GGLUCK TWIST AND UNKNOTTING OF SATELLITE -KNOTS SEUNGWON KIM
Abstract.
In this paper, we show that the Gluck twist of certain satellite 2-knots in a 4-manifold do not change the diffeomorphism type in three differentways: one is directly from the definition of the satellite 2-knot, and the other twoare by finding an equivalent description of the satellite 2-knot. Furthermore, usingthe new description, we gave infinite number of new examples of 2-knots whichare unknotted by connected summing a single standard real projective plane. Introduction
Let K be an embedded 2-sphere in a 4-manifold X with a product neighborhood.Consider an operation which cuts a neighborhood ν ( K ) of K and glues it back ina different way. By Gluck [2], there are only two ways to glue it back, one is justthe trivial gluing, and the other is called the Gluck twist . Gluck [2] showed that theGluck twist of S is a homotopy 4-sphere, which might give a potential counterex-ample to the smooth 4-dimensional Poincar´e conjecture. (See Kirby’s problem 4.23[8].)Many knotted spheres (i.e., 2-knots) in S are known to be Gluck twist trivial,such as ribbon 2-knots [2, 16], twist spun knots [3, 12], certain union of two ribbondisks [10], twist roll spun knots [11], tube sums of all such 2-knots [4], and moregenerally, 2-knots 0-concordant to all such knots [9, 14].In this paper, we consider the Gluck twist problem of a satellite -knot , which isdefined below: Definition 1.1.
Let P and C be 2-knots embedded in S and a 4-manifold X respectively. Assume C has a product neighborhood in X . Consider a simple loop γ ⊂ S − ν ( P ). Then there exists a diffeomorphism ρ : S − ν ( γ ) → ν ( C ), where ν ( · ) denotes a neighborhood of · in a 4-manifold. Let K = ρ ( P ) ⊂ X . We call K the satellite -knot in X of companion C with pattern ( P, V ). Equivalently,(
X, K ) = (( X − ν ( C )) (cid:91) ∂ρ ( S − ν ( γ )) , P ) , where ∂ρ = ρ (cid:22) ∂ ( S − ν ( γ )) : ∂ ( S − ν ( γ )) → ∂ν ( C ) (cid:39) ∂ ( X − ν ( C ))and P ⊂ S − ν ( γ ) ⊂ ( X − ν ( C )) (cid:91) ∂ρ ( S − ν ( γ )) (cid:39) X. a r X i v : . [ m a t h . G T ] O c t SEUNGWON KIM
We say a satellite 2-knot is degree n if [ P ] = n ∈ H ( S − ν ( γ )) (cid:39) Z . Especially, wecall a satellite 2-knot is a cable -knot if its pattern 2-knot P is the unknot.In [5], Hughes, Miller and the author studied the Gluck twist of a satellite 2-knotand showed that when the pattern is 0-concordant to a tube sum of twist spun knotsand the degree in H ( S − ν ( γ )) (cid:39) H ( S × D ) (cid:39) Z is zero, then the Gluck twistalong the satellite 2-knot is trivial. Furthermore, if the degree is one, then the Glucktwist of the satellite 2-knot is the same as the Gluck twist along its companion.In this paper, we extend the above result to a more general setting: Theorem 1.2.
Let K be a satellite -knot in a -manifold X of companion C withpattern ( P, V ) . Then the following holds: (i) If the degree of K is even, then the Gluck twist of X along K is diffeomorphicto the Gluck twist of X along P ⊂ D ⊂ X . (ii) If the degree of K is odd then the Gluck twist of X along K is diffeomorphicto the Gluck twist of X along C P . Theorem 1.2 has the following immediate corollary:
Theorem 1.3.
Let K be a satellite -knot in a -manifold X of companion C withpattern ( P, V ) . Suppose that the Gluck twist of S along P is trivial. Then thefollowing holds: (i) If the degree of K is even, then the Gluck twist of X along K is diffeomorphicto X . (ii) If the degree of K is odd and the Gluck twist of X along C is trivial, thenthe Gluck twist of X along K is diffeomorphic to X . Hence, for example, the Gluck twist of S along a satellite 2-knot of a twist spun2-knot companion with a twist spun 2-knot pattern is trivial.We give three different proofs. The first proof is directly from the definition ofthe satellite 2-knot, and the other two are by finding an equivalent description of thesatellite 2-knot. Especially, third proof is done by unknotting cable 2-knots using astandard real projective plane.Viro [15] found the first example of a 2-knot in S which can be unknotted byconnected summing a single standard real projective plane. As far as the author isaware, there were no previously known example of non-ribbon 2-knots which can beunknotted by connected summing with a single standard real projective plane.In this paper, We extend the Viro’s result to non-ribbon 2-knots. Theorem 1.4.
There exist an infinite number of non-ribbon -knots which are un-knotted by connected summing a single standard real projective plane. Acknowledgements.
The author would like to thank Hongtaek Jung, MaggieMiller, Jason Joseph and Hannah Schwartz for many helpful conversations aboutthe earlier draft. The author was supported by the Institute for Basic Science (IBS-R003-D1) at the time of this project.
LUCK TWIST AND UNKNOTTING OF SATELLITE 2-KNOTS 3
Figure 1: First figure is a banded unlink diagram of a spun trefoil in S . From thisbanded unlink, we can get the natural handle decomposition of its complement, byputting a dot on each unlink, and changing each band to a 0-framed circle as inthe second figure. If we put a +1-framed circle to a meridian of one of the dottedcircles, then we get a Kirby diagram of the Gluck twist of S along the spun trefoilknot. 2. Proofs of Main Theorem
First proof of Theorem 1.2.
Consider a banded unlink diagram of P . γ can be iso-toped so that it can be seen as the unknot in the banded unlink diagram of P . Notethat the obvious disk bounded by γ intersects the banded unlink diagram in n timeswhere n ≡ [ P ] (mod 2). Without loss of generality, we also can assume that theobvious disk does not intersect the bands.We can think of the Gluck twist in the following way: First, get the natural handledecomposition from a banded unlink diagram of a 2-knot in a 4-manifold. Then,we add the +1-framed circle to a meridian of one of the dotted circles. Then, thisKirby diagram represents the Gluck twist of the given 4-manifold along the 2-knotwith the given banded unlink diagram. See Figure 1 for an example.Also, the Gluck twist along K can be thought as follows: We first do the Glucktwist S − ν ( γ ) along P and glue it to X − ν ( C ) without twist. See the belowequations.( X − ν ( K )) (cid:91) φ ( S × D ) = (cid:16)(cid:0) ( X − ν ( C )) (cid:91) ∂ρ ( S − ν ( γ )) − ν ( P ) (cid:17) (cid:91) φ ( S × D )= (cid:0) ( X − ν ( C )) (cid:91) ∂ρ ( S − ν ( γ ) − ν ( P )) (cid:1) (cid:91) φ ( S × D )= ( X − ν ( C )) (cid:91) ∂ρ (cid:0) ( S − ν ( γ ) − ν ( P )) (cid:91) φ ( S × D ) (cid:1) . Here, φ is the self-diffeomorphism of S × S which gives the Gluck twist. Notethat ∂ρ is the identity map of S × S .Consider an isotopy of γ through the +1-framed circle. This isotopy will link γ and the +1-framed circle. However, we can push γ down along the gradient flow SEUNGWON KIM
Figure 2: We slide the curve γ through the +1-framed circle to get the figure in themiddle from the figure in the left. Then, we push γ below the index-2 critical pointsto sit on the 1-handlebody. Then, we can do small isotopy which does not intersectthe dotted circles. Then, we push γ back to the original level to get the figure inthe right.Figure 3: In the first figure, we slide the blue curve γ along the +1-framed circleto get the second figure. In the second figure, we push γ down to the level belowthe index-2 critical points, do small isotopy in that level, and push it back to theoriginal level to get the third figure. We can always move the +1-framed circle to ameridian of any dotted circles, so we can move it like the fourth figure. Then we dothe same moves to get the last figure.of the Morse function so that γ is sitting inside the 1-handle body, do an isotopyof γ in the 1-handlebody which does not touch the dotted circles, and push it backto the level of the Kirby diagram so that γ is unlinked from +1-framed circle. SeeFigure 2 for the actual moves in a Kirby diagram, and Figure 3 for an example.The +1-framed circle can be moved to any meridian of the dotted circles since it isisotopic to a meridian of P . Therefore, we can keep doing it so that γ is unlinkedfrom the every dotted circle and every attaching circle of the 2-handles. LUCK TWIST AND UNKNOTTING OF SATELLITE 2-KNOTS 5
Figure 4: A banded unlink diagram of a tube.We can figure out the result of the Gluck twist if we can specify the framingof γ . Each time we push γ through the +1-framed circle, framing changes by 1.All the other isotopies such as pushing up and down along the gradient flow, andthe isotopy in the level below the index-2 critical points which does not touch thedotted circles do not change the framing. Hence, the final framing of γ differs fromthe original framing by the number of times that it passes through the +1-framedcircle mod 2, since γ ’s framing is either 0 or 1. This number is same as the parityof the degree of K . Hence, the Gluck twist of X along K is only depend on thedegree of K . Hence, the Gluck twist of X along an even degree satellite 2-knot isdiffeomorphic to the Gluck twist of X along a degree zero satellite 2-knot, which isjust the pattern 2-knot. Also, the Gluck twist of X along an odd degree satellite2-knot is diffeomorphic to the Gluck twits of X along a degree one satellite 2-knot,which is just a connected sum of the companion and the pattern by [6]. This provesour main theorem. (cid:3) Definition 2.1.
Let J and K be disjoint two 2-knots in a 4-manifold X . Considera 3-ball B = D × I embedded in X − ( J ∪ K ) such that J ∩ B = D × { } and K ∩ B = D ×{ } . Consider a 2-knot L = ( J − D ×{ } ) ∪ ( ∂D × I ) ∪ ( K − D ×{ } .We call L a tube sum of J and K and L ∩ B = ∂D × I a tube connecting J and K .We call the procedure to make L from J and K a tubing J and K . See Figure 4 fora banded unlink diagram of a tube. Remark 2.2.
The tube sum is sometimes called the band sum (e.g., [5]), since it isa higher dimensional analogue of the band sum of 1-knots. It is a generalization ofthe connected sum.
Definition 2.3.
Let C be a 2-knot in a 4-manifold X with ν ( C ) (cid:39) S × D . Assume C = S × { } . Parallel copies of C are disjoint union of 2-spheres (cid:83) i S × { p i } ⊂ ν ( C ) ⊂ X (see Figure 5 for a banded unlink diagram in S × D , represented by anexterior of γ ∈ S ). Definition 2.4.
Let (cid:83) i C i , i = 1 , · · · , n be a set of parallel copies of a 2-knot C in a 4-manifold X . Without loss of generality, assume that D is a unit diskin C , C i = S × { p i } , where p i ∈ [ − , ⊂ D , and if i < j then p i < p j . Let δ = { q j }× [ p j , p j +1 ] ⊂ S × [ p j , p j +1 ] be an arc which connects C j and C j +1 . Consideran (cid:15) -neighborhood of δ in X and a tube in that neighborhood connecting C j and C j +1 which its core is δ . We call a tube is untwisted , if a tube is ∂D (cid:15) × [ p j , p j +1 ], SEUNGWON KIM
Figure 5: A banded unlink diagram of parallel copies of a 2-knot C . This figure isdrawn in S × D (cid:39) ν ( C ), represented by an exterior of γ ∈ S .Figure 6: Left: A banded unlink diagram of untiwsted tube. Right: A bandedunlink diagram of half-twisted tube.where D (cid:15) ⊂ S is a small disc neighborhood of q j . Otherwise, we call a tube is half-twisted . See Figure 6 for their banded unlink diagrams. LUCK TWIST AND UNKNOTTING OF SATELLITE 2-KNOTS 7
Figure 7: Equivalent banded unlink diagrams of (
P, V ). From the right most figure,we can stabilize sufficient number of times to get the middle figure, and then isotope γ as in the right most figure. Remark 2.5.
A banded unlink diagram of a tube can be twisted many times as inFigure 4, but we always can cancel full twist by local isotopies of bands. Therefore,there is no ambiguity in the definition of half-twisted.
Lemma 2.6.
Every satellite -knot K is a tube sums of a pattern -knot P ⊂ D ⊂ X and parallel copies of a companion -knot C .Proof. We only need to show that the pattern (
P, V ) is a tube sum of a pattern2-knot P ⊂ D ⊂ V and parallel copies of the core 2-sphere of V . Then theseparallel copies of the core 2-sphere becomes parallel copies of C after identifying V to ν ( C ), and P becomes a 2-knot in D ⊂ X , hence, we proved the lemma. We drawa banded unlink diagram of ( P, V ) as the right most figure of Figure 7. Then we cansee that the surface represented by the right most figure is isotopic to the surfacerepresented by the left most figure of Figure 7, which is a tube sum of P ⊂ D ⊂ V and parallel copies of the core of V . (cid:3) Remark 2.7.
This lemma is true for n -knot with n ≥
1, with suitable generaliza-tions of the notions, such as the tube sum and parallel copies.
Second proof of Theorem 1.2.
In [4, 9], it is shown that if a 2-knot is obtained bytubing a 2-link, then the Gluck twist along a 2-knot is same as Gluck twist alonga 2-link. Hence, by [4, 9] and Lemma 2.6, the Gluck twist of X along K is sameas the Gluck twist of X along a 2-link, which is a split union of parallel copies of C and P ⊂ D ⊂ X . Consider a pair of 2-spheres in parallel copies of C . Wecan tube them with a untwisted tube and isotope resulting 2-sphere to a unknotted2-sphere as in Figure 8. (There is another way to understand this isotopy. Thesetwo 2-spheres cobound a S × I . Tubing these two with untwisted tubes will removea 3-ball D × I from S × I , which makes the resulting 2-sphere bounds a 3-ball, soit is unknotted 2-sphere in a 4-ball.)If [ P ] is even, then we can pair them all to get a split union of unknotted 2-spheres, and if [ P ] is odd, then we can pair all 2-spheres except one to get a split SEUNGWON KIM
Figure 8: We can pair adjacent 2-spheres using a untwisted tube. Then we canisotope γ through the tube, and isotope tubed pair to one unknotted 2-sphere in aball.union of unknotted 2-spheres and C . Then by [4, 9] again, the Gluck twist along aparallel copies of C is is trivial if [ P ] is even, and is same as the Gluck twist along C if [ P ] is odd. Therefore, the Gluck twist of X along K is diffeomorphic to theGluck twist of X along P if the degree of K is even, and the Gluck twist of X along K is diffeomorphic to the Gluck twist of X along the split union of C and P if thedegree of K is odd. By [4, 9] again, the Gluck twist of X along the split union of C and P is diffeomorphic to the Gluck twist of X along C P . (cid:3) Lemma 2.8.
A cable -knot can be obtained from parallel copies of a companion byconnecting adjacent copies with half-twisted tubes.Proof. We only need to show that the pattern can be obtained from parallel copiesof unknotted 2-spheres by connecting adjacent copies with half-twisted tubes. Since π ( S − U ) = Z , where U is a unknot in S , the pattern of the cable is only deter-mined by the [ γ ] ∈ π ( S − U ) = Z . Let [ γ ] = n, n ∈ Z . Consider a banded unlinkdiagram as in Figure 9, which is obtained by taking n -parallel copies of unknot-ted 2-spheres and connecting them with half-twisted tubes. Note that without γ this banded unlink diagram is just a different banded unlink diagram of an unknot-ted 2-sphere U . Then γ links the banded unlink diagram n -times, which impliesthat [ γ ] = n . Hence, the satellite of the pattern represented by this banded unlinkdiagram is just a cable 2-knot with degree n . (cid:3) Theorem 2.9.
Let K be a cable -knot in a -manifold X of companion C withunknotted pattern ( P, V ) . Let R be a standard projective plane in D ⊂ X . Then, LUCK TWIST AND UNKNOTTING OF SATELLITE 2-KNOTS 9
Figure 9: A banded unlink diagram of a unknot pattern with [ γ ] = 3 ∈ Z . if the degree of K is even, then K R is isotopic to R and if the degree is odd, then K R is isotopic to C R .Proof. The effect of connected summing R in K is just adding half-twisted band to abanded unlink diagram of K . We can put this half-twisted band near a half-twistedtube of a cable 2-knot. Then by Figure 10, we always can untwist the tube using R P . Then by the similar argument as in the second proof of Theorem 1.2, eachadjacent pair of 2-spheres connected by a tube is isotoped to an unknotted 2-spherein a small ball in X . Hence, if the degree is even, then K R can be isotoped to R ,and if the degree is odd, then K R is isotoped to C R . (cid:3) Note that if a companion is ribbon 2-knot, then its cable is ribbon. It is naturalto expect that a cable is not ribbon if its companion is not ribbon.
Theorem 2.10.
If a companion is a n -twist spun -knot τ n ( k ) of a -knot k with | n | ≥ , then its cable is not a ribbon.Proof. By Zeeman [17], any τ n ( k ) is fibered by a once-punctured n -fold cyclicbranched covering of S over k .By Kanenobu [6], the degree m cable of τ n ( k ) is fibered by a once-punctured m -fold connect sum of the n -fold cyclic branched cover of S over k .To be a fibered ribbon 2-knot, Cochran [1] showed that its fiber should be aonce punctured q S × S , however, Plotnick [13] showed that a cyclic branchedcoverings of S over a non-trivial knot does not admit S × S summands. Hence,a cable of a τ n ( k ) is not a ribbon 2-knot. (cid:3) Figure 10: Half-twisted tubes can be untwisted by a half-twisted band.
Proof of Theorem 1.4.
Consider τ ( k ) where k is a 2-bridge knot, which its 2-foldbranched covering is a lens space L ( p, q ). Then 2 n -cable of τ ( k ) is fibered by n L ( p, q ) − D . Then 2 n i -cable and 2 n j -cable of τ ( k ) are not isotopic if n i (cid:54) = n j since the infinite cyclic cover of a fibered 2-knot has the homotopy type of the fiber.Since by Theorem 2.9 and 2.10, 2 n -cables of τ ( k ) are non-ribbon 2-knots which areunknotted by connected summing a single standard real projective plane, the set of2 n -cables of τ ( k ) is a desired set of 2-knots. (cid:3) The following theorem is Theorem 0.1 in [7].
Theorem 2.11. [7]
The Gluck twist of a -manifold X along a -knot K is diffeo-morphic to the Price twist of X along a K R where R ⊂ D ⊂ X is a standardprojective plane in D . The following theorem is an immediate consequence of the above theorem.
Corollary 2.12. [7]
If a -knot is unknotted by connected summing a standardprojective plane, then its Gluck twist is trivial.Third Proof of Theorem 1.2. Before going into the proof, we first define some ter-minology.
LUCK TWIST AND UNKNOTTING OF SATELLITE 2-KNOTS 11
Figure 11: Banded unlink diagrams of ( C P , C P ) P, V )(left) and( C P , C P ) U, W )
P, S )(right). Note two parallel 2-spheres connectedby a untwisted tube can be isotoped to a unknotted 2-sphere as in Figure 8 andcancelled. Definition 2.13.
A 2-knot K (cid:48) in X C P is called unit -knot if a pair ( K (cid:48) , X C P )is obtained from a pairwise connected sum of the standard pair ( C P , C P ) and a2-knot pair ( K, X ). Lemma 2.14.
Let K (cid:48) be unit -knot obtained from a satellite -knot K with com-panion C with pattern ( P, V ) . Then K (cid:48) is isotopic to a unit -knot obtained from aconnected sum of a pattern -knot and a cable -knot C with companion C .Proof. Let (
U, W ) be the pattern of C . It is suffices to show that ( C P , C P ) P, V ) (see left figure of Figure 11 for its banded unlink diagram) is isotopic to( C P , C P ) U, W )
P, S ) (see right figure of Figure 11). We need to show thatthese two figures represent isotopic surfaces. If we achieve this isotopy, we can canceltwo 2-spheres which are tubed by an untwisted tube, hence, cobound a 3-ball, sothat each adjacent 2-spheres are tubed by half-twisted tubes. Then by Lemma 2.8,we prove the lemma.To acheive the isotopy, we first do band slides so that only one tube is connecting P to the rest of the surface. During the isotopy, tubes can be tangled with thediagram. Then as in 2, we can untangle tubes from P . After we untangle all suchtubes, we move +1-framed circle to the left, and do further isotopy as before, toarrange every tube as we desired. (cid:3) In [9], Melvin showed that the two 2-knots have the same Gluck twist if unit2-knots obtained from them are isotopic. Then by Lemma 2.14, the Gluck twistof X along K is diffeomorphic to the Gluck twist of the connected sum of C and P ⊂ D ⊂ X . By Theorem 2.11, the Gluck twist of X along C P is diffeomorphicto the Price twist of X along C P R , where R is a standard projective plane in D ⊂ X . Then by Theorem 2.9, if the degree of K is even, K R is isotopic to P R ,and if the degree is odd, then K R is isotopic to C P R . Hence, the Gluck twist v ∪ v , . . . , v k v i v (cid:48) i A v i ∆Figure 12: The figure from [5]. The pair v i and v (cid:48) i in the figure represents the tube.We always can change crossings between bands and tubes/2-handles. Also, if wehave geometric dual sphere (in our case, C P represented by +1-framed circle), thenwe can change crossings between unlinks and tubes.of X along K is diffeomorphic to the Gluck twist of X along P if the degree is even,otherwise, it is diffeomorphic to the Gluck twist of X along C P . (cid:3) References
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