aa r X i v : . [ m a t h . G T ] A ug CONCORDANCE PROPERTIES OF PARALLEL LINKS
DANIEL RUBERMAN AND SAˇSO STRLE
Abstract.
We investigate the concordance properties of ‘parallel links’ P ( K ), given by the (2 ,
0) cable of a knot K . We focus on the question: if P ( K ) is concordant to a split link, is K necessarily slice? We show that if P ( K ) is smoothly concordant to a split link, then many smooth concordanceinvariants of K must vanish, including the τ and s -invariants, and suitablynormalized d -invariants of Dehn surgeries on K . We also investigate the(2 , ℓ ) cables P ℓ ( K ), and find obstructions to smooth concordance to thesum of the (2 , ℓ ) torus link and a split link. Introduction
A central theme in the study of link theory has long been the relationshipbetween the properties of a link and those of its individual components. Recallthat a split link is one whose components lie in disjoint balls; such a linkis certainly determined by its individual components. It is classical [2, 12]that not all links are split. However, showing that not every link (say withlinking number 0) is concordant to a split link takes more work; in the classicaldimension this was first done using Milnor’s ¯ µ -invariants [20, 21]. Examples inhigher odd dimensions were first constructed by S. Cappell and J. Shaneson [3]and independently by A. Kawauchi [17]. The even dimensional case remainsstubbornly out of reach.We investigate Kawauchi’s construction in the classical dimension, where theadditional distinction between the smooth and topological categories comesinto play. Kawauchi considers the (2 ,
0) cable of a knot K ; we denote this‘parallel’ link by P ( K ). Kawauchi showed that if P ( K ) is concordant to asplit link, then K is algebraically slice. We show that if P ( K ) is concordantto a split link, then a host of knot concordance invariants of K must vanish.Our results suggest the conjecture that P ( K ) is smoothly (resp. topologically)concordant to a split link if and only if K is smoothly (resp. topologically)slice. In the last section, we consider the (2 , ℓ ) cables, denoted P ℓ ( K ), and The first author was partially supported by NSF Grant 0804760. The second author wassupported in part by the Slovenian Research Agency program No. P1-0292-0101. Visits ofthe authors were supported by a Slovenian-U.S.A. Research Project BI-US/09-12-004, andby NSF Grant 0813619. give examples of knots K for which P ℓ ( K ) is topologically, but not smoothlyconcordant to the corresponding cable of the unknot.Before stating our main theorem, we describe a convenient normalizationof the Ozsv´ath-Szab´o d -invariant [23] of surgery on an oriented knot. WriteSpin c ( Y ) for the set of Spin c structures on the manifold Y . As we will detailbelow in Section 2.1, for any oriented knot K in S , one can canonically labelSpin c structures on S p/q ( K ) by elements i ∈ Z p . We use the notation O todenote the unknot. Definition 1.1.
Let K be a knot in S , and let i ∈ Z p . Then the normalized d -invariant ˜ d ( S p/q ( K ) , i ) is defined to be d ( S p/q ( K ) , i ) − d ( S p/q ( O ) , i ) . By Gordon’s classic paper [14] and the homology cobordism invariance ofthe d -invariant [23], d ( S p/q ( K ) , i ) (and hence ˜ d ( S p/q ( K ) , i )) is a smooth knotconcordance invariant. (A recent preprint [27] of T. Peters considers the specialcase when the surgery coefficients are ± Theorem A.
Suppose that P ( K ) is smoothly concordant to a split link. Thenthe following concordance invariants of K must vanish: (1) τ ( K ) , the Ozsv´ath-Szab´o τ -invariant [24] . (2) s ( K ) , Rasmussen’s invariant [28] . (3) δ n ( K ) , the d -invariant of the n -fold branched cover of K [19] . (4) For any p/q ∈ Q − { } , and any Spin c structure i on S p/q ( K ) , thenormalized d -invariant ˜ d ( S p/q ( K ) , i ) . The vanishing of each of the first three invariants follows directly from asimple geometric observation, Lemma 3.1. According to Tim Cochran, thissame observation leads to the vanishing of invariants derived from the workof [10, 11], even if one only assumes a topological locally flat concordance toa split link. The last item in Theorem A is our main result, and makes useof some Dehn surgery manipulations, as well as Ozsv´ath-Szab´o’s recipe forcomputing [26] the Heegaard-Floer homology of rational surgery on a knot.In fact, it follows from recent work of Ni and Wu [22] that the last item isequivalent to(4 ′ ) d ( S ± ( K )) = 0;see section 2.Theorem A, combined with some known computations of d -invariants, leadsdirectly to the following result. ONCORDANCE PROPERTIES OF PARALLEL LINKS 3
Theorem B.
There are infinitely many concordance classes of two componentlinks L such that: (1) L is topologically slice (2) L is a boundary link (3) L is not smoothly concordant to a split link. Chuck Livingston has pointed out that links as described in Theorem Bmay also be obtained by Bing doubling a topologically slice knot. We willexplain the argument, which is basically that of Cimasoni [6], after the proofof Theorem B.It is a standard conjecture that the Bing double of K is smoothly slice (whichaccording to Lemma 2.1 below is equivalent to B ( K ) being concordant to asplit link) if and only if K is. The evidence for this is that many concordanceinvariants of K vanish if B ( K ) is slice [4, 6, 9, 18, 31]. We add one more tothis list of invariants by showing the vanishing of the ˜ d -invariants of a knot K ,given that B ( K ) is smoothly slice. This is deduced by a geometric argument(related to [4]) establishing a link between concordance properties of B ( K )and those of P ( K ). Theorem C.
Suppose that B ( K ) is slice. Then P ( K ) is concordant, in a Z -homology S × I , to a split link. Using Theorem B, this implies
Corollary D. If B ( K ) is slice, then ˜ d ( S p/q ( K ) , i ) = 0 for any p/q ∈ Q − { } ,and any Spin c structure i on S p/q ( K ) . With regard to the other parts of Theorem B, it was shown in [4] that if B ( K ) is slice, then τ ( K ) = δ ( K ) = 0 (the same argument would apply to δ n ( K )). On the other hand, it is not known if s ( K ) would have to vanish.In the last section we consider concordance of two component links withlarger linking numbers. A natural generalization of the question of whether P ( K ) is concordant to a split link to the setting of links with nontrivial link-ing number ℓ is the question of whether P ℓ ( K ) is concordant to P ℓ ( O ) withknots tied in the individual components. We will refer to that process as lo-cal knotting of a link. Generalizing the above results we find links that aretopologically concordant to the (2 , ℓ ) torus link P ℓ ( O ) but not smoothly con-cordant to a locally knotted P ℓ ( O ). Acknowledgments:
We thank Jae Choon Cha, Tim Cochran, Matt Hedden,Chuck Livingston, Adam Levine, and Peter Ozsv´ath for helpful discussions.
DANIEL RUBERMAN AND SAˇSO STRLE Preliminaries
All links will be assumed to be oriented. Generally speaking, concordancewill refer to smooth concordance, with the adjective ‘topological’ (alwaysmeaning locally flat) added as appropriate. We will generally use the sameletters for a link and its components, so that for example L and L wouldindicate the components of a two-component link L .For a link L = ( L , . . . , L n ) let S ( L ) denote the split link with the samecomponents as L , thus S ( L ) = L ⊔ · · · ⊔ L n , where the symbol ‘ ⊔ ’ indicatesthat the components are in disjoint balls. A local knotting of a link J isany link obtained by tying knots in the components of J , or more formallyby connected sum of J with a split link. We will make use of the followingobservation regarding concordance to split links; a more general version of thisargument is given in [5, Lemma 2.5]. Lemma 2.1. If L is concordant to a split link, then L is concordant to S ( L ) .Proof. Let C be a concordance from L to the split link K ⊔ · · · ⊔ K n . Denoteby C j the concordance from the component L j of L to the correspondingcomponent K j . Then ˜ C , defined by turning C ⊔ · · · ⊔ C n upside down, givesa concordance from K ⊔ · · · ⊔ K n to S ( L ). Composing C with ˜ C gives aconcordance from S ( L ) to L . (cid:3) PSfrag replacements L L L L K K LS ( L ) K ⊔ K C C f C f C Figure 1.
Schematic illustration of proof of Lemma 2.12.1.
Heegaard-Floer invariants.
We will make extensive use of the correc-tion term, or d -invariant, introduced by Ozsv´ath-Szab´o in [24]. The d -invariantof a rational homology sphere Y depends on the choice of Spin c structure s ONCORDANCE PROPERTIES OF PARALLEL LINKS 5 on Y , and will be denoted d ( Y, s ). Our main theorem is a comparison of the d -invariants of the Dehn surgery manifold S p/q ( K ) and those of the lens space S p/q ( O ) = − L ( p, q ). We describe, briefly, a canonical way to enumerate Spin c structures on Dehn surgery on a knot in S , and hence a canonical corre-spondence between Spin c ( S p/q ( K )) and Spin c ( S p/q ( O )). The enumeration isin terms of relative Spin c structures on S − ◦ ν ( K ), which by definition [30,Chapter I.4] are equivalence classes of non-singular vectors fields on S − ◦ ν ( K )that point outward at the boundary. These are determined by their relativefirst Chern classes in H ( S − ◦ ν ( K ) , ∂ν ( K )). Hence [25] there is a one-to-one correspondence between the relative Spin c structures on S − ◦ ν ( K ) and S − ◦ ν ( K ′ ) for any two oriented knots, where two relative Spin c structurescorrespond if c has the same evaluation on a Seifert surface. This relativeSpin c structure is labelled by i ∈ Z if the evaluation of the relative Chern classon a Seifert surface is 2 i . Finally, a relative Spin c structure determines a Spin c structure on S p/q ( K ), by extending a vector field on S − ◦ ν ( K ) so that it istangent to the core of the solid torus glued in by the Dehn surgery. This Spin c structure is labelled by i (mod p ) in Z p .The d -invariants of a general rational homology sphere Y are difficult tocompute, but there is a good deal known in the case when Y is p/q Dehnsurgery on a knot K . Ozsv´ath-Szab´o [26] give a chain complex X i,p/q , de-scribed in terms of the knot chain complex CF K ∞ ( S , K ), whose homologyis HF + ( S p/q ( K ) , i ). The chain complex X i,p/q is the mapping cone of a map D i,p/q between chain complexes A i,p/q and B i,p/q . In turn, these are sums ofsubquotient complexes of CF K ∞ ( S , K ), and the components of D i,p/q aredefined in terms of certain maps v k and h k between those subcomplexes.This description gives a good deal of information about d ( S p/q ( K ) , s ), andthe implications have been elucidated in a recent preprint of Ni and Wu [22].The U -equivariance of v k and h k implies that they are determined, in suffi-ciently high gradings, by non-negative integers V k and H k . Moreover, V k aredecreasing and vanish for k ≥ g , H k are increasing and vanish for k ≤ − g (where g = g ( K ) denotes the genus of K ), and V = H . The d -invariants arethen determined by these integers. Proposition 2.2 ([22, Proposition 2.11]) . Suppose p, q > , and ≤ i ≤ p − .Then (1) ˜ d ( S p/q ( K ) , i ) = − { V ⌊ iq ⌋ , H ⌊ i − pq ⌋ } . In view of the proposition the interesting range of numbers V k and H k is V i and H − i for 0 ≤ i ≤ g −
1. This implies that the d -invariants of any largeenough surgery on K determine the d -invariants of all positive Dehn surgeries.The second item below was also observed by Ni and Wu. DANIEL RUBERMAN AND SAˇSO STRLE
Corollary 2.3. (1)
Given knots
K, K ′ ⊂ S , suppose that for some integer n ≥ { g ( K ) , g ( K ′ ) } − the normalized d -invariants satisfy ˜ d ( S n ( K ) , i ) = ˜ d ( S n ( K ′ ) , i ) for all ≤ i ≤ n − . Then for any p, q > and all ≤ i ≤ p − d ( S p/q ( K ) , i ) = ˜ d ( S p/q ( K ′ ) , i ) . (2) If ˜ d ( S ( K )) = 0 , then ˜ d ( S p/q ( K ) , i ) = 0 for all p, q > , and ≤ i ≤ p − .Proof. If n ≥ g −
1, then for each i satisfying 0 ≤ i ≤ n −
1, at most one of V i or H i − n can be nonzero. Hence the normalized d -invariants determine allof these numbers, which in turn determine the normalized d -invariants of allpositive Dehn surgeries.If ˜ d ( S ( K )) = 0, then V = H = 0 and all the relevant V k and H k vanishas well. (cid:3) A Dehn surgery move.
In section 3, we will use some moves on surgerydiagrams with rational coefficients [29]. The first is the well-known slam-dunk move [8, 13], pictured below.PSfrag replacements n r n − r Figure 2.
Slam-dunk move; n ∈ Z and r ∈ Q .The second move is similar to the first one, and looks sort like a sidewaysslam-dunk. In keeping with the sporting terminology, we call this the slap-shot move. The move simplifies certain rational surgeries on the (2 , ℓ ) cableof a knot K , which we label P ℓ ( K ) (for related discussion see [7]). We do p/q surgery on one copy, say K , and (for an integer q ′ ) we do ℓ + 1 /q ′ surgery onthe other. The slap shot move asserts the diffeomorphism indicated in Figure3. The proof is quite similar to the proof of the slam-dunk. First do the ℓ + 1 /q ′ surgery on K . By an isotopy, K may be moved inside the solidtorus that is glued in during that first surgery; in fact, it will be isotopicto the core of that solid torus. But then the p/q surgery on K turns thatsolid torus into another solid torus, which means that the surgery on P ℓ ( K ) isdiffeomorphic to a surgery on K . It is easy to compute the surgery coefficientto be ℓ + p/ ( q + q ′ p ). One may alternatively establish the slap-shot move bya sequence of slam-dunks (and the inverse of the slam-dunk) and ordinaryRolfsen twists. ONCORDANCE PROPERTIES OF PARALLEL LINKS 7
PSfrag replacements
KKK K p/qℓ ℓ + 1 /q ′ ℓ + pq + q ′ p Figure 3.
Slap-shot move on P ℓ ( K )3. Concordance to split links
In this section, we investigate the smooth concordance properties of thelink P ( K ) given by the (2 ,
0) cable of a knot K in S . Note that if K is(topologically) slice, then P ( K ) is (topologically) concordant to P ( O ), andhence is slice. Also, P ( K ) bounds two parallel copies of a Seifert surfacefor K , and hence is a boundary link. As mentioned in the introduction, thework of Kawauchi [17, Theorem 5.1] suggests the conjecture that if P ( K ) issmoothly (resp. topologically) concordant to a split link, then K is smoothly(resp. topologically) slice. We show that if P ( K ) is smoothly concordant toa split link, then various gauge-theoretic concordance invariants of K mustvanish. Such results may be viewed as evidence for that conjecture.For a k -component link L , denote by S r ,...,r k ( L ) the result of surgery on alink L , with surgery coefficients r , . . . , r k ∈ Q . Likewise, we indicate by M n ( L )the n -fold cyclic branched cover of S , branched along all components of L ,corresponding to the homomorphism π ( S − L ) → Z n taking all meridiansto 1 ∈ Z n . For an oriented knot K , the notation K r indicates the same knotwith reversed orientation, and K ρ the image of K under a reflection of S .The first observation about P ( K ) is close to saying that K has order 2 inthe concordance group. Lemma 3.1. If P ( K ) is concordant to a split link, then K is concordant to K ρ . The oriented meridian of K ρ is homologous to the oriented meridian of K in the complement of the concordance.Proof. By Lemma 2.1, if P ( K ) were concordant to a split link, it would beconcordant to K ⊔ K . Orient P ( K ) so that the components have oppositeorientation, which implies that those components cobound an oriented annu-lus. Glue that annulus to a cobordism between P ( K ) and K ⊔ K to obtainan annulus in B with oriented boundary K ⊔ K r . The knots K and K r lie in DANIEL RUBERMAN AND SAˇSO STRLE disjoint 3-balls in S ; viewing B as a product of one of those with an interval,the annulus becomes a cobordism between K and ( K r ) ρr = K ρ . (cid:3) The reader who is confused about the two sorts of orientations should com-pare this argument with the standard proof that the inverse of K in the con-cordance group is given by − K = K ρr .Lemma 3.1 gives rise to additional restrictions to P ( K ) being concordantto a split link, via any Z –valued homomorphism from the concordance groupthat is insensitive to the orientation on K . So for instance, we immediatelydeduce the first 3 parts of Theorem A. Corollary 3.2. If P ( K ) is concordant to a split link, then τ ( K ) = s ( K ) = δ k ( K ) = 0 , where s is Rasmussen’s invariant [28] , and δ k ( K ) is the d -invariant for the spin structure on the k -fold cover of S branched along K (cf. [19] ).Proof of Theorem B. Consider the parallel P ( K ), where K = D + ( J,
0) is theuntwisted, positive-clasped Whitehead double of J . The algorithm of Akbulutand Kirby [1] exhibits the double branched cover M ( K ) as surgery on a link oftwo components, one of which is J J r with framing 0, and the other of whichis a meridian of that knot, with framing −
2. An application of the slam-dunkmove then identifies M ( K ) with S / ( J J r ), hence if P ( K ) is concordant toa split link, then by Corollary 3.2 d ( S / ( J J r )) = 0.At this point, there are many choices for the knot J that will lead to acontradiction. For example, we may take J to be the connected sum of n copiesof the right-handed trefoil. Alternatively, we choose J to be the (2 , n + 1)torus knot. From this point the argument proceeds as in the proof of Theorem4.1 of [5]. In particular, the d -invariant of S / ( J J r ) is − n from which italso follows that different choices for K are not concordant and hence neitherare the corresponding links. (cid:3) Chuck Livingston pointed out an alternate construction that proves Theo-rem B, with the additional feature that the components are unknotted. Asobserved in [6, Propositions 1.1 and 3.2] Bing doubles are always boundarylinks, and there is a genus-0 cobordism between the Bing double of a knot K and the untwisted positive Whitehead double of K . This proves that the Bingdoubles of many knots (i.e. those whose Whitehead doubles are not smoothlyslice) are not smoothly slice. By Lemma 2.1, such a Bing double will not beconcordant to a split link, because its components are unknotted.Another collection of gauge-theoretic concordance invariants of a knot K isgiven by the set of d -invariants [23] of all of the Dehn surgeries on K . Thesehave been investigated for surgery coefficients ± ONCORDANCE PROPERTIES OF PARALLEL LINKS 9 coefficient and any Spin c structure. In seeking to show that a knot K is notsmoothly slice, one would want to show that the d -invariants of S p/q ( K ) differfrom those of p/q surgery on the unknot (for corresponding Spin c structures).Hence we would like to show that if P ( K ) is concordant to a split link, thenthe d -invariants of S p/q ( K ) are equal to those of S p/q ( O ). We start by deducingsome homology cobordisms from a concordance of P ( K ) to a split link. Proposition 3.3. If P ( K ) is concordant to a split link, then for any p/q ∈ Q and q ′ ∈ Z (1) S − p/q ( K ) is homology cobordant to − S p/q ( K ) , and (2) S p/q ( K ) S /q ′ ( K ) is homology cobordant to S p/ ( q + q ′ p ) ( K ) .Proof. In general, if a link L is concordant to a split link there is a homol-ogy cobordism between S r ,...,r k ( L ) and S r ( L ) · · · S r k ( L k ). This followsby Lemma 2.1, together with Gordon’s paper [14]. We use the orientation-preserving diffeomorphism S − p/q ( K ) ∼ = − S p/q ( K ρ ) for any p/q ∈ Q ∪ {∞} combined with Lemma 3.1 to get the homology cobordism of item (1).The proof of the second item again starts by surgering a concordance be-tween P ( K ) and K ⊔ K to get a homology cobordism between surgery on S p/q, /q ′ ( P ( K )) and S p/q ( K ) S /q ′ ( K ). Then an application of the slap-shotmove to S p/q, /q ′ ( P ( K )) gives (2). (cid:3) Proposition 3.3 leads to obstructions, by using the Ozsv´ath-Szab´o d -invariant[23]. We note that a homology cobordism induces a canonical bijection on theset of Spin c structures. In fact, in the case of interest, the orientation of aknot induces an orientation of any knot to which it is concordant, hence thelabelling of relative Spin c structures of the two knots agrees. Proposition 3.4. If L = P ( K ) is concordant to a split link, then for any p/q ∈ Q − { } and any q ′ ∈ Z , (1) d ( S − p/q ( K ) , i ) = − d ( S p/q ( K ) , i ) , and (2) d ( S p/q ( K ) , i ) + d ( S /q ′ ( K )) = d ( S p/ ( q + q ′ p ) ( K ) , i ) for all Spin c structures i ∈ Z p .Proof. These follow from the homology cobordism invariance of the d -invariant. (cid:3) We note that the d -invariants of the 0 surgery on K are determined by theabove (see [23, Proposition 4.12]).Proposition 3.4 can be used by itself to obstruct P ( K ) being concordantto a split link. It gains strength by being coupled with relations between d -invariants for different surgeries on an arbitrary knot. This is how we provepart (4) of Theorem A. Theorem 3.5. If P ( K ) is concordant to a split link, then d ( S ± ( K )) = 0 . Moreover, for any p/q ∈ Q − { } , and any Spin c structure i on S p/q ( K ) , thenormalized d -invariant ˜ d ( S p/q ( K ) , i ) vanishes.Proof. Part (2) of Proposition 3.4 for p = q = q ′ = 1 yields2 d ( S ( K )) = d ( S / ( K )) . Combining this with d ( S ( K )) = d ( S / ( K )) (see [23, Corollary 9.14]) we ob-tain the result for 1 surgery on K . Vanishing of other normalized d -invariantsnow follows from part (2) of Corollary 2.3 and part (1) of Proposition 3.4. (cid:3) Bing doubles
The relationship between B ( K ) and Kawauchi’s results on cables of K isa key ingredient in [4]. In this section we develop this parallel a bit more,yielding new restrictions on a knot whose Bing double is slice. Proof of Theorem C.
We make use of the covering link technique in [4, § B ( K ) is slice, consider the lift of one component of B ( K ) tothe 4-fold branched cyclic cover over the other component, drawn in [4, Figure3]. By lifting the corresponding component of the concordance, we obtain aconcordance in a Z -homology S × I (say W ) between the 2-component link( J , J ) in that figure (consisting of two ‘adjacent’ lifts) and the unlink. Directinspection of that figure shows that ( J , J ) is, in our notation, P ( K ) K r ⊔ K r ).Let C be a concordance in W between P ( K ) K r ⊔ K r ) and the unlink, andconsider embedded arcs α , α in the two components of C . Take the connectedsum of C , along α , α , with two copies of the product concordance between K and itself. The result is then a concordance, in W , between P ( K ) K r K ) ⊔ ( K r K )) and the split link K ⊔ K . On the other hand, P ( K ) K r K ) ⊔ ( K r K )) is concordant to P ( K ), since K r K is slice. Composing the twoconcordances proves the theorem. (cid:3) For the proof of Corollary D, we need only observe that the proofs of thevanishing of the ˜ d -invariants given in Section 3 require only that P ( K ) beconcordant to a split link in a Z -homology S × I , both of whose boundarycomponents are S . 5. Other linking numbers
For any integer ℓ and knot K denote by P ℓ ( K ) the (2 , ℓ ) cable of K ; notethat the parallel P ( K ) is equal to P ( K ). In this section we consider the ONCORDANCE PROPERTIES OF PARALLEL LINKS 11 question whether P ℓ ( K ) is concordant to a locally knotted P ℓ ( O ). In a similarvein as for ℓ = 0 we give examples of links for which this holds in the topologicalcategory but not in the smooth. We start with two simple results analogousto the first two parts of Theorem A that yield single examples where thereis a topological concordance to a locally knotted P ℓ ( O ), but no such smoothconcordance. The main result of this section is Theorem 5.3, which givesinfinitely many examples. We conjecture that if P ℓ ( K ) is concordant to alocally knotted P ℓ ( O ) then K must be a slice knot. Proposition 5.1.
Suppose that P ℓ ( K ) is concordant to a locally knotted P ℓ ( O ) .Then, for any ℓ , both τ ( K ) and s ( K ) must vanish.Proof. According to [5, Lemma 2.5] if P ℓ ( K ) is concordant to a locally knotted P ℓ ( O ), then we may assume that this locally knotted link is P ℓ ( O ) K ⊔ K ).Orient P ℓ ( K ) so that the components go in opposite directions, or in otherwords so that P ℓ ( K ) bounds an annulus. It follows that P ℓ ( O ) K ⊔ K )bounds an annulus in B .Do a band sum of the two components of P ℓ ( O ) K ⊔ K ) indicated inFigure 5 to obtain K K r . We conclude that K K r is the boundary of aPSfrag replacements KK ℓ genus-1 surface in B . From the adjunction inequalities satisfied by the twoinvariants [24, 28], we get that2 | τ ( K ) | = | τ ( K K r ) | ≤ , and2 | s ( K ) | = | s ( K K r ) | ≤ . Since s ( K ) and τ ( K ) are both integers, they must vanish. (cid:3) Choosing K to be the untwisted positive Whitehead double of a knot with τ <
0, we conclude from [15] that while P ℓ ( K ) is topologically concordantto P ℓ ( O ), it is not smoothly concordant to a locally knotted P ℓ ( O ). To getinfinitely many concordance classes of such examples we make use of the d -invariants, which have a greater range than the Z -valued s and τ -invariants.We begin by establishing a useful way to estimate the d -invariants of Dehnsurgery on a knot. Lemma 5.2.
For any nonnegative integers n, p and any r ∈ Q there is aconstant d ( n, p, r ) such that for any knot K that can be unknotted by changing p positive and n negative crossings the correction terms of the r surgery on K satisfy d ( S r ( K ) , s ) ≥ − d ( n, p, r ) for r [0 , p ] d ( S r ( K ) , s ) ≤ d ( n, p, r ) for r [ − n, for all Spin c structures s .Proof. Suppose K can be unknotted by changing p positive and n negativecrossings. Realizing the crossing changes by surgeries on − K at the appropriate crossings yields the unknot in S with framing r + 4 n . The cobordism W corresponding to the surgeries is negative definitefor r [ − n, S r ( K ) along the J pictured in blue in Figure 4; its framing in S is −
1. If the crossing is positive, then J is null-homologous in S r ( K ), andthe generator of H ( W ) is represented by a null-homology for J in S − K together with the core of the two-handle, and has self-intersection −
1. So thecobordism is always negative definite in this case.In case of a negative crossing consider the cobordism W ′ corresponding tothe surgeries on both K and J . Suppose first r > r = a/b intoa continued fraction. Using the slam-dunk move this gives a presentation ofthe r surgery on K as an integral surgery on a link. Denote the intersectionform of the resulting positive definite cobordism by A and the minor of A obtained by deleting the generator corresponding to K by B . Then det A = a and det B = b . The determinant of the intersection pairing on W ′ is then − a − b = − b ( r + 4) which is negative. This proves that the cobordism W isnegative definite. If r < − r = a/b . In this case A is negative definite,det A = ( − k a , det B = ( − k − b and the determinant of the intersectionpairing on W ′ is ( − k b ( r + 4). This has the opposite sign as det A for r < − r the cobordism W is negative definite.Alternatively, we can identify the generator of H ( W ) in the case of neg-ative crossing as well. The situation is a bit more complicated, as the self-intersection of the generator is a rational number. Assume p is odd; if p iseven then replace p by p/ p times J in S r ( K ), union p copies of the core of the twohandle. The self-intersection of the generator is a rational number, given by1 /p times the intersection of a push-off of J (using the given framing) withsuch a null-homology. We may compute this via a formula of Hoste [16] (hetreats the setting where r = ±
1, but the method works in general) and get
ONCORDANCE PROPERTIES OF PARALLEL LINKS 13 blow down
PSfrag replacements
JJ KK rrrr + 4 − − Figure 4.
Changing crossings − − /r . The sign of this is negative for r [ − , r .Then by [23, Theorem 9.6] for any Spin c structure s on the cobordism wehave 4 d ( S r ( K ) , s ′ ) + c ( s ) + p + n ≤ d ( S r +4 n ( O ) , s ′′ ) , where the primes denote the restrictions of Spin c structures to the boundarycomponents. Note that the restriction map H ( W ; Z ) → H ( S r ( K ); Z ) is ontosince H ( W, S r ( K ); Z ) ∼ = H ( W, S r +4 n ( O ); Z ) = 0 hence the above inequalityyields the desired upper bounds independent of K .To obtain the lower bounds one could consider the positive definite cobor-dism corresponding to unknotting K by +1 surgeries and then changing theorientation. Alternatively we note that reversing the roles of the unknot O and K one can obtain K from an appropriate diagram for O by changing n positive and p negative crossings. This gives a negative definite cobordismfrom S r − p ( O ) to S r ( K ) for r [0 , p ]. (cid:3) Theorem 5.3.
For any integer ℓ there are infinitely many concordance classesof two component links L that are topologically concordant to P ℓ ( O ) but notsmoothly concordant to a locally knotted P ℓ ( O ) .Proof. Consider the links P ℓ ( K ( n )) for K ( n ) = D + ( J ( n ) ,
0) the positive White-head double of the (2 , n + 1) torus knot J ( n ). Orient P ℓ ( K ( n )) so that thecomponents have opposite orientation and choose ℓ < P ℓ ( K ( n )) is | ℓ | . (The case when ℓ > link.) The branched double cover M ( n ) of P ℓ ( K ( n )) is S ℓ ( K ( n ) K ( n ) r ).Since K ( n ) can be unknotted by changing one positive crossing it follows fromLemma 5.2 that the d -invariants of M ( n ) are bounded below independent of n . Suppose P ℓ ( K ( n )) were concordant to a locally knotted P ℓ ( O ). As above,we may assume that that this locally knotted link is P ℓ ( O ) K ( n ) ⊔ K ( n )),whose branched double cover M ( n ) ′ is L (2 ℓ, M ( K ( n )). The brancheddouble cover M ( K ( n )) of K ( n ) is S / ( J ( n ) J ( n ) r ) whose d -invariant is − n (see proof of [5, Theorem 4.1]) which yields a contradiction for large n . (cid:3) References [1] Selman Akbulut and Robion Kirby,
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