Free quandles and knot quandles are residually finite
aa r X i v : . [ m a t h . G T ] S e p FREE QUANDLES AND KNOT QUANDLES ARE RESIDUALLY FINITE
VALERIY G. BARDAKOV, MAHENDER SINGH, AND MANPREET SINGH
Abstract.
In this note, residual finiteness of quandles is defined and investigated. It is provedthat free quandles and knot quandles of tame knots are residually finite and Hopfian. Residualfiniteness of quandles arising from residually finite groups (conjugation, core and Alexanderquandles) is established. Further, residual finiteness of automorphism groups of some residuallyfinite quandles is also discussed. Introduction
In the recent years, quandles have been a subject of intensive investigation due to theirappearance in various areas of mathematics. These objects first appeared in the work of Joyce[8] under the name quandle, and that of Matveev [19] under the name distributive groupoid.A quandle is a set with a binary operation that satisfies three axioms modelled on the threeReidemeister moves of diagrams of knots in S . Joyce and Matveev independently proved thateach oriented diagram D ( K ) of a tame knot K (in fact, tame link) gives rise to a quandle Q ( K ),called the knot quandle, which is independent of the diagram D ( K ). Further, they showedthat if K and K are two tame knots with Q ( K ) ∼ = Q ( K ), then there is a homeomorphismof S mapping K onto K , not necessarily preserving the orientations. We refer the readerto the survey articles [5, 10, 20] for more on the historical development of the subject and itsrelationships with other areas of mathematics.Although the knot quandle is a strong invariant for tame knots, it is usually difficult tocheck whether two knot quandles are isomorphic. This motivates search for newer properties ofquandles, in particular, of knot quandles. Over the years, various ideas have been transferredfrom other algebraic theories to that of quandles and their analogues (racks, shelves, etc).The notion of residual finiteness (and other residual properties) of groups plays a crucial role incombinatorial group theory and low dimensional topology. In this note, we define and investigateresidual finiteness of quandles. We begin by proving some closure properties of residual finitenessof quandles. We then investigate residual finiteness of conjugation, core, Alexander quandles ofresidually finite groups. Further, we discuss residual finiteness of automorphism groups of someresidually finite quandles. Our first main result is that free quandles are residually finite, andfinitely generated residually finite quandles are Hopfian. Our next main result is that the knotquandles of tame knots are residually finite. The key idea in its proof is the notion of finiteseparability of a subgroup of a group, and a result of Long and Niblo [14] on finite separabilityof π ( X, p ) in π ( M, p ), where M is an orientable irreducible compact 3-manifold and X anincompressible connected subsurface of a component of the boundary ∂ ( M ) of M containing thebase point p . As a consequence, we obtain that knot quandles of tame knots are Hopfian. Mathematics Subject Classification.
Primary 57M25; Secondary 20E26, 57M05, 20N05.
Key words and phrases. Preliminaries on quandles
We begin with the definition of the main object of our study, namely, a quandle.
Definition 2.1. A quandle is a non-empty set X with a binary operation ( x, y ) x ∗ y satisfyingthe following axioms:(1) x ∗ x = x for all x ∈ X ;(2) For any x, y ∈ X there exists a unique z ∈ X such that x = z ∗ y ;(3) ( x ∗ y ) ∗ z = ( x ∗ z ) ∗ ( y ∗ z ) for all x, y, z ∈ X .A non-empty set with a binary operation satisfying only the axioms (2) and (3) is called a rack . Obviously, every quandle is a rack, but not conversely. Example 2.2.
Although tame knots are rich sources of quandles, many interesting examples ofquandles come from groups. • If G is a group, then the set G equipped with the binary operation a ∗ b = b − ab gives aquandle structure on G , called the conjugation quandle , and denoted by Conj( G ). • If G is a group and we take the binary operation a ∗ b = ba − b , then we get the corequandle , denoted as Core( G ). In particular, if G is additive abelian, then Core( G ) is theTakasaki quandle of G . • Let G be a group and ϕ ∈ Aut( G ), then the set G with binary operation a ∗ b = ϕ ( ab − ) b gives a quandle structure on G , which is denoted by Alex( G, ϕ ). These quandles arecalled as generalized Alexander quandles .The quandle axioms are equivalent to saying that for each x ∈ X , the map S x : X → X givenby S x ( y ) = y ∗ x is an automorphism of the quandle X fixing x , called an inner automorphism of X . The group generated by all such automorphisms is denoted by Inn( X ). The fact that S x isa bijection for each x ∈ X is equivalent to existence of another binary operation on X , written( x, y ) x ∗ − y , and satisfying x ∗ y = z if and only if x = z ∗ − y for all x, y, z ∈ X . Further, the map S : X → Conj (cid:0)
Inn( X ) (cid:1) given by S ( x ) = S x is a quandleanti-homomorphism. In other words,(2.0.1) S x ∗ y = S x ∗ − S y = S y ◦ S x ◦ S − y for x, y ∈ X . It is easy to see that X and Y are quandles and f : X → Y a map, then f ( x ∗ x ) = f ( x ) ∗ f ( x ) if and only if f ( x ∗ − x ) = f ( x ) ∗ − f ( x ) for all x , x ∈ X .A quandle X is called trivial if x ∗ y = x for all x, y ∈ X . Note that a trivial quandle cancontain arbitrary number of elements.3. Residually finite quandles and some properties
Recall that a group G is called residually finite if for each g ∈ G with g = 1, there exists afinite group F and a homomorphism φ : G → F such that φ ( g ) = 1.It is easy to see that a group G being residually finite is equivalent to saying that for g, h ∈ G with g = h , there exists a finite group F and a homomorphism φ : G → F such that φ ( g ) = φ ( h ).The preceding observation motivates the definition of residually finite quandles. REE QUANDLES AND KNOT QUANDLES ARE RESIDUALLY FINITE 3
Definition 3.1.
A quandle X is said to be residually finite if for all x, y ∈ X with x = y , thereexists a finite quandle F and quandle homomorphism φ : X → F such that φ ( x ) = φ ( y ).In [17], Mal’cev gave the definition of a residually finite algebra, and proved that for somealgebras residual finiteness implies that the word problem is solvable. The preceding definitionis a particular case of Mal’cev’s definition.Obviously, every finite quandle is residually finite, and every subquandle of a residually finitequandle is residually finite. Further, we have the following. Proposition 3.2.
Every trivial quandle is residually finite.Proof.
Let X be a trivial quandle. If X has only one element, then there is nothing to prove.Suppose that X has at least two elements. Let x, y ∈ X with x = y . Consider the trivialsubquandle { x, y } of X and define φ : X → { x, y } by φ ( x ) = x and φ ( z ) = y for all z = x . Thenit is easy to see that φ is a quandle homomorphism with φ ( x ) = φ ( y ), and hence X is residuallyfinite. (cid:3) Next, we investigate some closure properties of residually finite quandles. Let { X i } i ∈ I bean indexed family of quandles and X = Q i ∈ I X i their cartesian product. Then X is itself aquandle, called product quandle , with binary operation given by( x i ) ∗ ( y i ) = ( x i ∗ y i )for ( x i ) , ( y i ) ∈ X . Further, for each j ∈ I , the projection map π j : X → X j given by π j (cid:0) ( x i ) (cid:1) = x j is a quandle homomorphism. Proposition 3.3.
Let { X i } i ∈ I be an indexed family of residually finite quandles. Then theproduct quandle X = Q i ∈ I X i is residually finite.Proof. Let x = ( x i ) , y = ( y i ) ∈ X such that x = y . Then there exists an i ∈ I such that x i = y i . Since X i is residually finite, there exists a finite quandle F and a homomorphism φ : X i → F such that φ ( x i ) = φ ( y i ). The homomorphism φ ′ := φ ◦ π i satisfy φ ′ ( x ) = φ ′ ( y ),and hence X is a residually finite quandle. (cid:3) Proposition 3.4.
The following statements are equivalent for a quandle X : (1) X is residually finite; (2) there exists a family { W i } i ∈ I of finite quandles such that the quandle X is isomorphicto a subquandle of the product quandle Q i ∈ I W i .Proof. The implication (2) = ⇒ (1) follows from Proposition 3.3 and the fact that a subquandleof a residually finite quandle is residually finite. Conversely, suppose that X is residually finite.For each pair ( x, y ) ∈ X × X such that x = y , there exists a finite quandle W ( x,y ) and ahomomorphism φ ( x,y ) : X → W ( x,y ) such that φ ( x,y ) ( x ) = φ ( x,y ) ( y ). Now consider the quandle W = Y ( x,y ) ∈ X × X, x = y W ( x,y ) , VALERIY G. BARDAKOV, MAHENDER SINGH, AND MANPREET SINGH and define a homomorphism ψ : X → W by ψ = Y ( x,y ) ∈ X × X, x = y φ ( x,y ) , which is clearly injective. Hence X is residually finite being isomorphic to a subquandle of W . (cid:3) An inverse system of quandles { X i , π ij , I } consists of a directed set I , a family of quandles { X i } i ∈ I , and a collection of quandle homomorphisms π ij : X j → X i for i ≤ j in I satisfying thefollowing conditions:(1) π ii = id X i for each i ∈ I ;(2) π ij ◦ π jk = π ik for all i ≤ j ≤ k in I .Given an inverse system { X i , π ij , I } of quandles, as discussed above, we construct the productquandle X = Q i ∈ I X i . Let lim ←− X i be the subset of X consisting of elements ( x i ) ∈ X such that x i = π ij ( x j ) for i ≤ j in I . It is easy to see that lim ←− X i is subquandle of X called the inverse limitof the inverse system { X i , π ij , I } . In view of Proposition 3.3 and the fact that every subquandleof a residually finite quandle is residually finite, we obtain the following: Corollary 3.5.
The inverse limit of an inverse system of residually finite quandles is residuallyfinite. Residual finiteness of quandles arising from groups
In this section, we investigate residual finiteness of conjugation, core and Alexander quandlesof residually finite groups. We also discuss residual finiteness of certain automorphism groupsof residually finite quandles.
Proposition 4.1. If G is a residually finite group, then Conj( G ) and Core( G ) are both residuallyfinite quandles.Proof. If g , g ∈ G with g = g , then there exists a finite group F and a group homomorphism φ : G → F such that φ ( g ) = φ ( g ). The map Conj( φ ) : Conj( G ) → Conj( F ) given byConj( φ )( g ) = φ ( g ) for g ∈ Conj( G ) is a quandle homomorphism with Conj( φ )( g ) = Conj( φ )( g ).Similarly, Core( G ) is residually finite. (cid:3) For generalised Alexander quandles, we prove
Proposition 4.2.
Let G be a residually finite group. If α : G → G is an inner automorphism,then Alex(
G, α ) is a residually finite quandle.Proof. Let α be the inner automorphism induced by g ∈ G . If g , g ∈ G such that g = g , thenthere exists a finite group F and a group homomorphism ψ : G → F such that ψ ( g ) = ψ ( g ).Let β be the inner automorphism of F induced by ψ ( g ). It follows that ψ viewed as a map ψ : Alex( G, α ) → Alex(
F, β ) is a quandle homomorphism with ψ ( g ) = ψ ( g ), and henceAlex( G, α ) is residually finite. (cid:3)
It is well-known that the automorphism group of a finitely generated residually finite groupis residually finite [16, p.414]. For the inner automorphism group of residually finite quandles,we have the following result.
REE QUANDLES AND KNOT QUANDLES ARE RESIDUALLY FINITE 5
Theorem 4.3. If X is a residually finite quandle, then Inn( X ) is a residually finite group.Proof. If X = h x i | i ∈ I i , then Inn( X ) = h S x i | i ∈ I i . Let S e a ◦ S e a ◦ · · · ◦ S e m a m = 1 be anelement of Inn( X ), where a j ∈ { x i | i ∈ I } and e j ∈ { , − } . Then there exists an element x ∈ X such that S e a ◦ S e a ◦ · · · ◦ S e m a m ( x ) = x, equivalently (cid:0)(cid:0) ( x ∗ e m a m ) ∗ e m − a m − (cid:1) · · · (cid:1) ∗ e a = x. Since X is residually finite, there exists a finite quandle F and a quandle homomorphism φ : X → F such that(4.0.1) φ (cid:0)(cid:0)(cid:0) ( x ∗ e m a m ) ∗ e m − a m − (cid:1) · · · (cid:1) ∗ e a (cid:1) = φ ( x ) . Define a map e φ : (cid:8) S ± x i | i ∈ I (cid:9) → Inn( F )by setting e φ (cid:0) S ± x i (cid:1) = S ± φ ( x i ) . We claim that e φ preserves relations in Inn( X ), and hence extends to a group homomorphism.Observe that relations in Inn( X ) are induced by relations in X . If x ∗ y = z is a relation in X ,by (2.0.1), the induced relation in Inn( X ) is S z ◦ S y = S y ◦ S x . Since φ is a quandle homomorphism, we have φ ( x ) ∗ φ ( y ) = φ ( z ) in F . Again, by (2.0.1), wehave S φ ( z ) ◦ S φ ( y ) = S φ ( y ) ◦ S φ ( x ) . This proves our claim, and hence e φ extends to a group homomorphism e φ : Inn( X ) → Inn( F ). If e φ (cid:0) S e a ◦ S e a ◦ · · · ◦ S e m a m (cid:1) = 1, then evaluating both the sides at φ ( x ) contradicts (4.0.1). Hence,Inn( X ) is a residually finite group. (cid:3) Next, we present some observations for automorphism groups of core and conjugation quandlesof residually finite groups.
Proposition 4.4. If G is a finitely generated abelian group with no -torsion, then Aut (cid:0)
Core( G ) (cid:1) is residually finite group.Proof. Since G is a finitely generated abelian group, it is residually finite, and hence Aut( G )is also residually finite. Moreover, semi-direct product of residually finite groups is residuallyfinite. By [1, Theorem 4.2], Aut (cid:0) Core( G ) (cid:1) ∼ = G ⋊ Aut( G ), and hence Aut (cid:0) Core( G ) (cid:1) is residuallyfinite. (cid:3) Proposition 4.5. If G is a finitely generated residually finite group with trivial centre, then Aut (cid:0)
Conj( G ) (cid:1) is residually finite.Proof. Since G has trivial center, by [2, Corollary 4.2], Aut (cid:0) Conj( G ) (cid:1) = Aut( G ), which isresidually finite as G is so. (cid:3) VALERIY G. BARDAKOV, MAHENDER SINGH, AND MANPREET SINGH Residual finiteness of free quandles
In this section, we consider residual finiteness of free quandles, the free objects in the categoryof quandles.
Definition 5.1.
A free quandle on a non-empty set S is a quandle F Q ( S ) together with a map φ : S → F Q ( S ) such that for any other map ρ : S → X , where X is a quandle, there exists aunique quandle homomorphism ¯ ρ : F Q ( S ) → X such that ¯ ρ ◦ φ = ρ .A free rack is defined analogously. It follows from the definition that a free quandle (a freerack) is unique up to isomorphism, and every quandle (rack) is quotient of a free quandle (rack).The following construction of a free rack is due to Fenn and Rourke [6, p.351]. Let S be a setand F ( S ) the free group on S . Define F R ( S ) := S × F ( S ) = (cid:8) a w := ( a, w ) | a ∈ S, w ∈ F ( S ) (cid:9) with the operation defined as a w ∗ b u := a wu − bu . It can be seen that
F R ( S ) is a free rack on S .A model of the free quandle on the set S is due to Kamada [11, 12], who defined the freequandle F Q ( S ) on S as a quotient of F R ( S ) modulo the equivalence relation generated by a w = a aw for a ∈ S and w ∈ F ( S ). It is not difficult to check that F Q ( S ) is quandle satisfying the aboveuniversal property.There is another model of free quandle on a set S [21, Example 2.16], which is defined asthe subquandle of Conj (cid:0) F ( S ) (cid:1) consisting of all conjugates of elements of S . For the benefit ofreaders, we present an explicit isomorphism between the two models. Proposition 5.2.
The map
Φ :
F Q ( S ) → Conj (cid:0) F ( S ) (cid:1) given by Φ( a w ) = w − aw is an embed-ding of quandles.Proof. Let a w , a w ∈ F Q ( S ). Then Φ( a w ) = w − a w , Φ( a w ) = w − a w and a w ∗ a w = a w w − a w . Further, Φ( a w ∗ a w ) = Φ( a w w − a w )= ( w w − a w ) − a ( w w − a w )= w − a − w w − a w w − a w = ( w − a w ) − ( w − a w )( w − a w )= Φ( a w ) ∗ Φ( a w ) , and hence Φ is a quandle homomorphism. Let a w , a w ∈ F Q ( S ) such that a w = a w .Case 1: Suppose a = a . If Φ( a w ) = Φ( a w ), then w − a w = w − a w , which contradictsthe fact that F ( S ) is a free group. Hence Φ( a w ) = Φ( a w ).Case 2: Suppose a = a = a . If Φ( a w ) = Φ( a w ), then w − aw = w − aw , which furtherimplies that w w − commutes with a in F ( S ). Since F ( S ) is a free group, only powers of a cancommute with a , and hence w w − = a i for some integer i . Thus w = a i w , which implies that REE QUANDLES AND KNOT QUANDLES ARE RESIDUALLY FINITE 7 a w = a a i w = a w in F Q ( S ), a contradiction. Hence Φ( a w ) = Φ( a w ), and Φ is an embeddingof quandles. (cid:3) Theorem 5.3.
Every free quandle is residually finite.Proof.
Let
F Q ( S ) be the free quandle on the set S . It is well-known that the free group F ( S ) isresidually finite [24, Theorem 2.3.1]. By Proposition 4.1, the quandle Conj (cid:0) F ( S ) (cid:1) is residuallyfinite. Since F Q ( S ) is a subquandle of Conj (cid:0) F ( S ) (cid:1) , it follows that F Q ( S ) is residually finite. (cid:3) The following is a well-known result for free groups [13, p.42].
Theorem 5.4. If F ( S ) is a free group on a set S and g = 1 an element of F ( S ) , then thereis a homomorphism ρ : F ( S ) → S n for some n such that ρ ( g ) = 1 , where S n is the symmetricgroup on n elements. We prove an analogue of the preceding result for free quandles.
Theorem 5.5.
Let
F Q ( S ) be a free quandle on a set S and x, y ∈ F Q ( S ) such that x = y . Thenthere is a quandle homomorphism φ : F Q ( S ) → Conj(S n ) for some n such that φ ( x ) = φ ( y ) .Proof. Recall that the map Φ :
F Q ( S ) → Conj (cid:0) F ( S ) (cid:1) in Theorem 5.3 is an injective quandlehomomorphism. Let a w = a w ∈ F Q ( S ). Then g = g ∈ F ( S ), where g = Φ( a w ) and g =Φ( a w ). Thus, g − g is a non-trivial element of F ( S ). By Theorem 5.4, there exists a symmetricgroup S n for some n and a group homomorphism ρ : F ( S ) → S n such that ρ ( g ) = ρ ( g ). LetConj( ρ ) : Conj (cid:0) F ( S ) (cid:1) → Conj(S n ) be the induced map with Conj( ρ )( g ) = Conj( ρ )( g ). Taking φ := Conj( ρ ) ◦ Φ :
F Q ( S ) → Conj(S n ), we see that φ ( a w ) = φ ( a w ). (cid:3) Definition 5.6.
A quandle X is called Hopfian if every surjective quandle endomorphism of X is injective.It is well-known that finitely generated residually finite groups are Hopfian [18]. We prove asimilar result for quandles. Theorem 5.7.
Every finitely generated residually finite quandle is Hopfian.Proof.
Let X be a finitely generated residually finite quandle and φ : X → X a surjectivequandle homomorphism. Suppose that φ is not injective. Let x , x ∈ X such that x = x and φ ( x ) = φ ( x ). Since X is residually finite, there exist a finite quandle F and a quandlehomomorphism τ : X → F such that τ ( x ) = τ ( x ).We claim that the maps τ ◦ φ n : X → F are distinct quandle homomorphisms for all n ≥ ≤ m < n be integers. Since φ m : X → X is surjective, there exist y , y ∈ X such that φ m ( y ) = x and φ m ( y ) = x . Thus, we have τ ◦ φ m ( y ) = τ ◦ φ m ( y ) , whereas τ ◦ φ n ( y ) = τ ◦ φ n ( y ) , which proves our claim. Thus, there are infinitely many quandle homomorphisms from X to F , which is a contradiction, since X is finitely generated and F is finite. Hence, φ is anautomorphism, and X is Hopfian. (cid:3) VALERIY G. BARDAKOV, MAHENDER SINGH, AND MANPREET SINGH
By theorems 5.3 and 5.7, we obtain
Corollary 5.8.
Every finitely generated free quandle is Hopfian.
Remark 5.9.
The preceding result is not true for infinitely generated free quandles (free racks).Indeed, if
F Q ∞ is a free quandle that is freely generated by an infinite set { x , x , . . . } , then wecan define a homomorphism ϕ : F Q ∞ → F Q ∞ by setting ϕ ( x ) = x and ϕ ( x i ) = x i − for i ≥
2. It is easy to see that ϕ is an epimorphism which is not an automorphism since ϕ ( x ) = ϕ ( x ).The enveloping group of a quandle Q , denoted by G Q , is the group with Q as the set ofgenerators and defining relations x ∗ y = y − xy for all x, y ∈ Q . For example, if Q is a trivial quandle, then G Q is the free abelian group of rankthe cardinality of Q .Since every quandle is quotient of a free quandle, a quandle Q can be defined by a set ofgenerators and defining relations as Q = (cid:10) X || R (cid:11) . For example, knot (link) quandles have such presentations.
Proposition 5.10.
Let
F Q ( S ) and F Q ( T ) be free quandles on sets S and T , respectively. If F Q ( S ) ∼ = F Q ( T ) , then | S | = | T | .Proof. By [26, p.106, Theorem 5.1.7], if Q is a quandle with a presentation Q = h X || R i ,then its enveloping group has presentation G Q ∼ = h X || ¯ R i , where ¯ R consists of relations in R with each expression x ∗ y replaced by y − xy . Consequently, since F Q ( S ) and F Q ( T ) arefree quandles, it follows that G F Q ( S ) = F ( S ) and G F Q ( T ) = F ( T ) are free groups on the sets S and T , respectively. Since F Q ( S ) ∼ = F Q ( T ), we must have G F Q ( S ) ∼ = G F Q ( T ) , and hence | S | = | T | . (cid:3) In view of Proposition 5.10, we can define the rank of a free quandle as the cardinality of itsany free generating set.Analogous to groups, we define the word problem for quandles as the problem of determiningwhether two given elements of a quandle are the same. The word problem is solvable for finitelypresented residually finite groups [22, p.55, 2.2.5]. Below is a similar result for quandles.
Theorem 5.11.
Every finitely presented residually finite quandle has a solvable word problem.Proof.
Let Q = h X || R i be a finitely presented residually finite quandle, and w , w two wordsin the generators X . We describe two procedures which tell us whether or not w = w in Q .The first procedure lists all the words that we obtain by using the relations of Q on the word w . If the word w turns up at some stage, then w = w , and we are done.The second procedure lists all the finite quandles. Since Q is finitely generated, for eachfinite quandle F , the set Hom( Q, F ) of all quandle homomorphisms is finite. Now for eachhomomorphism φ ∈ Hom(
Q, F ), we look for φ ( w ) and φ ( w ) in F , and check whether or not REE QUANDLES AND KNOT QUANDLES ARE RESIDUALLY FINITE 9 φ ( w ) = φ ( w ). Since Q is residually finite, the above procedure must stop at some time. Thatis, there exists a finite quandle F and φ ∈ Hom(
Q, F ) such that φ ( w ) = φ ( w ) in F , and hence w = w in Q . (cid:3) Remark 5.12.
In a recent work [3], Belk and McGrail showed that the word problem forquandles is unsolvable in general by giving an example of a finitely presented quandle withunsolvable word problem. In view of Theorem 5.11, such a quandle cannot be residually finite.6.
Residual finiteness of knot quandles
In this section, we prove that the knot quandle of a tame knot is residually finite. We recallthe following definition from [17].
Definition 6.1.
A subgroup H of a group G is said to be finitely separable in G if for each g ∈ G \ H , there exists a finite group F and a group homomorphism φ : G → F such that φ ( g ) φ ( H ).Let H be a subgroup of a group G and G/H the set of right cosets of H in G . For g ∈ G , wedenote its right coset by ¯ g . Let z ∈ C G ( H ), the centraliser of H in G , be a fixed element. Thenit is easy to see that the set G/H with the binary operation given by¯ x ∗ ¯ y = ¯ z − ¯ x ¯ y − ¯ z ¯ y for ¯ x, ¯ y ∈ G/H forms a quandle, denoted (
G/H, z ). Proposition 6.2.
Let H be a subgroup of a group G and z ∈ C G ( H ) . If H is finitely separablein G , then the quandle ( G/H, z ) is residually finite.Proof. Let ¯ g , ¯ g ∈ G/H such that ¯ g = ¯ g , that is, g = hg for any h ∈ H . Since H is finitelyseparable in G , there exists a finite group F and a group homomorphism φ : G → F such that φ ( g ) = φ ( hg ) for each h ∈ H . Let H := φ ( H ) and ¯ z := φ ( z ) ∈ C F ( H ). Then ( F/H, ¯ z ) is afinite quandle. Further, the group homomorphism φ : G → F induces a well-defined map¯ φ : ( G/H, z ) → ( F/H, ¯ z )given by ¯ φ (¯ x ) = Hφ ( x ) , which is a quandle homomorphism. Also, ¯ φ (¯ g ) = ¯ φ (¯ g ), otherwise φ ( g ) = φ ( hg ) for some h ∈ H , which is a contradiction. Hence the quandle ( G/H, z ) is residually finite. (cid:3)
Definition 6.3.
A subquandle Y of a quandle X is said to be finitely separable in X if for each x ∈ X \ Y , there exists a finite quandle F and a quandle homomorphism φ : X → F such that φ ( x ) φ ( Y ).The following result might be of independent interest. Proposition 6.4.
Let X be a residually finite quandle and α ∈ Aut( X ) . If Fix( α ) := { x ∈ X | α ( x ) = x } is non-empty, then it is a finitely separable subquandle of X . Proof.
Clearly Fix( α ) is a subquandle of X . Let x ∈ X \ Fix( α ), that is, α ( x ) = x . Since X is residually finite, there exists a finite quandle F and a quandle homomorphism φ : X → F such that φ ( α ( x )) = φ ( x ). Define a map η : X → F × F by η ( x ) = (cid:0) φ ( x ) , φ ( α ( x )) (cid:1) . Clearly η is a quandle homomorphism with η ( x ) η ( X ), and hence Fix( α ) is finitely separable in X . (cid:3) Answering a question raised by Jaco [7, V.22], Long and Niblo [14] proved the following resultusing the fact that doubling a 3-manifold along its boundary preserves residual finiteness.
Theorem 6.5.
Suppose that M is an orientable, irreducible compact 3-manifold and X anincompressible connected subsurface of a component of ∂ ( M ) . If p ∈ X is a base point, then π ( X, p ) is a finitely separable subgroup of π ( M, p ) . A group G is said to be subgroup separable if every finitely generated subgroup of G is finitelyseparable in G . It is well-known that not all 3-manifold groups are subgroup separable (see [14]).We refer the reader to [23] for relation between subgroup separability and geometric topology.Let V ( K ) be a tubular neighbourhood of a knot K in S . Then the knot complement C ( K ) := S \ V ( K ) has boundary ∂C ( K ) a torus. Let x ∈ ∂C ( K ) a fixed base point. Then the inclusion ι : ∂C ( K ) −→ C ( K )induces a group homomorphism ι ∗ : π (cid:0) ∂C ( K ) , x (cid:1) −→ π (cid:0) C ( K ) , x (cid:1) , which is injective unless the knot K is trivial [4, p.41, Proposition 3.17]. In fact, π (cid:0) ∂C ( K ) , x (cid:1) ∼ = Z ⊕ Z , and if K is trivial, then π (cid:0) C ( K ) , x (cid:1) ∼ = Z . The group P := ι ∗ (cid:0) π ( ∂C ( K ) , x ) (cid:1) is calledthe peripheral subgroup of the knot group π (cid:0) C ( K ) , x (cid:1) . Now, an immediate consequence ofTheorem 6.5 is the following result. Corollary 6.6.
The peripheral subgroup of a non-trivial tame knot is finitely separable in theknot group.
We need the following result of Joyce [9, Section 4.9], which follows by observing that theknot group G of a tame knot K acts transitively on its knot quandle Q ( K ) with the stabiliserof an element of Q ( K ) being isomorphic to the peripheral subgroup P . Proposition 6.7.
Let K be a tame knot with knot group G and knot quandle Q ( K ) . Let P be the peripheral subgroup of G containing the meridian m . Then the knot quandle Q ( K ) isisomorphic to the quandle ( G/P, m ) . We now have our main result.
Theorem 6.8.
The knot quandle of a tame knot is residually finite.Proof.
Let K be a tame knot. If K is an unknot, then the knot quandle Q ( K ) is vacuously resid-ually finite being a trivial quandle with one element. If K is non-trivial, then using Proposition6.7, Corollary 6.6 and Proposition 6.2 it follows that Q ( K ) is residually finite. (cid:3) As a consequence of Theorem 5.11 and 6.8, it follows that the word problem is solvable inknot quandles of tame knots.Theorems 5.7 and 6.8 yield the following.
REE QUANDLES AND KNOT QUANDLES ARE RESIDUALLY FINITE 11
Corollary 6.9.
The knot quandle of a tame knot is Hopfian.
An immediate consequence of Theorem 6.8 is the following
Corollary 6.10.
Let K be a non-trivial tame knot. Then there exists a finite quandle X suchthat Hom (cid:0) Q ( K ) , X (cid:1) has a non-constant homomorphism. Remark 6.11.
Joyce [8, pp. 47–48] used a result of Waldhausen [25] to prove that quandlesassociated to tame knots are complete invariants up to orientation. To use [25] the knot com-plements are required to be irreducible 3-manifolds. But, this is not always true for tame linkssince there are tame links whose complements in S are reducible 3-manifolds. Thus, quandlesassociated to tame links are not complete invariants. For the same reason, Theorem 6.5 is notapplicable, and hence we are not able to prove an analogue of Theorem 6.8 for tame links withmore than one component. Problem 6.12.
We conclude with the following problems which might shed more light on theideas pursued in this paper.(1) Let L be a tame link with more than one component. Is it true that the link quandle Q ( L ) is residually finite? We note that if L n is a trivial n -component link, then the linkquandle Q ( L n ) is isomorphic to the free quandle on n generators, and hence is residuallyfinite by Theorem 5.3.(2) Is it true that any subquandle of a free quandle is free? Acknowledgement.
The authors thank the anonymous referee for useful comments and forthe references [6, 11, 12, 21]. Bardakov acknowledges support from the Russian Science Founda-tion project N 16-41-02006. Mahender Singh acknowledges support from INT/RUS/RSF/P-02grant and SERB Matrics Grant. Manpreet Singh thanks IISER Mohali for the PhD ResearchFellowship.
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