Normal closures of slope elements in knot groups and the peripheral Magnus property
Let be a nontrivial knot in with the exterior . A slope element in is a nontrivial element represented by a simple closed curve on . Each slope element defines a normal subgroup , the normal closure of . We investigate their relations such as intersection and inclusion. In particular, we establish the “peripheral” Magnus property which asserts that if and only if or .
Let be a nontrivial knot in with its exterior . Then by the loop theorem the inclusion map induces a monomorphism , where we take a base point in . We denote the knot group by and its peripheral subgroup by . A slope element in is a primitive element in , which is represented by an oriented simple closed curve on . Denote by the normal closure of in . Taking a standard meridian-longitude pair of , each slope element is expressed as for some relatively prime integers . As usual we use the term slope to mean the isotopy class of an unoriented simple closed curve on . A slope element and its inverse give the same normal subgroup , and they represent the same slope (by forgetting orientations), which may be identified with , so in the following we denote by . Thus each slope defines the normal subgroup , which will be referred to as the normal closure of the slope for simplicity. A slope is trivial if , i.e. is represented by a meridian of .
Recall that a group possesses the Magnus property, if two elements , of have the same normal closure, then is conjugate to or . Magnus  established this property for free groups, and recently [3, 4, 9, 16] prove the fundamental groups of closed surfaces have this property. However, in general knot groups do not satisfy this property; see Remark 1.2 below. Restricting our attention to normal closures of slopes of a knot, we introduce the following:
Let be a nontrivial knot in . The knot group has the peripheral Magnus property if implies for two slopes and .
A nontrivial element is called a killer or a pseudo-meridian if , but it is not conjugate to [33, 34]. It is known by [33, 34] that the knot groups of -bridge knots, torus knots, hyperbolic knots with unknotting number one, and a certain family of satellite knots admit such elements. Hence these groups do not satisfy the Magnus property. See also . It is conjectured that any nontrivial knot group admits a pseudo-meridian , in particular, does not satisfy the Magnus property.
In this context, Property P  says that if and only if . We first establish:
Theorem 1.3 (peripheral Magnus property).
Any nontrivial knot satisfies the peripheral Magnus property, namely if and only if .
Hence there is a one to one correspondence between the set of slopes, which is identified with , and the set of normal closures of slopes. Let us investigate for which slope , is finitely generated.
If has a finite surgery slope , i.e. –surgery on yields a –manifold with finite fundamental group, then is finitely generated. (See the proof of Theorem 1.4.)
If is a torus knot , then is an infinite cyclic normal subgroup of , hence finitely generated.
Actually we have:
Theorem 1.4 (finitely generated normal closures of slopes).
Let be a nontrivial knot. The normal closure is finitely generated if and only if is a finite surgery slope, or is a torus knot and .
Thus normal closures of slopes are generically infinitely generated, and it seems to be interesting to ask:
Let be a nontrivial knot in . For how many slopes of , the intersection of their normal closures intersect nontrivially? Furthermore, if it is nontrivial, how big is this subgroup?
Theorem 1.6 (nontrivial intersection property).
Let be a nontrivial knot in and a finite set of slopes of . If is a torus knot , we assume that . Then is nontrivial. Moreover, this subgroup is finitely generated if and only if all the are finite surgery slopes.
This result should be compared with the following result .
Theorem 1.7 ().
Let be a hyperbolic knot in . Then for any infinite family of slopes .
Let be a nontrivial torus knot . Then , which is the free group of rank . See Proposition 6.5.
Let be a hyperbolic knot in . For any infinite family of slopes, their normal closures intersect trivially, while for any finite subfamily, the intersection of their normal closures contains infinitely many elements.
A normal closure of a slope of naturally arises via Dehn surgery on . Denote by the 3-manifold obtained by -surgery on . Then we have the following short exact sequence which relate and .
is the subgroup of consisting elements which become trivial after -surgery on .
Thus, we immediately obtain the following interpretation of Corollary 1.9.
Corollary 1.10 (vanishing elements by Dehn fillings).
Let be a hyperbolic knot in . Then for any nontrivial element , there are only finitely many Dehn surgeries on trivializing , while for any finitely many Dehn surgeries on , we can find infinitely many nontrivial elements which become trivial after these surgeries.
Let us turn to inclusion relations among normal closures of slopes. Since , for any slope .
Theorem 1.11 (chains for non-torus knots).
Let be a non-torus knot in . If for mutually distinct slopes , then . In particular, there is no infinite descending chain nor ascending chain of normal closures of slopes.
On the contrary, for torus knots we have:
Theorem 1.12 (chains for torus knots).
Let be a torus knot .
There is no infinite ascending chain .
For each finite surgery slope , there exists an infinite descending chain
2. Inclusions between two normal closures of slopes
In this section we study inclusions between two normal closures of slopes. We say that a slope is a reducing surgery slope if is a reducible 3-manifold.
Let be a nontrivial knot in with meridian . If for some integer , then is a finite surgery slope or a reducing surgery slope.
Without loss of generality we may assume . If , then and , and thus is a finite surgery slope. (Actually, Property P  implies .) So in the following we assume . Since , and we have the canonical epimorphism . Note that is a nontrivial torsion element in . Hence, if is nontrivial, then it is a nontrivial torsion element in . Assume . Then and as above . Thus we may assume , i.e. it is a nontrivial torsion element in . Recall that an irreducible –manifold with infinite fundamental group is aspherical [1, p.48 (C.1)] and hence has no torsion element . Hence is finite or must be a reducible manifold. Accordingly is a finite surgery slope or a reducing surgery slope. ∎
Let be a nontrivial knot in with meridian , and a nontrivial slope. If for some integer , then is not a reducing surgery slope.
Assume to the contrary that is a reducing surgery slope. Following , , and thus is a connected sum of two closed –manifolds other than . By the Poincaré conjecture, they have nontrivial fundamental groups, and for some nontrivial groups and .
As we have seen in the proof of Lemma 2.1, , the image of under the canonical epimorphism is a nontrivial torsion element in . By [25, Corollary 4.1.4], a nontrivial torsion element in a free product is conjugate to a torsion element of or . Thus we may assume that there exists such that .
On the other hand, is normally generated by since is normally generated by . This implies that is normally generated by an element . In particular, the normal closure of in is equal to , and . However, ([25, p.194]). This is a contradiction. ∎
Let be a nontrivial knot in . Assume that for distinct slopes and . Then is a finite surgery slope.
Let be a non-torus knot in . Assume that for distinct nontrivial slopes and . Then is not a finite surgery slope.
By the assumption and Proposition 2.3, is a nontrivial finite surgery slope. Assume to the contrary that is also a finite surgery slope. Write and with . Since , we have a canonical epimorphism from to , which induces an epimorphism
This then implies that and is a multiple of .
By the assumption is a hyperbolic knot or a satellite knot. Furthermore, in the latter case, since admits a nontrivial finite surgery, is a –cable of a torus knot , where .
Case 1. is a hyperbolic knot.
Recall that the distance between finite surgery slopes of a hyperbolic knot is at most two . Hence . Since is a multiple of , the inequality implies . Since , we have . A finite surgery is also an L-space surgery, so by [32, Corollary 1.4], , where is the genus of . This implies . Since a knot admitting an L-space surgery is fibered [27, 28, 12, 21], is a trefoil knot (or ) or the figure-eight knot. By the assumption, is not a torus knot, and hence would be the figure-eight knot. However, the figure-eight knot has no nontrivial finite surgery, a contradiction.
Case 2. is a –cable of a torus knot .
Finite surgeries on iterated torus knots are classified by [2, Table 1]. For any cable of a torus knot which admits two finite surgeries and , is not a multiple of .
This completes a proof of Proposition 2.4. ∎
3. Peripheral Magnus property for knot groups
Now we are ready to prove the peripheral Magnus property for knot groups.
Any nontrivial knot satisfies the peripheral Magnus property, namely if and only if .
The “if” part is obvious. Let us prove the “only if” part. Recall first that if , then can happen only when by Property P . If , then holds only if for homological reason. So in the following we assume is neither nor . We divide the argument into two cases depending upon is a torus knot or a non-torus knot.
Case 1. is a non-torus knot.
Suppose for a contradiction that we have mutually distinct slopes and which satisfy . Then it follows from Proposition 2.3 that and are finite surgery slopes. Since is not a torus knot, Proposition 2.4 shows that neither nor is a finite surgery slope, a contradiction.
Case 2. is a torus knot .
Without loss of generality, we assume . Assume that for mutually distinct slopes and . By Proposition 2.3 and are finite surgery slopes. Let us write and ; we may assume . Then . Recall that has a Seifert fibration with base orbifold .
Assume first that is cyclic, i.e. . Then since is finite cyclic, we have . A simple computation shows that , which is impossible, because .
Suppose next that is finite, but non-cyclic. Then (where is an odd integer), or .
Subcase 1. Assume that . Then is a torus knot and . Since , and are homeomorphic . Thus has a base orbifold , where . By the assumption , and hence , which is an integer. This means or , a contradiction.
Subcase 2. Assume that . Then and . Similarly, we have . Thus is a multiple of and since and are coprime to , neither nor is divided by . Furthermore, the equalities gives . Write for some integer . Then and is written as . Hence we have a unique solution . This gives , . However, in this case by Proposition 3.2 below.
Subcase 3. Assume that . Then we have two possibilities: and , or and . In the former case, , a contradiction. In the latter case, , a contradiction.
Subcase 4. Assume that . Then we have three possibilities: and , and , or and . In either case cannot be an integer and we have a contradiction.
This completes a proof of Theorem 1.3. ∎
Let and be slope elements in . Assume that in . By the assumption , which coincides with . Then the peripheral Magnus property (Theorem 1.3) says or . Hence if and are conjugate in , then or .
Let be the trefoil knot . In the knot group ,
More precisely, and .
Figure 3.1 shows that is a Seifert fibered manifold with Seifert invariant .
Let us choose so that . Then has a Seifert invariant:
Similarly if we choose , then , and has a Seifert invariant:
This shows that is orientation reversingly homeomorphic to .
To obtain a presentation of their fundamental groups, we fix a section for the circle bundle arising from the top left picture of Figure 3.1, where is a fibered tubular neighborhood. Note that is the surgery dual to and is a regular fiber.
Then, with this section has a presentation:
where is represented by a meridian of , a boundary of the section, and is represented by a regular fiber.
Using the same bases , has a presentation:
Note that the element is central and generates a cyclic normal subgroup in both and . Let us consider the quotient groups and , which have the same presentation:
This group is the tetrahedral group (spherical triangle group ) of order . Hence has order in both and .
The slope element with slope does not belong to .
We first observe that the slope element with slope is expressed as ; see Figure 3.1. In , the above presentation shows that . Hence . Assume for a contradiction that . Then in . Hence would be divided by . This is impossible. ∎
Similarly we have:
The slope element with slope does not belong to .
We first observe that the slope element with slope is expressed as ; see Figure 3.1. In , the above presentation shows that . Hence . Assume for a contradiction that . Then in , and we have a contradiction. ∎
The peripheral Magnus property (Theorem 1.3) asserts that if and only if . Are there any distinct slopes and for which there is an automorphism such that ? For instance, if is amphicheiral, i.e. admits an orientation revering homeomorphism , then induces an automorphism such that .
Obviously an existence of an automorphism with implies . We will prove that the converse is not true, in general. To prove this, we begin by observing the following.
Let be a prime knot and an automorphism of . Then for any slope , or . Furthermore, if is not amphicheiral, then .
When is a prime knot, [35, Corollary 4.2] shows that any automorphism of is induced by a homeomorphism of . Let be a standard meridian-longitude pair of . Since () by  and () for homological reasons, . This implies that or . If is non-amphicheiral, then the homeomorphism preserves the orientation of , and hence . Thus . ∎
There is a knot in with a pair of distinct slopes and such that , yet no automorphism of sends to , in particular .
Recall that for the torus knot , is (orientation reversingly) homeomorphic to . Hence, we have:
Our example of a knot in Theorem 3.6 is the torus knot . It seems to be interesting to find a similar example for hyperbolic knots since for hyperbolic knots the outer automorphism group of is equal to the isometry group of .
Let be distinct slopes on a nontrivial knot in . The cosmetic surgery conjecture asserts that and are distinct –manifolds as oriented manifolds. Although this conjecture is still open, Ni and Wu  showed that if and are orientation preservingly homeomorphic, then . On the other hand, the peripheral Magnus property (Theorem 1.3) says that , in particular, . Furthermore, if is non-amphicheiral, then as shown in Proposition 3.5, there is no automorphism sending to .
Does the distinctness imply non-existence of orientation preserving homeomorphism between and ?
4. Finitely generated normal closures of slopes
In this section we prove Theorem 1.4. The next proposition gives a classification of finitely generated, normal subgroups of knot groups of infinite index.
Let be a finitely generated, normal subgroup of of infinite index. Then either
is Seifert fibered (i.e. is a torus knot) and is a subgroup of Seifert fiber subgroup, the subgroup generated by a regular fiber of a Seifert fibration (i.e. is a subgroup of the center of the torus knot group ) or,
fibers over with surface fiber and is a subgroup of of finite index.
This essentially follows from the classification of finitely generated, normal subgroups of –manifold groups of infinite index , [1, p.118 (L9)]. Suppose for a contradiction that is neither (1) nor (2). Then it follows from , [1, p.118 (L9)] that is the union of two twisted -bundle over a compact connected (possibly non-orientable) surface and is a subgroup of of finite index. Then is written as an extension
However, this would imply has an epimorphism to , the abelianization of , which is impossible. ∎
Proof of Theorem 1.4.
Let us assume is a finite surgery slope of . Then is finite. Hence is a subgroup of the finitely generated group of finite index, so it is finitely generated [25, Corollary 2.7.1].
If is a torus knot and is a cabling slope , then is represented by a regular fiber in the Seifert fiber space , and is the infinite cyclic normal subgroup generated by . This means that .
To prove the converse, we suppose that is finitely generated. We divide into two cases depending upon has finite index in or not. If it has finite index, then is a finite group, and hence is a finite surgery slope.
If has infinite index in , then we have two possibilities described in Proposition 4.1. If we have the case (1) in Proposition 4.1, then is a torus knot and . Now suppose for a contradiction that the case (2) in Proposition 4.1 occurs. Then is a subgroup of the normal subgroup of finite index, in particular, lies in the Seifert surface subgroup . This implies is a preferred longitude, i.e. . Since is a normal subgroup of , it is also normal in . Hence is a normal subgroup of of finite index, and hence would be a finite group, where is a closed orientable surface of genus obtained by capping off along . This is a contradiction. ∎
5. Finite family of normal closures of slopes and their intersection
The goal of this section is to establish Theorem 1.6.
For an element , we denote its centralizer by . We call a central element of if , and denote the center of , the normal subgroup consisting of all the central elements, by .
Let be a slope of . Then, throughout this section, we use to denote also a slope element representing . So means . (Note that also represents and .)
Recall that happens for some slope if and only if is a torus knot and ; see  and [1, Theorem 2.5.2]. If and , then given non-central element , obviously for any . Except this very restricted situation, we have:
Let be a nontrivial knot and a nontrivial slope of . If is a torus knot , we assume . Then for every non-central element , we can take an element so that .
If , then take to obtain the desired conclusion. So in the following we assume . By a structure theorem of the centralizer of 3-manifold groups [1, Theorem 2.5.1], either is an abelian group of rank at most two, or is a conjugate to a subgroup of a Seifert fibered piece of with respect to the torus decomposition of [19, 20].
First assume that that is an abelian group of rank at most two. (In fact, , because .) Assume, to the contrary that for all . Then the normal closure of in is a nontrivial normal subgroup of contained in . Thus is finitely generated. If has finite index in then is virtually abelian, which cannot happen for non-trivial knot groups since the knot group contains a free group of rank , the fundamental group of minimum genus Seifert surface. So is an abelian normal subgroup of of infinite index. By Proposition 4.1, either is a torus knot and is a subgroup of the center of , or fibers over with torus fiber. In the former case, is a central element, contradicting the choice of . In the latter case so this cannot happen, either.
Next we assume that is not an abelian group of rank at most two. Then is a torus knot, or a satellite knot which has a Seifert fibered piece with respect to its torus decomposition. Assume first that