Annulus twist and diffeomorphic 4-manifolds II
aa r X i v : . [ m a t h . G T ] A ug ANNULUS TWIST AND DIFFEOMORPHIC 4-MANIFOLDS II
TETSUYA ABE AND IN DAE JONG
Abstract.
We solve a strong version of Problem 3.6 (D) in Kirby’s list, that is, we showthat for any integer n , there exist infinitely many mutually distinct knots such that 2-handleadditions along them with framing n yield the same 4-manifold. Introduction
For a knot K in the 3-sphere S = ∂B , we denote by M K ( n ) the 3-manifold obtainedfrom S by n -surgery on K , and by X K ( n ) the smooth 4-manifold obtained from B byattaching a 2-handle along K with framing n . The symbol ≈ stands for a diffeomorphism.In [1], the authors, Omae, and Takeuchi asked the following problem, a strong version ofProblem 3.6 (D) in Kirby’s list [9] (see also Problem 2 in [10]). Problem 1.1.
Let γ be an integer. Find infinitely many mutually distinct knots K , K , . . . such that X K i ( γ ) ≈ X K j ( γ ) for each i, j ∈ N . In [3, 4], Akbulut gave a partial answer to Problem 1.1 by finding a pair of distinct knots K and K ′ such that X K ( γ ) ≈ X K ′ ( γ ) for each γ ∈ Z . Using an annulus twist introduced byOsoinach [12], Problem 1.1 was solved affirmatively for γ = 0 , ± Theorem 1.2.
For every n ∈ Z , there exist distinct knots J , J , J , . . . such that X J ( n ) ≈ X J ( n ) ≈ X J ( n ) ≈ · · · . The knots J and J in Theorem 1.2 (for n >
0) are depicted in Figure 1. In the figure,the rectangle labelled n stands for n times right-handed full twists. Note that J is the knot8 in Rolfsen’s table [14].This paper is organized as follows: In Section 2, we recall the definition of an annuluspresentation of a knot and introduce the notion of a “simple” annulus presentation. We definea new operation ( ∗ n ) on an annulus presentation, which is a generalization of an annulustwist. For a knot K with an annulus presentation and an integer n , we construct a knot K ′ (with an annulus presentation) such that M K ( n ) ≈ M K ′ ( n ) by using the operation ( ∗ n )(Theorem 2.7). In Section 3, for a knot K with a simple annulus presentation and any integer n , we construct a knot K ′ (with a simple annulus presentation) such that X K ( n ) ≈ X K ′ ( n )by using the operation ( ∗ n ) (Theorem 3.2). Note that the two knots K and K ′ are possibly Mathematics Subject Classification.
Key words and phrases.
Kirby calculus; Annulus twist; Dehn surgery; 2-handle addition. J J n Figure 1.
The knots J and J such that X J ( n ) ≈ X J ( n ).the same. In Section 4, we introduce the notion of a “good” annulus presentation, andshow that, for a given knot with a good annulus presentation, the infinitely many knotsconstructed by using the operation ( ∗ n ) have mutually distinct Alexander polynomials when n = 0 (Theorem 4.2). This yields Theorem 1.2 as an immediate corollary. In Appendix A,we give a potential application of Theorem 2.7 to the cabling conjecture. Acknowledgments.
The authors would like to express their gratitude to Yuichi Yamadaand other participants of handle seminar organized by Motoo Tange. This paper would notbe produced without Yamada’s interest to annulus twists. The first author was supportedby JSPS KAKENHI Grant Number 25005998.2.
Construction of knots
Annulus presentation.
We recall the definition of an annulus presentation of a knot.Let A ⊂ R ∪ {∞} ⊂ S be a trivially embedded annulus with an ε -framed unknot c in S asshown in the left side of Figure 2, where ε = ±
1. Take an embedding of a band b : I × I → S such that • b ( I × I ) ∩ ∂A = b ( ∂I × I ), • b ( I × I ) ∩ int A consists of ribbon singularities, and • b ( I × I ) ∩ c = ∅ ,where I = [0 , A ∪ b ( I × I ) is orientable. Thisassumption implies that the induced framing is zero (see [1]). Unless otherwise stated,we also assume for simplicity that ε = −
1. If a knot K in S is isotopic to the knot( ∂A \ b ( ∂I × I )) ∪ b ( I × ∂I ) in M c ( − ≈ S , then we say that K admits an annuluspresentation ( A, b, c ). It is easy to see that a knot admitting an annulus presentation isobtained from the Hopf link by a single band surgery (see [1]). A typical example of a knotadmitting an annulus presentation is given in Figure 2.For an annulus presentation (
A, b, c ), ( R ∪ {∞} ) \ int A consists of two disks D and D ′ ,see Figure 3. Assume that ∞ ∈ D ′ . Definition 2.1.
An annulus presentation (
A, b, c ) is called simple if b ( I × I ) ∩ int D = ∅ . In [1], it was called a band presentation . NNULUS TWIST AND DIFFEOMORPHIC 4-MANIFOLDS II 3
A ε c b ( I × I ) b ( ∂I × I ) − c Figure 2.
The knot depicted in the center admits an annulus presentationas in the right side.For example, in Figure 3, the annulus presentation depicted in the center is simple, andthe right one is not. DA − − Figure 3.
The position of D , a simple annulus presentation and a non-simpleannulus presentation.Let ( A, b, c ) be an annulus presentation of a knot. In a situation where it is inessentialhow the band b ( I × I ) is embedded, we often indicate ( A, b, c ) in an abbreviated form as inFigure 4. A − c Figure 4.
Thick arcs stand for b ( ∂I × I ). TETSUYA ABE AND IN DAE JONG
Operations.
To construct knots yielding the same 4-manifold by a 2-handle attaching,we define operations on an annulus presentation.
Definition 2.2.
Let (
A, b, c ) be an annulus presentation, and n an integer. • The operation ( A ) is to apply an annulus twist along the annulus A . • The operation ( T n ) is defined as follows:(1) Adding the ( − /n )-framed unknot as in Figure 5, and(2) (after isotopy) blowing down along the ( − /n )-framed unknot. • The operation ( ∗ n ) is the composition of ( A ) and ( T n ).In the operation ( T n ), the added ( − /n )-framed unknot is lying on the neighborhood of c and ∂A , and does not intersect b ( I × I ). The intersection of A and the added unknot is justone point.The operation ( ∗ n ) is a generalization of an annulus twist, in particular, ( ∗
0) = ( A ). − − − nA Ac c Figure 5.
Add the ( − /n )-framed unknot in the operation ( T n ).2.3. Construction.
For a given knot K with an annulus presentation, we can obtain anew knot K ′ with a new annulus presentation by applying the operation ( ∗ n ). By abuse ofnotation, we call K ′ the knot obtained from K by the operation ( ∗ n ). Here we give examples. Example 2.3.
Let J be the knot with the simple annulus presentation as in Figure 6. Let J be the knot obtained from J by the operation ( ∗ n ). Then J is as in Figure 6. Remark . Let K be a knot with an annulus presentation ( A, b, c ), and K ′ the knot obtainedfrom K by ( ∗ n ). If ( A, b, c ) is simple, then the resulting annulus presentation of K ′ is alsosimple. Example 2.5.
For the knot J in Example 2.3 with n = 1, let J be the knot obtained from J by applying the operation ( ∗ J is as in Figure 7.The following lemma is obvious, however, important in our argument. For the definition of an annulus twist, see [2, Section 2].
NNULUS TWIST AND DIFFEOMORPHIC 4-MANIFOLDS II 5 − − − n n ( A )blowdownblowdown ( T n )( ∗ n ) J J − n Figure 6.
By the operation ( ∗ n ), the knot J with the annulus presentationis deformed into the knot J with the annulus presentation. Lemma 2.6.
Let L be a -component framed link which consists of L with framing ( − /n ) and L with framing as in the left side of Figure 8. Suppose that the linking number of L and L is ± (with some orientation). Then two Kirby diagrams in Figure 8 represent thesame -manifold. Theorem 2.7.
Let K be a knot with an annulus presentation and K ′ be the knot obtainfrom K by the operation ( ∗ n ) . Then M K ( n ) ≈ M K ′ ( n ) . Proof.
First, we consider the case where K = J = 8 with the usual annulus presentationas in Figure 6. Figure 9 shows that M K ( n ) is represented by the last diagram in Figure 9,and this is diffeomorphic to M K ′ ( n ) by Figure 10. The moves in Figure 10 correspond to theoperation ( ∗ n ).Next we consider a general case. Let ( A, b, c ) be an annulus presentation of K . As seenin Figure 11, M K ( n ) is represented by the last diagram in Figure 11. Now it is not difficultto see that this is diffeomorphic to M K ′ ( n ). (cid:3) Remark . Let K be a knot with an annulus presentation ( A, b, c ) and K ′ be the knotobtain from K by the operation ( ∗ n ). In general, K ′ is much complicated than K . If theannulus presentation ( A, b, c ) is simple, then K ′ is not too complicated. Indeed, let ( A, b A , c )be the annulus presentation obtained from ( A, b, c ) by applying the operation ( A ) as in theleft side of Figure 12. Then the knot K ′ is indicated as in the right side of Figure 12. TETSUYA ABE AND IN DAE JONG − −
11 1= J J Figure 7.
An annulus presentation of the knot J (lower half) obtained from J by applying ( ∗
1) two times. nL L · · · · · · − n n Figure 8.
Two Kirby diagrams represent the same 3-manifold.3.
Extension of a diffeomorphism between 3-manifolds
In his seminal work, Cerf [7] proved that Γ = 0, that is, any orientation preserving selfdiffeomorphism of S extends to a self diffeomorphism of B . As an application of Γ = 0,Akbulut obtained the following lemma. Lemma 3.1 ([4]) . Let K and K ′ be knots in S = ∂D with a diffeomorphism g : ∂X K ( n ) → ∂X K ′ ( n ) , and let µ be a meridian of K . Suppose that(1) if µ is -framed, then g ( µ ) is the -framed unknot in the Kirby diagram representing X K ′ ( n ) , and NNULUS TWIST AND DIFFEOMORPHIC 4-MANIFOLDS II 7 − − − − − − − n − n − n − n − − n − − n − n − − − − − isotopy slide Figure 9.
A proof of M K ( n ) ≈ M K ′ ( n ) when K = 8 .0 0 n − n − n − − − − n blowdown Figure 10.
Moves which correspond to the operation ( ∗ n ). (2) the Kirby diagram X K ′ ( n ) ∪ h represents D , where h is the -handle representedby (dotted) g ( µ ) .Then g extends to a diffeomorphism e g : X K ( n ) → X K ′ ( n ) such that ˜ g | ∂X K ( n ) = g . This technique is called “carving” in [5]. For a proof, we refer the reader to [1, Lemma2.9]. Applying Lemma 3.1, we show the following.
Theorem 3.2.
Let K be a knot with a simple annulus presentation and K ′ be the knot obtainfrom K by the operation ( ∗ n ) . Then X K ( n ) ≈ X K ′ ( n ) .Proof. First, we consider the case where K = 8 with the usual simple annulus presentation.Let f : ∂X K ( n ) → ∂X K ′ ( n ) be the diffeomorphism given in Figures 9 and 10. Let µ be themeridian of K . If we suppose that µ is 0-framed, then we can check that f ( µ ) is the 0-framedunknot in the Kirby diagram of X K ′ (0) as in Figure 13. Let W be the 4-manifold D ∪ h ∪ h ,where h is the dotted 1-handle represented by f ( µ ) and h is the 2-handle represented by K ′ with framing n . Sliding h over h , we obtain a canceling pair (see Figure 14), implyingthat W ≈ B . By Lemma 3.1, we have ˜ f : X K (0) ≈ X K ′ (0).Next, We consider a general case. Let g : ∂X K ( n ) → ∂X K ′ ( n ) be the diffeomorphismgiven in the proof of Theorem 2.7 in a general case (see Figure 15), and µ the meridian of TETSUYA ABE AND IN DAE JONG n − − − − − − − − n − n − n − − n − − n − n −
11 1 − − − − Figure 11.
A proof of M K ( n ) ≈ M K ′ ( n ) for a general case. b ( I × I ) c − − n Figure 12.
The annulus presentation (
A, b A , c ) and the knot K ′ . ∂X K ( n ). In Figure 15, the annulus presentation in the right hand side represents K ′ , seeRemark 2.8. If we suppose that µ is 0-framed, then we can check that g ( µ ) is the 0-framedunknot in the Kirby diagram of X K ′ (0) as in Figure 15. Let W be the 4-manifold D ∪ h ∪ h ,where h is the dotted 1-handle represented by g ( µ ) and h is the 2-handle represented by K ′ with framing n . Sliding h over h , we obtain a canceling pair (see Figure 16), implyingthat W ≈ B . By Lemma 3.1 again, we have ˜ g : X K (0) ≈ X K ′ (0). (cid:3) Remark . It is important which knot admits a simple annulus presentation. An answeris a knot with unknotting number one (see [1, Lemma 2.2]).
NNULUS TWIST AND DIFFEOMORPHIC 4-MANIFOLDS II 9 µ n f ( µ ) ≈ f Figure 13.
The image of µ under f .1 n n n − n ≈ B Figure 14.
The 4-manifold W is diffeomorphic to B .0 µ n g ( µ )1 n n ≈ Figure 15.
The image of µ under g .4. Proof of Theorem 1.2
For a knot K , we denote by ∆ K ( t ) the Alexander polynomial of K . We assume that ∆ K ( t )is of the symmetric form ∆ K ( t ) = a + d X i =1 a i ( t i + t − i ) . We call the integer d the degree of ∆ K ( t ), and denote it by deg ∆ K ( t ).In this section, we define a “good” annulus presentation. Theorem 1.2 will be shownas a typical case of the argument in this section. The following technical lemma plays animportant role. n n n n − ≈ B Figure 16.
The 4-manifold W is diffeomorphic to B . Lemma 4.1.
Let n be a positive integer. Let K be a knot with a good annulus presentation,and K ′ be the knot obtained from K by applying the operation ( ∗ n ) . Then (i) K ′ also admits a good annulus presentation, and (ii) deg ∆ K ( t ) < deg ∆ K ′ ( t ) . We will prove Lemma 4.1 later. Using Lemma 4.1, we show the following which yieldsTheorem 1.2 as an immediate corollary.
Theorem 4.2.
Let n be a positive integer. Let K be a knot with a good annulus presentationand K i ( i ≥ the knot obtained from K i − by applying the operation ( ∗ n ) . Then(1) X K ( n ) ≈ X K ( n ) ≈ X K ( n ) ≈ · · · , and(2) the knots K , K , K , · · · are mutually distinct.Let K i be the mirror image of K i . Then(3) X K ( − n ) ≈ X K ( − n ) ≈ X K ( − n ) ≈ · · · , and(4) the knots K , K , K , · · · are mutually distinct.Proof. By the definition (Definition 4.3), any good annulus presentation is simple. Thus, byTheorem 3.2, we have X K ( n ) ≈ X K ( n ) ≈ X K ( n ) ≈ · · · . By Lemma 4.1 (i), each K i ( i ≥
1) also admits a good annulus presentation. Thus, byLemma 4.1 (ii), we havedeg ∆ K ( t ) < deg ∆ K ( t ) < deg ∆ K ( t ) < · · · . This implies that the knots K , K , K , · · · are mutually distinct.Since X K i ( n ) ≈ X K i ( − n ) and deg ∆ K i ( t ) = deg ∆ K i ( t ), we have X K ( − n ) ≈ X K ( − n ) ≈ X K ( − n ) ≈ · · · , anddeg ∆ K ( t ) < deg ∆ K ( t ) < deg ∆ K ( t ) < · · · . This completes the proof of Theorem 4.2. (cid:3)
NNULUS TWIST AND DIFFEOMORPHIC 4-MANIFOLDS II 11
Good annulus presentation and the Alexander polynomial.
Let K be a knotwith a simple annulus presentation ( A, b, c ). Note that the knot ( ∂A \ b ( ∂I × I )) ∪ b ( I × ∂I )is trivial in S if we ignore the ( − c . We denote by U this trivial knot. Since( A, b, c ) is simple, U ∪ c can be isotoped so that U bounds a “flat” disk D (contained in R ∪ {∞} ). This isotopy, denote by ϕ b , is realized by shrinking the band b ( I × I ). Forexample, see Figure 17. In the abbreviated form, ϕ b is represented as in Figure 18. Herewe note that the linking number of U and c is zero since we assumed that A ∪ b ( I × I ) isorientable (see subsection 2.1). Let Σ be the disk bounded by c as in Figure 18. We assumethat Σ stays during the isotopy ϕ b . cU c U isotopy ϕ b Figure 17.
By the isotopy ϕ b (shrinking the band b ( I × I )), U ∪ c (the leftside) is changed to the right side. c c U isotopy ϕ b • • p ∗ p ′∗ Σ Σ
Figure 18.
The isotopy ϕ b in the abbreviated form of ( A, b, c ). Assume thatΣ stays during the isotopy.After the isotopy ϕ b , cutting along the disk D , the loop c is separated into arcs whoseendpoints are in D . Furthermore, choosing orientations on c and U , these arcs are oriented.Unless otherwise noted, we choose the orientations of c and U as in Figure 18. These orientedarcs are classified into four types as follows: For p ∈ c ∩ D , let sign( p ) = ± according to thesign of the intersection between D and c at p . For an oriented arc α , let p s (resp. p t ) be thestarting point (resp. terminal point) of α . Then we say that α is of type (sign( p s ) sign( p t )).That is, the oriented arc α is of type (++), ( −− ), (+ − ), or ( − +). For example, see Figure 19. K is the knot ( ∂A \ b ( ∂ × I )) ∪ b ( I × ∂I ) in M c ( − − + − + (+ − ) arc( −− ) arc( − +) arc(++) arc Figure 19.
The four types of arcs.Here we consider the infinite cyclic covering ˜ E ( U ) of E ( U ). Notice that ˜ E ( U ) consists ofinfinitely many copies of a cylinder obtained from E ( U ) by cutting along D . Thus ˜ E ( U ) isdiffeomorphic to D × R ≈ ∪ i ∈ Z ( D × [ i, i + 1]). Each oriented arc is lifted in ˜ E ( U ) as shownin Figure 20. Hereafter, for simplicity, we say an arc instead of an oriented arc. D × { i } D × { i + 1 } Figure 20.
Lifts of oriented arcs of type (++) , ( −− ) , (+ − ), and ( − +) respectively. Figure 21.
Lifts of the arcs of type (+ − ) and ( − +) from a good annulus presentation. Definition 4.3.
We say that a simple annulus presentation (
A, b, c ) is good if the set of arcs A obtained as above satisfies the following up to isotopy.(1) A contains just one (+ − ) arc and one ( − +) arc, and they are lifted as in Figure 21.(2) For α ∈ A , if Σ ∩ α = ∅ , then α is of type (++) (resp. ( −− ) arc) and the sign ofeach intersection point in Σ ∩ α is + (resp. − ).(3) b ( I × ∂I ) ∩ int A = ∅ . NNULUS TWIST AND DIFFEOMORPHIC 4-MANIFOLDS II 13
Remark . For a simple annulus presentation (
A, b, c ), after the isotopy ϕ b , the intersection c ∩ D corresponds to the intersection b ( I × ∂I ) ∩ int A and further two points p ∗ and p ′∗ depictedin Figure 18. Notice that b ( I × ∂I ) ∩ int A = ⊔ i b ( { t i } × ∂I )for some 0 < t < · · · < t r <
1. For each i , b ( { t i } × ∂I ) consists of two points whose signsare differ. Furthermore, with the orientation as in Figure 18, we havesign( p ∗ ) = − and sign( p ′∗ ) = + . Example 4.5.
The annulus presentation obtained by applying ϕ b on Figure 19 is not goodsince the condition (2) does not hold. In such a case, changing the position of an intersectionas in Figure 22 by an isotopy, we can obtain a good annulus presentation. We often applysuch an argument in the proof of Lemma 4.1. − + − + (+ − ) arc( −− ) arc( − +) arc(++) arc − + − +(+ − ) arc ( −− ) arc( − +) arc(++) arc Figure 22.
By an isotopy, we move the intersection point of c ∩ D .Considering a surgery description of the infinite cyclic covering of the exterior of K , wecan easily show the following. Lemma 4.6.
If a knot K admits a good annulus presentation, then deg ∆ K ( t ) = { arcs of type (++) } + 1 . (4.1)For the details of a surgery description of ˜ E ( K ) and the Alexander polynomial, we referthe reader to Rolfsen’s book [14, Chapter 7]. Remark . To show Lemma 4.6, we do not need the conditions (2) and (3) in Defitnition 4.3.These conditions are used to prove Lemma 4.1.
Remark . If a knot K admits a good annulus presentation, then we can see that ∆ K ( t )is monic.Now we are ready to prove the main result in this paper. Proof of Theorem 1.2.
The case where n = 0 was proved in [1]. We can check that the simpleannulus presentation of the knot 8 in Figure 2 is good. Thus the proof for the case where n = 0 is obtained by Theorem 4.2 immediately. (cid:3) Proof of Lemma 4.1.
We start the proof of Lemma 4.1. Let (
A, b, c ) be a goodannulus presentation of a knot K . Recall that the operation ( ∗ n ) is a composition of thetwo operations ( A ) and ( T n ) for an annulus presentation. Let ( A, b A , c ) be the annuluspresentation obtained from ( A, b, c ) by applying the operation ( A ), and ( A, b ′ , c ) the annuluspresentation obtained from ( A, b A , c ) by applying the operation ( T n ). That is,( A, b, c ) ( A ) −→ ( A, b A , c ) ( T n ) −→ ( A, b ′ , c ) . Note that K ′ admits the annulus presentation ( A, b ′ , c ).First we show that ( A, b A , c ) is good. The operation ( A ) preserves the number of arcsand type of each arc. Furthermore we can suppose that the (+ − ) arc and ( − +) arc arefixed by the operation ( A ) up to isotopy. Therefore ( A, b A , c ) satisfies the condition (1)of Definition 4.3. We can also check that ( A, b A , c ) satisfies the conditions (2) and (3) ofDefinition 4.3. Therefore ( A, b A , c ) is good.Next we show that ( A, b ′ , c ) is good. The operation ( T n ) may increase the number of arcs.Indeed a (++) (resp. ( −− )) arc through Σ is changed to n + 1 (++) (resp. ( −− )) arcssince ( A, b A , c ) is good, in particular, a (++) arc (resp. ( −− ) arc) intersects Σ positively(resp. negatively). Note that the (+ − ) arc and the ( − +) arc are fixed by the operation( T n ). Hence (+ − ) arcs and ( − +) arcs do not produced by the operation ( T n ). Therefore( A, b ′ , c ) satisfies the condition (1). We can also check that ( A, b ′ , c ) satisfies the conditions(2) and (3). Therefore ( A, b ′ , c ) (of K ′ ) is good. This completes the proof of the claim (i) ofLemma 4.1.Let δ = A ∩ b ( I × ∂I )) / σ = ∩ b ( I × ∂I )) /
2. Then we see that A ∩ b A ( I × ∂I )) / δ , ∩ b A ( I × ∂I )) / σ + δ . Then we have A ∩ b ′ ( I × ∂I )) / A ∩ b A ( I × ∂I )) / n · ∩ b A ( I × ∂I )) /
2= ( n + 1) δ + nσ , and ∩ b ′ ( I × ∂I )) = ∩ b A ( I × ∂I )) . These are equivalent to (cid:18) δ ′ σ ′ (cid:19) = (cid:18) n + 1 n (cid:19) (cid:18) δσ (cid:19) . Since n ≥ δ ≥
1, we have δ < δ ′ . (4.2)By the condition that ( A, b, c ) and (
A, b ′ , c ) is good, and by Remark 4.4, we see that δ = { (++) arcs of ( A, b, c ) } , δ ′ = { (++) arcs of ( A, b ′ , c ) } . Therefore, by Lemma 4.6, we havedeg ∆ K = δ + 1 , deg ∆ K ′ = δ ′ + 1 . (4.3) NNULUS TWIST AND DIFFEOMORPHIC 4-MANIFOLDS II 15
By (4.2) and (4.3), we have deg ∆ K ( t ) < deg ∆ K ′ ( t ). This completes the proof of the claim(ii) of Lemma 4.1, and thus, the proof of Lemma 4.1. Appendix A. Potential application
We introduce a potential application of our technique to the cabling conjecture.
Conjecture A.1 (cabling conjecture, [8]) . Let K be a knot in S . If M K ( γ ) is a reduciblemanifold for some integer γ , then K is a ( p, q ) -cable of a knot and γ = pq . It is known that this conjecture is true for many knots, in particular, torus knots and satel-lite knots. Therefore if any hyperbolic knot admits no reducible surgery, then Conjecture A.1is true. For details, we refer the reader to [11], [6]. Here we explain a potential applicationof Theorem 2.7 to Conjecture A.1. Suppose that K is a ( p, q )-cable of a knot admitting anannulus presentation, and K ′ the knot obtain from K by the operation ( ∗ pq ). Then M K ′ ( pq )is a reducible manifold since M K ( pq ) is reducible and M K ( pq ) ≈ M K ′ ( pq ) by Theorem 2.7.If K ′ is a hyperbolic knot, then it is a counterexample to the cabling conjecture. Thereforeit is interesting to determine whether a ( p, q )-cable of a knot admits an annulus presentationor not.Recall that a ( p, q )-torus knot is a ( p, q )-cable of the trivial knot. Lemma A.2.
A torus knot admits an annulus presentation with ε = − (resp. ε = 1 ) ifand only if it is the unknot or the negative (reps. positive) trefoil knot .Proof. We only show the case where ε = − ε = 1 isachieved in a similar way. Let T be a torus knot which admits an annulus presentation and H the negative Hopf link. Then T and H are related by a single band surgery. Therefore | σ ( T ) − σ ( H ) | ≤ . That is, 0 ≤ σ ( T ) ≤ . This implies that T is the unknot or the negative trefoil knot. On the other hand, the unknotand the negative trefoil knot have annulus presentations, see Figure 23. (cid:3) − − Figure 23.
Annulus presentations of the unknot and the negative trefoil knot.Let K be the unknot (resp. the negative trefoil knot). If M K ( γ ) ≈ M K ′ ( γ ) for some knot K ′ and an integer γ , then K ′ is the unknot (resp. the trefoil knot), see [13]. Therefore, by using an annulus presentation of the trivial knot, we can not obtain a counterexample ofthe cabling conjecture by using Theorem 2.7 unfortunately. Then we propose the followingquestion. Question A.3.
Let K be a ( p, q ) -cable knot of a non-trivial knot. Then does K admit anyannulus presentation?Remark A.4 . If the 4-ball genus of a knot K is greater than one, then K does not admit anyannulus presentations. Therefore, for example, the (2,1)-cable of the trefoil knot does notadmit any annulus presentations. On the other hand, it is not known whether the (2 , References [1] T. Abe, I. D. Jong, Y. Omae, and M. Takeuchi,
Annulus twist and diffeomorphic 4-manifolds , Math.Proc. Cambridge Philos. Soc. (2013), 219–235.[2] T. Abe and M. Tange,
A construction of slice knots via annulus twists ,arXiv:1305.7492v2 [math.GT], preprint.[3] S. Akbulut,
Knots and exotic smooth structures on -manifolds , J. Knot Theory Ramifications (1993),no. 1, 1–10.[4] S. Akbulut, On -dimensional homology classes of -manifolds , Math. Proc. Cambridge Philos. Soc. (1977), no. 1, 99–106.[5] S. Akbulut, 4 -manifolds , draft of a book (2012),available at [6] S. Boyer, Dehn surgery on knots , Chapter 4 of the Handbook of Geometric Topology, R.J. Daverman,R.B. Sher, ed., Amsterd am, Elsevier, 2002.[7] J. Cerf,
Sur les diffeomorphismes de la sphere de dimension trois (Γ = 0), Lecture Notes in Mathe-matics, No. 53, Springer-Verlag, Berlin-New York (1968) xii+133 pp.[8] F. Gonz´alez-Acu˜na and H. Short, Knot surgery and primeness , Math. Proc. Cambridge Philos. Soc. (1986), no. 1, 89–102.[9] R. Kirby, Problems in Low-Dimensional Topology , AMS/IP Stud. Adv. Math., (2), Geometric topol-ogy (Athens, GA, 1993), 35–473, Amer. Math. Soc., Providence, RI, 1997.[10] J. Luecke and J. Osoinach, Infinitely many knots admitting the same integer surgery , arXiv:1407.1529.[11] T. Mrowka and P. Ozsvath,
Low Dimensional Topology .[12] J. Osoinach
Manifolds obtained by surgery on an infinite number of knots in S , Topology (2006),725–733.[13] P. Ozsvath and Z. Szabo, The Dehn surgery characterization of the trefoil and the figure eight knot ,math.GT/0604079.[14] D. Rolfsen,
Knots and Links , Mathematics Lecture Series, No. 7. Publish or Perish, Inc., Berkeley,Calif., 1976.
Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan
E-mail address : [email protected] Department of Mathematics, Kinki University, 3-4-1 Kowakae, Higashiosaka City, Osaka577-0818, Japan
E-mail address ::