makes sense to say that an equivalence class of link diagrams representsa µ -component link type. Reidemeister [12] showed that an equivalence classof classical diagrams represents a link type in R , and Kuperberg [8] showedthat an equivalence class of virtual link diagrams represents a link type in athickened surface.Linking numbers are defined using the notion of writhe of a classical crossing,illustrated in Fig. 1. w ( c ) = − w ( c ) = 1Figure 1: Classical crossings of negative and positive writhe. Definition 1.
Let D be a diagram of an oriented link, L = K ∪ · · · ∪ K µ . If i (cid:54) = j ∈ { , . . . , µ } , let C j/i ( D ) be the set of classical crossings in D at which K i passes under K j . Then the linking number of K i under K j is (cid:96) j/i ( K i , K j ) = (cid:96) j/i ( K j , K i ) = (cid:88) c ∈ C j/i ( D ) w ( c ) . We refer to Ricca and Nipoti [13] for the history of the classical notionof linking numbers. The extension to virtual links given in Definition 1 wasmentioned by Goussarov, Polyak and Viro [4]. It is a simple exercise to verifythat these linking numbers are invariant under Reidemeister moves and de-tour moves, so they are link type invariants. In the classical case, it is al-ways true that (cid:96) i/j ( K i , K j ) = (cid:96) j/i ( K i , K j ), so we can use the simpler nota-tion (cid:96) ( K i , K j ) = (cid:96) ( K j , K i ). For virtuals, instead, the two linking numbers (cid:96) i/j ( K i , K j ) and (cid:96) j/i ( K i , K j ) are independent of each other.When Kauffman [8] introduced virtual links, he extended many notions ofclassical knot theory to them. Among these notions were quandles, which hadbeen introduced to classical knot theory by Joyce [7] and Matveev [10]. Werecall the definition. Definition 2. A quandle on a set Q is specified by a binary operation (cid:46) , withthe following properties.1. For each x ∈ Q , x (cid:46) x = x.
2. For each y ∈ Q , the formula β y ( x ) = x (cid:46) y defines a bijection β y : Q → Q . . For all w, x, y ∈ Q , ( w (cid:46) x ) (cid:46) y = ( w (cid:46) y ) (cid:46) ( x (cid:46) y ) . If Q and Q (cid:48) are quandles, then a function f : Q → Q (cid:48) is a quandle map if f ( x (cid:46) y ) = f ( x ) (cid:46) f ( y ) ∀ x, y ∈ Q . A bijective quandle map Q → Q (cid:48) is anisomorphism, a bijective quandle map Q → Q is an automorphism, and theautomorphisms of Q form a group Aut( Q ) under function composition. The β y maps mentioned in part 2 of Definition 2 are the translations of Q . Notice thatthe third requirement of Definition 2 guarantees that the translations of Q areautomorphisms of Q . The subgroup of Aut( Q ) generated by the translations isthe translation group of Q ; we denote it β ( Q ). a a a Figure 2: The arcs incident at a crossing.
Definition 3.
Let D be a diagram of an oriented link L . The quandle of L is the quandle generated by { q a | a ∈ A ( D ) } , subject to the requirement that ateach classical crossing as indicated in Fig. 2, q a = q a (cid:46) q a . We denote thisquandle Q ( L ) . It is not hard to see that Q ( L ) is invariant under Reidemeister moves anddetour moves, up to isomorphism, so it provides a link type invariant.Here is a central definition of the paper. Definition 4.
A quandle Q is translation-commutative if its translation group β ( Q ) is commutative. As β ( Q ) is generated by the translations, Q is translation-commutative ifand only if ( x (cid:46) y ) (cid:46) z = ( x (cid:46) z ) (cid:46) y ∀ x, y, z ∈ Q. (1)It follows that translation-commutative quandles constitute a variety of quan-dles, in the sense discussed by Joyce [7, Sec. 10]. Therefore, every quandle has acanonical translation-commutative quotient, determined up to isomorphism bythe original quandle.In Sec. 2, we show that translation-commutative quandles are much lesscomplicated than general quandles. A translation-commutative quandle is com-pletely determined by a family of subgroups of a free abelian group, and itis not hard to determine whether or not two families of subgroups determineisomorphic translation-commutative quandles. Definition 5. If L is an oriented link, then the translation-commutative quan-dle Q tc ( L ) is the translation-commutative quotient of Q ( L ) . That is, if D is adiagram of L then Q tc ( L ) is the largest translation-commutative quandle gener-ated by { q a | a ∈ A ( D ) } , such that q a = q a (cid:46) q a at each classical crossing asindicated in Fig. 2. Q tc ( L ) is strongly related to the linking numbers in L . Theorem 6.
Suppose L and L (cid:48) are links. Then Q tc ( L ) ∼ = Q tc ( L (cid:48) ) if and onlyif both of the following requirements are satisfied.(a) L and L (cid:48) have the same number of components.(b) The components of L and L (cid:48) can be indexed so that L = K ∪ · · · ∪ K µ , L (cid:48) = K (cid:48) ∪ · · · ∪ K (cid:48) µ , and there are (cid:15) , . . . , (cid:15) µ ∈ {− , } such that (cid:96) j/i ( K i , K j ) = (cid:15) i (cid:96) j/i ( K (cid:48) i , K (cid:48) j ) for all i (cid:54) = j ∈ { , . . . , µ } . Notice in particular that if µ = 1 then Q tc ( L ) ∼ = Q tc ( L (cid:48) ), as the require-ment regarding linking numbers in L and L (cid:48) is satisfied vacuously. That is, thetranslation-commutative quandles of knots are all isomorphic to each other. Infact, they are all trivial one-element quandles.As mentioned above, every quandle has a canonical translation-commutativequotient. Therefore, Theorem 6 directly implies the following. Corollary 7.
Let (cid:101) Q ( L ) be any invariant quandle of a link L , whose canonicaltranslation-commutative quotient is Q tc ( L ) . If L and L (cid:48) are links with (cid:101) Q ( L ) ∼ = (cid:101) Q ( L (cid:48) ) , then L and L (cid:48) satisfy the two requirements of Theorem 6. There are at least two well-known invariant quandles that can play the role of (cid:101) Q in Corollary 7: the full link quandle, Q ( L ), and the medial quandle, AbQ( L )in Joyce’s notation [7].In general, the sign changes (cid:15) , . . . , (cid:15) µ in Theorem 6 can vary independently.When L and L (cid:48) are both classical links, however, the sign changes are tiedtogether: if (cid:96) ( K i , K j ) (cid:54) = 0, then (cid:15) i must equal (cid:15) j . One way to make thisinterdependence explicit involves the following construction. Definition 8.
A link L = K ∪· · ·∪ K µ has a linking graph (cid:96)g ( L ) , with a vertex v i for each component K i of L such that (cid:96) j/i ( K i , K j ) (cid:54) = 0 or (cid:96) i/j ( K i , K j ) (cid:54) = 0 for some j , and an edge v i v j whenever i (cid:54) = j and (cid:96) j/i ( K i , K j ) (cid:54) = 0 . Recall that a connected component of a graph is an equivalence class ofvertices under the equivalence relation ∼ generated by v i ∼ v j whenever v i v j isan edge. We always use the phrase “connected component” for this notion, toavoid confusion with the components of a link.As classical links always have (cid:96) j/i ( K i , K j ) = (cid:96) i/j ( K i , K j ), Theorem 6 imme-diately implies the following. Theorem 9.
Let L and L (cid:48) be classical links. Then Q tc ( L ) ∼ = Q tc ( L (cid:48) ) if andonly if both of the following conditions hold.(a) L and L (cid:48) have the same number of components.(b) The components of L and L (cid:48) can be indexed in such a way that for each con-nected component C of (cid:96)g ( L ) , there is an (cid:15) C ∈ {− , } such that (cid:96) ( K i , K j ) = (cid:15) C (cid:96) ( K (cid:48) i , K (cid:48) j ) whenever v i is a vertex of C .
4n particular, if (cid:96)g ( L ) has only one connected component then there is an (cid:15) ∈ {− , } such that (cid:96) ( K i , K j ) = (cid:15)(cid:96) ( K (cid:48) i , K (cid:48) j ) ∀ i (cid:54) = j ∈ { , . . . , µ } .In the classical case, Theorems 6 and 9 are the strongest results one couldhope for. Even the full quandle Q ( L ) of a classical link – a very sensitive linkinvariant – does not detect linking numbers absolutely. The reason is that Q ( L ) is invariant under the combination of reversing the orientations of all thecomponents of L , and replacing L with its mirror image. Moreover, for a splitlink L , Q ( L ) is not changed if the component orientations are reversed in justone split portion of L , and that portion is replaced with its mirror image. Theeffect of replacing a split portion with the orientation-reversed mirror imageis to multiply all linking numbers from the connected component(s) of (cid:96)g ( L )corresponding to that portion of the link by − G ( L ) /G ( L ) of a classical link group. In Sec. 5 we briefly discuss the connectionbetween our results and those of Harrell and Nelson [5], who related linkingnumbers of two-component links to quandle counting invariants. In this section, we give a structure theory for translation-commutative quandles.If B is a nonempty set, we use Z B to denote the free abelian group on B . Definition 10.
Let S = { S b | b ∈ B } be a family of subgroups of Z B , with b ∈ S b ∀ b ∈ B . For each b ∈ B , let A b = Z B /S b . Then Q ( S ) is the disjointunion Q ( S ) = (cid:91) b ∈ B A b ,equipped with the operation (cid:46) defined as follows: if x ∈ A b and y ∈ A c , then x (cid:46) y = x + ( c + S b ) ∈ A b . As a minor abuse of notation we often write x (cid:46) y = x + c + S b , even though x + c is not well defined. Proposition 11. Q ( S ) is a translation-commutative quandle.Proof. If x ∈ A b then as b ∈ S b , x (cid:46) x = x + b + S b = x + 0 + S b = x. If y ∈ A c then for each b ∈ B , the map A b → A b defined by x (cid:55)→ x (cid:46) y is abijection, with inverse function given by x (cid:55)→ x − ( c + S b ) ∈ A b .If x ∈ A b , y ∈ A c and z ∈ A d , then y (cid:46) z ∈ A c , so( x (cid:46) y ) (cid:46) z = x + c + d + S b = x + d + c + S b x + d + S b ) (cid:46) ( y (cid:46) z ) = ( x (cid:46) z ) (cid:46) ( y (cid:46) z ) . Also, formula (1) is satisfied because( x (cid:46) y ) (cid:46) z = x + c + d + S b = x + d + c + S b = ( x (cid:46) z ) (cid:46) y. Recall that if x is an element of a quandle Q , then the orbit of x in Q is thesmallest subset Q x ⊆ Q such that x ∈ Q x and β ± y ( z ) ∈ Q x ∀ y ∈ Q ∀ z ∈ Q x .It is easy to see that for a quandle of type Q ( S ), the orbits are the subsets A b .Also, Definition 10 implies that if b ∈ B and y, z ∈ A b , then β y = β z . We use β b to denote the map β y for all y ∈ A b . Lemma 12.
Suppose n ≥ ∈ Z , b , . . . , b n ∈ B , and m , . . . , m n ∈ Z . Let y = n (cid:88) i =1 m i b i ∈ Z B ,and let β y = n (cid:89) i =1 β m i b i ∈ β ( Q ( S )) . Then for each b ∈ B , the following are equivalent to each other.1. y ∈ S b .2. For some x ∈ A b , β y ( x ) = x .3. For all x ∈ A b , β y ( x ) = x .Proof. According to Definition 10, if x ∈ A b then β y ( x ) = β m b · · · β m n b n ( x ) = x + y + S b . Therefore, β y ( x ) = x if and only if y + S b = S b .If f : B → B (cid:48) is a function of sets, then we also use f to denote thehomomorphism Z B → Z B (cid:48) defined by f . Proposition 13.
Suppose B and B (cid:48) are nonempty sets, { S b | b ∈ B } is a familyof subgroups of Z B , and { S b (cid:48) | b (cid:48) ∈ B (cid:48) } is a family of subgroups of Z B (cid:48) . Then Q ( S ) ∼ = Q ( S (cid:48) ) if and only if there is a bijection f : B → B (cid:48) with f ( S b ) = S (cid:48) f ( b ) ∀ b ∈ B .Proof. We modify notation used for Q ( S ) by using apostrophes when discussing Q ( S (cid:48) ). For instance, if b (cid:48) ∈ B (cid:48) then A (cid:48) b (cid:48) = Z B (cid:48) /S (cid:48) b (cid:48) . If there is a bijection f as described, then for each b ∈ B , f defines a group isomorphism A b → A (cid:48) f ( b ) .These group isomorphisms define a quandle isomorphism Q ( S ) → Q ( S (cid:48) ).For the converse, suppose g : Q ( S ) → Q ( S (cid:48) ) is a quandle isomorphism. Asan isomorphism, g must define a bijection between the orbits of Q ( S ) and the6rbits of Q ( S (cid:48) ), so g must define a bijection f : B → B (cid:48) with g ( A b ) = A (cid:48) f ( b ) ∀ b ∈ B .As g is a quandle isomorphism, g ( x (cid:46)y ) = g ( x ) (cid:46) g ( y ) ∀ x, y ∈ Q ( S ). It followsthat g ◦ β ± b = ( β (cid:48) f ( b ) ) ± ◦ g ∀ b ∈ B , and hence g ◦ β m b · · · β m n b n = ( β (cid:48) f ( b ) ) m · · · ( β (cid:48) f ( b n ) ) m n ◦ g (2)whenever n ≥ ∈ Z , b , . . . , b n ∈ B and m , . . . , m n ∈ Z . Now, suppose b ∈ B . According to (2), if x ∈ A b , b , . . . , b n ∈ B and m , . . . , m n ∈ Z then β m b · · · β m n b n ( x ) = x if and only if ( β (cid:48) f ( b ) ) m · · · ( β (cid:48) f ( b n ) ) m n ( g ( x )) = g ( x ). Wededuce from Lemma 12 that (cid:80) m i b i ∈ S b if and only if (cid:80) m i f ( b i ) ∈ S (cid:48) f ( b ) .That is, f ( S b ) = S (cid:48) f ( b ) .The next result will help us show that every translation-commutative quan-dle is isomorphic to a quandle of type Q ( S ). Lemma 14.
Suppose Q is a translation-commutative quandle, and y ∈ Q . Then β y = β z ∀ z ∈ Q y .Proof. If y ∈ Q then according to Definition 2, for any w, x ∈ Qβ β x ( y ) ( w ) = w (cid:46) ( y (cid:46) x ) = ( β − x ( w ) (cid:46) x ) (cid:46) ( y (cid:46) x ) = ( β − x ( w ) (cid:46) y ) (cid:46) x. According to formula (1), it follows that for every w ∈ Q , β β x ( y ) ( w ) = ( β − x ( w ) (cid:46) x ) (cid:46) y = w (cid:46) y = β y ( w ) . Hence β β x ( y ) = β y . Theorem 15.
Let Q be a translation-commutative quandle, and suppose thereis a bijection b ↔ Q b between a set B and the set of orbits of Q . Then there isa family S = { S b | b ∈ B } of subgroups of Z B such that Q ∼ = Q ( S ) .Proof. For each b ∈ B , choose a representative element q b ∈ Q b , and let β b = β q b .According to Lemma 14, { β b | b ∈ B } includes all the β maps of Q , so β ( Q ) isan abelian group generated by { β b | b ∈ B } . For each b ∈ B , then, there is asurjective function f b : Z B → Q b given by f b (cid:32) n (cid:88) i =1 m i b i (cid:33) = (cid:32) n (cid:89) i =1 β m i b i (cid:33) ( q b )whenever n ≥ ∈ Z , b , . . . , b n ∈ B and m , . . . , m n ∈ Z .For each b ∈ B , let S b = { (cid:80) m i b i ∈ Z B | f b ( (cid:80) m i b i ) = q b } . The fact that Q is translation-commutative implies that S b is a subgroup of Z B , and f b inducesa bijection between Z B /S b and Q b . Also, the fact that β b ( q b ) = β q b ( q b ) = q b implies that b ∈ S b . It follows that taken together, the surjections f b induce anisomorphism Q ( S ) → Q . 7 Theorems 6 and 9
Let D be a diagram of an oriented link L = K ∪ · · · ∪ K µ . The image in D of each component K i consists of finitely many arcs, separated at classicalcrossings where K i is the underpassing component. For each i , choose a fixedarc a i belonging to the image of K i in D . Lemma 16.
The quandle Q tc ( L ) has µ orbits. For each i ∈ { , . . . , µ } , thereis an orbit that contains all the elements q a such that a ∈ A ( D ) belongs tothe image of K i in D . This orbit does not contain any element q a such that a ∈ A ( D ) belong to the image of K j in D , where j (cid:54) = i .Proof. Consider a crossing of D as pictured in Fig. 2. Definition 3 tells us that q a = q a (cid:46) q a , so q a and q a belong to the same orbit of Q tc ( L ).Now, choose any i ∈ { , . . . , µ } . If we walk along the image of K i in D ,starting at a i , then we ultimately encounter every arc of D belonging to K i .The observation of the previous paragraph applies each time we pass from onearc of D to a different arc of D , so the q a elements corresponding to arcs of K i all belong to the same orbit of Q tc ( L ).To show that these orbits are distinct, consider the trivial quotient quandle Q of Q tc ( L ), i.e., the quandle obtained by adding x (cid:46) y = x ∀ x, y ∈ Q to therequirements of Definition 3. It is clear that the definition of this quotientquandle is satisfied by the set { , . . . , µ } , considered as a quandle with all β maps equal to the identity map. The fact that there is a well-defined quotientmap Q tc ( L ) → Q guarantees that Q tc ( L ) has µ different orbits.For each i ∈ { , . . . , µ } , let β i = β a i : Q tc ( L ) → Q tc ( L ). According toLemma 14, β , . . . , β µ are all the β maps of Q tc ( L ). It follows that Definition 3can be rephrased like this: Proposition 17. Q tc ( L ) is the translation-commutative quandle generated by { q a | a ∈ A ( D ) } , subject to these requirements. • Q tc ( L ) has β maps β , . . . , β µ , corresponding to the components of L . • At each crossing as indicated in Fig. 2, if the overpassing arc a belongsto the image of K κ in D , then q a = β κ ( q a ) . To describe Q tc ( L ) in greater detail, it is convenient to index the arcs andclassical crossings of D . Suppose that for each i ∈ { , . . . , µ } , n i is the number ofarcs of D belonging to K i . We index these arcs a i , a i , . . . , a i ( n i − , a in i = a i in order, as we walk along K i in the direction of its orientation. If 0 ≤ m < n i ,let c im be the classical crossing we traverse as we walk from a im to a i ( m +1) , let K κ ( c im ) be the overpassing component of L at c im , and let w ( c im ) ∈ {− , } bethe writhe of c im , as indicated in Fig. 3.Proposition 17 implies that if w ( c im ) = − q a im = β κ ( c im ) ( q a i ( m +1) ),and if w ( c im ) = 1 then q a i ( m +1) = β κ ( c im ) ( q a im ). In either case, q a i ( m +1) =8 im a i ( m +1) K κ ( c im ) a im K κ ( c im ) a i ( m +1) w ( c im ) = − w ( c im ) = 1Figure 3: K i passes under K κ ( c im ) at the classical crossing c im . β w ( c im ) κ ( c im ) ( q a im ). Therefore if 1 ≤ i ≤ µ and 0 ≤ m ≤ n i − q a i ( m +1) = (cid:32) m (cid:89) k =0 β w ( c ik ) κ ( c ik ) (cid:33) ( q a i ) . (3)An instance of (3) with 0 ≤ m ≤ n i − q a i ( m +1) in terms of q a , . . . , q a µ . After we remove all of these redundant q a i ( m +1) generators, only q a , . . . , q a µ remain. We rename these elements q , . . . , q µ , forsimplicity.The instances of (3) that have not been used to remove redundant generatorsare the ones with m = n i −
1. They tell us that for 1 ≤ i ≤ µ , q i = µ (cid:89) j =1 j (cid:54) = i β (cid:96) j/i ( K i ,K j ) j ( q i ) . (4)We deduce yet another equivalent description of Q tc ( L ): Proposition 18. Q tc ( L ) is the translation-commutative quandle generated by { q , . . . , q µ } , subject to the requirement that (4) holds for every i ∈ { , . . . , µ } . Let B = { b , . . . , b µ } , and for each i ∈ { , . . . , µ } let S i ( L ) = S b i ( L ) be thesubgroup of Z B generated by b i and (cid:96) i , where (cid:96) i = µ (cid:88) j =1 j (cid:54) = i (cid:96) j/i ( K i , K j ) b j . (5)Let S ( L ) = { S b i ( L ) | ≤ i ≤ µ } . Then Proposition 18 implies the following. Corollary 19.
The quandles Q tc ( L ) and Q ( S ( L )) are isomorphic. Now, suppose 1 ≤ i ≤ µ . Let ( b i ) be the subgroup of Z B generated by b i , andlet π i : Z B → Z B / ( b i ) be the canonical map onto the quotient. Then π i ( S i ( L ))is a cyclic subgroup of Z B / ( b i ), and it is cyclically generated by π i ( (cid:96) i ) or − π i ( (cid:96) i ).No other element cyclically generates π i ( S i ( L )). As Z B / ( b i ) is freely generatedby the elements π i ( b j ) with j (cid:54) = i , it follows that the coefficients (cid:96) j/i ( K i , K j )9hat appear in (5) are almost completely determined by π i ( S i ( L )); the onlyuncertainty is that they might all be multiplied by − (cid:96) j/i ( K i , K j ) = (cid:96) i/j ( K i , K j ) ∀ i (cid:54) = j ∈ { , . . . , µ } . G ( L ) /G ( L ) G ( L ) /G ( L ) of a link group. Webegin with three constructions. The first two are standard, and date back tothe introduction of quandles to knot theory by Joyce [7] and Matveev [10]. Definition 20. If Q is a quandle, then there is an associated group As( Q ) ,described using generators and relations as follows. There is a generator g q foreach q ∈ Q , and for each pair p, q ∈ Q there is a relation g p(cid:46)q = g q g p g − q . We should mention that Joyce [7] used the notation Adconj( Q ) instead ofAs( Q ). Also his quandle is the “opposite” of ours, i.e., it is defined by g p(cid:46)q = g − q g p g q instead of g p(cid:46)q = g q g p g − q . We use the convention of Definition 20 sothat if L is a link with a diagram D , then the presentation of As( Q ( L )) providedby Definitions 3 and 20 is the same as the presentation of the link group G ( L )given by Kauffman [8], and in the classical case by Fox [3].The next definition is an instance of the notion of a quandle variety, discussedby Joyce [7, Sec. 10]. Definition 21.
Let Q be a quandle. Then the translation-commutative quotient of Q is the quandle described using generators and relations as follows. Thereis a generator x q for each q ∈ Q , there is a relation x p(cid:46)q = x p (cid:46) x q for eachpair of elements p, q ∈ Q , and there is a relation ( x p (cid:46) x q ) (cid:46) x r = ( x p (cid:46) x r ) (cid:46) x q for each triple of (not necessarily distinct) elements p, q, r ∈ Q . We denote thisquandle Q tc . The mapping q (cid:55)→ x q defines a canonical surjective quandle map Q → Q tc .It is easy to see that if S ⊆ Q generates Q , then the image of S generates Q tc .Notice that if L is a link, then Q tc ( L ) = Q ( L ) tc .Recall that if G is a group and H is a subgroup of G , then [ G, H ] is thesubgroup of G generated by the set of commutators [ g, h ] = ghg − h − with g ∈ G and h ∈ H . The lower central series of G is the sequence of normalsubgroups G = G ⊇ G ⊇ G ⊇ . . . with G n +1 = [ G, G n ] for each n , and G is nilpotent of class n if G n = { } (cid:54) = G n − . Proposition 22.
Let Q be a quandle. Then the quandle surjection Q → Q tc induces a group surjection As( Q ) → As( Q tc ) whose kernel is the lower centralseries subgroup As( Q ) .Proof. The kernel of the induced surjection As( Q ) → As( Q tc ) is the normalsubgroup of As( Q ) generated by the images of relators derived from the quandlerelations ( x p (cid:46) x q ) (cid:46) x r = ( x p (cid:46) x r ) (cid:46) x q for p, q, r ∈ Q .10he quandle relation ( x p (cid:46)x q ) (cid:46)x r = ( x p (cid:46)x r ) (cid:46)x q provides the group relation g x r g x q g x p g − x q g − x r = g x q g x r g x p g − x r g − x q , which corresponds to the relator g x r g x q g x p g − x q g − x r g x q g x r g − x p g − x r g − x q . Conjugating by g − x p g − x r g − x q , we see that this relator is equivalent to g − x p g − x r g − x q g x r g x q g x p g − x q g − x r g x q g x r = [ g − x p , [ g − x r , g − x q ]] . As As( Q ) is generated by the inverses of the elements g x y for y ∈ Q , the com-mutators [ g − x p , [ g − x r , g − x q ]] generate As( Q ) as a normal subgroup of As( Q ). Corollary 23.
Let L be a link with group G ( L ) . Then the nilpotent quotientgroup G ( L ) /G ( L ) has a presentation with a generator g i for each i ∈ { , . . . , µ } and two kinds of relations. If i ∈ { , . . . , µ } , there is a relation g i = µ (cid:89) j =1 j (cid:54) = i g (cid:96) j/i ( K i ,K j ) j g i µ (cid:89) j =1 j (cid:54) = i g (cid:96) j/i ( K i ,K j ) j − ,and if i, j, k ∈ { , . . . , µ } , there is a relation [ g i , [ g j , g k ]] = 1 .Proof. Proposition 22 implies that the nilpotent quotient G ( L ) /G ( L ) is iso-morphic to As( Q tc ( L )). According to Proposition 18, the quandle Q tc ( L ) isdetermined by two properties: the quandle relations in (4) and the fact that Q tc ( L ) is translation-commutative. The quandle relations in (4) provide thefirst kind of group relation mentioned in the statement of the corollary, and asnoted in the proof of Proposition 22, the quandle relations in (1) provide grouprelations of the form [ g − i , [ g − j , g − k ]] = 1. The − g − i , [ g − j , g − k ]] generatethe same normal subgroup of the free group on { g , . . . , g µ } as elements of theform [ g i , [ g j , g k ]].Corollary 23 extends a well-known property of classical link groups to virtuallink groups. For classical links with µ = 2, the presentation of G ( L ) /G ( L ) wasmentioned by Reidemeister in his famous monograph [12, p. 45]. Reidemeister’sobservation was extended to classical links of arbitrarily many components byChen [1], who studied the lower central series of both G ( L ) and the metabelianquotient G ( L ) /G ( L ) (cid:48)(cid:48) . The lower central series quotients of G ( L ) /G ( L ) (cid:48)(cid:48) arecalled Chen groups in his honor. Only G ( L ) /G ( L ) is important here; we referto Hillman [6] and Papadima and Suciu [11] for more general discussions andfurther references to the literature.Sakuma and Traldi [14] used Chen’s presentation of G ( L ) /G ( L ) to proveTheorem 25 below. Recall that an articulation point in a graph is a vertexwhose removal results in a graph with strictly more connected components.11 efinition 24. Let L = K ∪ · · · ∪ K µ be an oriented classical link, and let (cid:96)g ( L ) be the linking graph defined in Definition 8. Then L has inseparablelinking numbers if (cid:96)g ( L ) is connected, and has no articulation point. Theorem 25. ([14]) Let L = K ∪ · · · ∪ K µ and L = K (cid:48) ∪ · · · ∪ K (cid:48) µ be µ -component classical links. Then these two properties are equivalent.1. There is an isomorphism between G ( L ) /G ( L ) and G ( L (cid:48) ) /G ( L (cid:48) ) , whichis “meridian-preserving” in the sense that for each i ∈ { , . . . , µ } , theimage of a Wirtinger generator of G ( L ) /G ( L ) corresponding to K i is aconjugate of a Wirtinger generator of G ( L (cid:48) ) /G ( L (cid:48) ) corresponding to K (cid:48) i .2. Whenever K i ∪· · ·∪ K i ν ⊆ L is a sublink with inseparable linking numbers,either (cid:96) ( K i j , K i k ) = (cid:96) ( K (cid:48) i j , K (cid:48) i k ) ∀ j (cid:54) = k ∈ { , . . . , ν } or (cid:96) ( K i j , K i k ) = − (cid:96) ( K (cid:48) i j , K (cid:48) i k ) ∀ j (cid:54) = k ∈ { , . . . , ν } . The notion of “meridian-preserving isomorphism” used in Theorem 25 canbe described using quandles. There are natural maps η : Q tc ( L ) → As( Q tc ( L ))and η (cid:48) : Q tc ( L (cid:48) ) → As( Q tc ( L (cid:48) )), and an isomorphism f : G ( L ) /G ( L ) → G ( L (cid:48) ) /G ( L (cid:48) ) is meridian-preserving if and only if it has the property thatfor each i ∈ { , . . . , µ } , f ( η ( q i )) = η (cid:48) ( x ) for some element x of the orbit of q (cid:48) i in Q tc ( L (cid:48) ).Comparing Theorem 25 to Theorem 9, we see that the group G ( L ) /G ( L ) is aless sensitive invariant of classical links than the quandle Q tc ( L ). The differenceis that Theorem 9 involves entire connected components of (cid:96)g ( L ), while Theorem25 involves only connected subgraphs of (cid:96)g ( L ) that have no articulation points.These subgraphs can be much smaller than connected components. a a a a µ − b b µ − b a µ Figure 4: A connected sum of µ − µ ≥
3, and consider the oriented versionsof the µ -component link pictured in Fig. 4. There are as many different ori-ented link types as there are lists of µ − ± (cid:96) ( K , K ) , . . . , (cid:96) ( K µ − , K µ ) considered equivalent to the reversed list (cid:96) ( K µ − , K µ ) , . . . , (cid:96) ( K , K ). It follows that there are always at least2 + (2 µ − − / µ − + 1oriented link types, because there are always at least two lists that remain thesame when reversed, namely 1 , . . . , − , . . . , −
1. For all of these links,12he entire graph (cid:96)g ( L ) is connected, but the only nontrivial subgraphs of (cid:96)g ( L )without articulation points are single edges. Theorem 9 tells us that the Q tc quandles corresponding to two lists of linking numbers are isomorphic if and onlyif the components of the two links can be indexed so that the linking numbersdiffer only by a common factor of ±
1. Therefore, there are always at least (cid:100) (2 µ − + 1) / (cid:101) = 2 µ − + 1nonisomorphic Q tc quandles among these links. In contrast, Theorem 25 tells usthat the nilpotent quotient groups G ( L ) /G ( L ) are all isomorphic to each other,through meridian-preserving isomorphisms. In fact, even the G ( L ) groups areall isomorphic through meridian-preserving isomorphisms: Proposition 26.
The groups of all oriented versions of the link pictured in Fig.4 are isomorphic to each other, through meridian-preserving isomorphisms.Proof.
Let L be one of these oriented links. Index the components of L as K , . . . , K µ in order from left to right, and index the arcs of the diagram as inFig. 4. For convenience, we use g a to denote the element of G ( L ) = As( Q ( L ))corresponding to an arc a ∈ A ( D ), rather than g q a .The two crossings involving K and K are on the far left. These twocrossings provide two relations in G ( L ); one relation is g b g a g − b = g a and theother is either g a g a g − a = g b or g a g b g − a = g a , depending on orientations.The first relation tells us that g a and g b commute with each other. Then eitherof the possible second relations tells us that g a = g b . With this equality inmind, one of the next two crossings tells us that g b g a g − b = g a , so g a and g b commute. The other crossing between K and K then provides a relation thatguarantees g a = g b . With this equality in mind, the relations provided by thenext two crossings tell us that g a and g b commute, and g a = g b .Continuing in this vein, we ultimately conclude that no matter how thelink components are oriented, G ( L ) is generated by g a , . . . , g a µ , subject to therelations g a i g a i +1 = g a i +1 g a i ∀ i ∈ { , . . . , µ − } . Generally speaking, the absence of a structure theory for arbitrary quandlesmakes it difficult to work with them. One way to deal with this difficulty isto study special varieties of quandles, which are easier to describe. For in-stance, knot theorists have considered Alexander, involutory, latin, medial and n -quandles; see [2] for a survey. The translation-commutative quandles we havediscussed in this paper constitute a subvariety of the medial quandles.Another way to deal with the intractability of quandles is to consider numer-ical invariants derived from them. One way to derive such invariants of a link L is to count the quandle maps from Q ( L ) into specified target quandles. In [5],Harrell and Nelson observed that if L = K ∪ K is a two-component link, thenthere is a connection between the linking numbers (cid:96) / ( K , K ) , (cid:96) / ( K , K )13nd the number of quandle maps from Q ( L ) into quandles X n of a particulartype.In the notation of Sec. 2, the quandle X n may be defined as follows. Suppose n ≥ ∈ Z , let B = { b , b } , and let S = { S b , S b } , where S b = Z B and S b is the subgroup of Z B generated by b and nb . Then X n = Q ( S ). Noticethat X n is a translation-commutative quandle with a very simple structure: thefirst orbit has only one element, the second orbit has n elements, β b cyclicallypermutes the elements of the second orbit, and β b is the identity map. Harrelland Nelson [5] proved the following. Proposition 27. ([5]) Suppose n ≥ ∈ Z . Let L be a two-component link, andlet k be the number of quandle maps Q ( L ) → X n .1. k = n + 1 if n divides neither (cid:96) / ( K , K ) nor (cid:96) / ( K , K ) .2. k = n + 1 + n if n divides precisely one of (cid:96) / ( K , K ) , (cid:96) / ( K , K ) .3. k = n + 1 + 2 n if n divides both (cid:96) / ( K , K ) and (cid:96) / ( K , K ) . As X n is translation-commutative, every quandle map Q ( L ) → X n factorsthrough the canonical surjection Q ( L ) → Q tc ( L ). Using this fact, it is notdifficult to deduce Proposition 27 from Corollary 19. For every two-componentlink L , there are n + 1 non-surjective quandle maps Q tc ( L ) → X n , which areconstant on each orbit of Q tc ( L ). Moreover, if i (cid:54) = j ∈ { , } and n divides (cid:96) j/i ( K , K ), then there are n surjective quandle maps Q tc ( L ) → X n , whichmap A i = Z B /S i ( L ) onto the n -element orbit of X n and map A j onto theone-element orbit of X n . Acknowledgments
We are indebted to Kyle Miller for illuminating correspondence.
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