1 Introduction
Suciu [Reference Suciu13] constructed a family of fibred ribbon n-knots in
$S^{n+2}$
for
$n>1$
, whose knot groups are all isomorphic to that of the trefoil. He distinguished these knots using the second homotopy groups of their complements. Later, Kanenobu and Sumi [Reference Kanenobu and Sumi7] computed the fundamental groups of their double branched covers and showed that these groups are mutually nonisomorphic via
$\operatorname {SL}(2,\mathbb {C})$
-representations, thereby distinguishing the knots. A quandle [Reference Joyce6, Reference Matveev10] is an abstract algebraic structure whose axioms correspond to the Reidemeister moves on
$1$
-knot diagrams. In knot theory, the most important example is the knot quandle, which is associated with an oriented n-knot and serves as a knot invariant. It is closely related to the knot group, in that information about the knot group can be extracted from the knot quandle. It is known that the converse does not hold in general (see [Reference Tanaka and Taniguchi14]).
Jabłonowski [Reference Jabłonowski5] proved that the knot quandles of Suciu’s n-knots are mutually nonisomorphic by comparing their core groups, and Yasuda [Reference Yasuda16] later provided an alternative proof. More specifically, his alternative proof can be described as follows. Winker [Reference Winker15] established for
$n=1$
that the knot quandle of an oriented n-knot carries information about the fundamental group of its double branched cover. Yasuda [Reference Yasuda16] extended this result to all
$n \ge 1$
. As a consequence, combining his result with the work of Kanenobu and Sumi [Reference Kanenobu and Sumi7], he reproved that the knot quandles of Suciu’s n-knots are mutually nonisomorphic and hence distinguish these knots. In this paper, we provide yet another proof of this result by analysing conjugacy classes of certain automorphisms of the free group of rank two.
Aside from nonisomorphism, a fundamental invariant of quandles is their type, which measures their complexity and takes values in
$\mathbb {N} \cup \{\infty \}$
. Yasuda [Reference Yasuda16] also showed that the types of these quandles are infinite by analysing linear representations of the
$3$
-strand braid group, which is isomorphic to the knot group of the trefoil. We give another proof of this infiniteness by computing the orders of the above automorphisms of the free group of rank two. We believe our alternative proof offers an additional perspective on the diversity of Suciu’s knots.
Tanaka and Taniguchi [Reference Tanaka and Taniguchi14] gave examples of
$2$
-knots whose knot groups are mutually isomorphic while their knot quandles are mutually nonisomorphic. In their examples, the types of the knot quandles differ and this difference distinguishes the quandles. From our result, we see that Suciu’s family of n-knots gives examples in which the knot quandles share the same type but are nevertheless nonisomorphic.
2 Suciu’s n-knots and the free group of rank two
Suciu [Reference Suciu13] constructed a family of fibred ribbon n-knots
$R_k$
in
$S^{n+2}$
for
$n>1$
, indexed by positive integers k. The fibre of
$R_k$
is a punctured
$(S^1 \times S^n) \mathbin {\#} (S^1 \times S^n)$
, whose fundamental group is isomorphic to the free group F of rank two, generated by a and b. For each positive integer k, let
$f_k$
be an element of the automorphism group
$\operatorname {Aut}(F)$
of F defined by
$f_k(a) = a^k b a^{-k}$
and
$f_k(b) = a^{k-1} b a^{-k}$
. We fix an isomorphism that sends the homotopy class of the loop
$S^1 \times \{*\}$
in the left
$S^1 \times S^n$
to a and that in the right
$S^1 \times S^n$
to b. With respect to this identification, the automorphism induced by the monodromy coincides with
$f_k$
(see [Reference Suciu13, page 488]). Since the knot quandle of a fibred oriented n-knot with
$n>1$
can be described by the automorphism induced by the monodromy of its fibre [Reference Inoue4], we examine the automorphisms
$f_k$
in detail.
Proposition 2.1. The order of
$f_k$
in
$\operatorname {Aut}(F)$
is infinite for any positive integer k.
Proof. The abelianisation map from F to
$\mathbb {Z}^2$
induces a natural homomorphism from
$\operatorname {Aut}(F)$
to
$\operatorname {GL}(2,\mathbb {Z})$
, and the image of
$f_k$
is
for all k. A simple calculation shows that this matrix has order
$6$
. Hence, if
$f_k$
has finite order, it must be a multiple of
$6$
. However, any nontrivial periodic automorphism from
$\operatorname {Aut}(F)$
has order
$2$
,
$3$
or
$4$
(see [Reference Meskin and Newman11]). Therefore, the order of
$f_k$
is infinite.
From the above argument, the map
$f_k^6$
lies in the kernel of the natural homomorphism from
$\operatorname {Aut}(F)$
to
$\operatorname {GL}(2,\mathbb {Z})$
. Since
$\operatorname {GL}(2,\mathbb {Z})$
is isomorphic to the outer automorphism group
$\operatorname {Out}(F)$
of the rank-two free group F, it follows that
$f_k^6$
belongs to the inner automorphism group
$\operatorname {Inn}(F)$
of F. Since the centre of F is trivial, the natural homomorphism
$I \colon F \to \operatorname {Inn}(F)$
is an isomorphism (see, for example, [Reference Lyndon and Schupp8, Ch. I]). At this point, it is natural to attempt a direct computation of
$f_k^6$
, which yields the following result.
Lemma 2.2. The map
$f_k^6$
is the inner automorphism
$I(x_k)$
of F, with
which lies in the commutator subgroup
$[F,F]$
of F.
Proof. Let f be the automorphism of F defined by
$f(a) = b$
and
$f(b) = a^{-1}b$
. Then, for each positive integer k, we have
$f_k = I(a^k) \circ f$
. By induction, for any positive integer
$\ell $
, we also have
$f_k^{\ell +1} = I(a^k f(a)^k \cdots f^{\ell }(a)^k) \circ f^{\ell +1}$
. Moreover, by computing up to
$f^6(a)$
and
$f^6(b)=f^7(a)$
, one sees that
$f^6 = I(b^{-1}ab a^{-1})$
, as can be verified from the following calculations:
Hence,
$f_k^6 = I(a^k f(a)^k \cdots f^5(a)^k \,(b^{-1}ab a^{-1}))$
. A straightforward computation shows that
Since the total exponents of both a and b in this element are zero, it lies in the commutator subgroup
$[F,F]$
.
Proposition 2.3. The automorphisms
$f_k$
are mutually nonconjugate in
$\operatorname {Aut}(F)$
.
Proof. It suffices to show that the elements
$f_k^6$
in
$\operatorname {Inn}(F)$
are mutually nonconjugate in
$\operatorname {Aut}(F)$
. Since
$f \circ I(x) \circ f^{-1} = I(f(x))$
for any
$x \in F$
and
$f \in \operatorname {Aut}(F)$
, the conjugacy class of
$f_k^6 = I(x_k)$
in
$\operatorname {Aut}(F)$
corresponds to the
$\operatorname {Aut}(F)$
-orbit of
$x_k$
in F. Hence, it is enough to show that the
$\operatorname {Aut}(F)$
-orbits of the elements
$x_k$
in
$[F,F]$
are distinct. We then focus on the
$\operatorname {Aut}(F)$
-equivariant map from
$[F,F]$
to
$[F,F]/[[F,F],F]$
and show that the
$\operatorname {Aut}(F)$
-orbits of the images of
$x_k$
are distinct. Note that the codomain is the centre of the
$2$
-step nilpotent quotient of F, whose structure is well understood. For instance, it is isomorphic to
$\wedge ^2 \mathbb {Z}^2 \cong \mathbb {Z}$
(see [Reference Magnus, Karrass and Solitar9]), where we note that the abelianisation of F is
$\mathbb {Z}^2$
with the equivalence class of the commutator
$[a,b]$
as a generator. Moreover, the action of
$\operatorname {Aut}(F)$
on the codomain factors through
that is, it is given by the sign of the determinant.
First, we rewrite
$x_k$
in
$[F,F]$
modulo
$[[F,F],F]$
as a product of commutators:
Next, we express each commutator in terms of the generator
$[a,b]$
. To do so, we use the isomorphism
$[F,F]/[[F,F],F] \cong \wedge ^2 \mathbb {Z}^2 \cong \mathbb {Z}$
. That is, we apply the abelianisation map from F into
$\mathbb {Z}^2$
to convert products into sums, and replace each commutator with the exterior product. The computations then yield:
In summary, we obtain
$x_k \equiv [a,b]^{2k(k-1)} \cdot [a,b]^{k^2} \cdot [a,b]^{-(k-1)} = [a,b]^{3k^2-3k+1}$
. Since
$3k^2-3k+1> 0$
is strictly increasing for
$k \ge 1$
, these are distinct integers up to sign. Hence, we see that the
$\operatorname {Aut}(F)$
-orbits of the images of
$x_k$
are distinct.
3 The knot quandles of Suciu’s n-knots
A typical example of quandles is the generalised Alexander quandle
$\operatorname {GAlex}(G,\varphi )$
, which is associated with a group G and an automorphism
$\varphi $
in
$\operatorname {Aut}(G)$
. Inoue [Reference Inoue4] proved that if K is a fibred oriented n-knot for
$n>1$
, then its knot quandle
$Q(K)$
is isomorphic to a generalised Alexander quandle
$\operatorname {GAlex}(G,\varphi )$
, where G is the fundamental group of the fibre of K and
$\varphi $
is the automorphism of G induced by the monodromy of the complement of K. From this, it follows in particular that the knot quandle
$Q(R_k)$
of Suciu’s fibred ribbon n-knot
$R_k$
is isomorphic to
$\operatorname {GAlex}(F,f_k)$
. With these preliminaries in place, we provide alternative proofs of [Reference Jabłonowski5, Theorem 5.1] and [Reference Yasuda16, Theorem 1.3], in that order. We remark that [Reference Jabłonowski5, Theorem 5.1] and [Reference Yasuda16, Theorem 1.2] state the same result, as mentioned in the introduction.
Theorem 3.1. The knot quandles
$Q(R_k)$
are mutually nonisomorphic.
Proof. When
$\operatorname {GAlex}(G,\psi )$
and
$\operatorname {GAlex}(G,\varphi )$
are connected (in the standard terminology of quandle theory), they are isomorphic if and only if
$\psi $
and
$\varphi $
are conjugate in
$\operatorname {Aut}(G)$
(see [Reference Clark, Dunning and Saito2, Lemma B.1] and [Reference Higashitani, Kamada, Kosaka and Kurihara3]). Since knot quandle
$Q(R_k)$
is connected [Reference Nosaka12, Lemma 2.27] and is isomorphic to
$\operatorname {GAlex}(F,f_k)$
, Proposition 2.3 immediately implies the statement of the theorem.
Theorem 3.2. The type of
$Q(R_k)$
is infinite for all positive integers k.
Proof. Since
$Q(R_k)$
is isomorphic to
$\operatorname {GAlex}(F,f_k)$
, it is enough to determine the type of the latter. It has been shown in [Reference Tanaka and Taniguchi14, Proposition 2.1] that the type of a generalised Alexander quandle coincides with the order of its defining automorphism. Therefore, Proposition 2.1 implies that the type of
$Q(R_k)$
is infinite.
We have mentioned that the knot quandle carries information about the knot group. As also noted in [Reference Yasuda16], we make here a remark concerning the converse. The second author, together with Taniguchi [Reference Tanaka and Taniguchi14], exhibited a triple of oriented
$2$
-knots whose knot groups are isomorphic, but whose knot quandles are mutually nonisomorphic. In that example, the types of the knot quandles differ and this difference distinguishes the knot quandles. Although it is not explicitly stated in their paper, we note that by applying Artin’s spinning construction [Reference Artin1], one can construct such triples of oriented n-knots for any
$n>1$
. This naturally raises the question of whether examples exist in which the types also coincide. In fact, combining the two theorems just proved shows that Suciu’s ribbon n-knots provide such examples, as stated in the following corollary.
Corollary 3.3. There exists an infinite family of oriented n-knots for
$n>1$
whose knot groups are mutually isomorphic and whose knot quandles have infinite type, but these knot quandles are mutually nonisomorphic.
Acknowledgment
The authors thank Michal Jabłonowski for bringing his preprint [Reference Jabłonowski5] to their attention.
