Linear Independence of Knots Arising from Iterated Infection Without the Use of Tristram Levine Signatures
LLINEAR INDEPENDENCE OF KNOTS ARISING FROMITERATED INFECTION WITHOUT THE USE OFTRISTRAM LEVINE SIGNATURE.
CHRISTOPHER DAVIS
Abstract.
We give an explicit construction of linearly independentfamilies of knots arbitrarily deep in the ( n )-solvable filtration of theknot concordance group using the ρ -invariant defined in [12]. A dif-ference between previous constructions of infinite rank subgroups in theconcordance group and ours is that the deepest infecting knots in theconstruction we present are allowed to have vanishing Tristram-Levinesignatures. Introduction
A knot K is an isotopy class of oriented locally flat embeddings of thecircle S into the 3-sphere S . A pair of knots K and J are called topo-logically concordant if there is a locally flat embedding of the annulus S × [0 ,
1] into S × [0 ,
1] mapping S × { } to a representative of K in S × { } and S × { } to a representative of J in S × { } . A knot is called slice if it is concordant to the unknot or equivalently if it is the boundaryof a locally flat embedding of the 2-ball B into the 4-ball B . The set of allknots modulo concordance under the operation of connected sum is a groupcalled the knot concordance group and is denoted by C .In [10], Cochran, Orr and Teichner define the solvable filtration of C : . . . F n. ⊆ F n ⊆ · · · ⊆ F . ⊆ F ⊆ F . ⊆ F ⊆ C . For k a half integer, the elements in F k are called ( k )-solvable . Theyshow that F is the set of Arf-invariant zero knots, F . is the set of alge-braically slice knots and that Casson-Gordon invariants vanish on F . . In[10, Section 6] Cochran-Orr-Teichner show that F / F . is infinite rank bystudying a satellite operation (called infection in [6, Section 8]). The quo-tient groups F n / F n. have been an active place of research ever since. In[9] Cochran, Harvey and Leidy begin with knots for which the integrals ofthe Tristram-Levine signature functions are linearly independent over Q anduse an iterated infection procedure to produce an infinite rank free Abeliansubgroup of F n / F n. . Date : November 7, 2018.2000
Mathematics Subject Classification. a r X i v : . [ m a t h . G T ] S e p CHRISTOPHER DAVIS
In [3] Cha constructs another infinite rank subgroup of F n / F n. startinginstead with knots whose Tristram-Levine signatures evaluate to sufficientlylarge values at particular finite sets.We present a variation on this idea, performing iterated infections to pro-duce linearly independent sets deep in the solvable filtration. A novel aspectof the construction we present is that the deepest infecting knots are allowedhave vanishing Tristram-Levine signature. A concrete advantage of this con-struction over previous ones is that its conditions may be directly verified,providing explicit infinite linearly independent sets rather than generatingsets for infinite rank subgroups.Given any closed oriented 3-manifold M and a homomorphism φ : π ( M ) → Γ, the von Neumann ρ -invariant, ρ ( M, φ ) ∈ R , is defined. It is an invariantof orientation preserving homeomorphism of the pair ( M, φ ). Restrictingthis invariant to the zero surgery of knots and links gives rise to an isotopyinvariant. We provide a brief overview of ρ -invariants in Section 2.In [12] the author defines a particular ρ -invariant, ρ , shows that in arestricted setting it provides a concordance obstruction and computes it foran infinite family of twist knots of order 2 in the algebraic concordancegroup. In this paper we bring this invariant to bear on an iterated infectionprocedure in order to produce examples whose deepest infecting knots havevanishing Tristram-Levine signature.For an overview of infection, see [6, section 8]. We denote the infection ofthe base knot R along the infecting curve η in S − R by the infecting knot J as either R η ( J ) or R ( η, J ) depending on notational convenience.Recall that for a knot K , the rational Alexander module of K , A ( K ),is given by the first homology with coefficients in Q of the infinite cycliccover of the exterior of K or equivalently of the zero surgery of K , M ( K ).The rational Alexander module of K is a module over the ring of Laurentpolynomials, Q [ t ± ]. With respect to the involution on Q [ t ± ] given by p ( t ) = p ( t − ), there is a sesquilinear form Bl : A ( K ) × A ( K ) → Q ( t ) Q [ t ± ] , called the Blanchfield form . A submodule P ⊆ A ( K ) is called isotropic if Bl ( x, y ) vanishes for all x, y ∈ P .We now give definitions of the concepts needed in the statement of Theo-rem 1.1, the main theorem of this paper. They will be recalled when needed.Two polynomials p ( t ) , q ( t ) are called strongly coprime if p ( t k ) and q ( t l )have no common roots in C for every choice of nonzero integers k and l . Apair ( R, η ) with R a knot and η a curve in its complement is called doublyanisotropic if η represents an element of A ( R ) for which there does notexist any α and β in A ( R ) with η = α + β and Bl ( α, α ) = Bl ( β, β ) = 0. Theorem 1.1.
Let { K i } be a possibly infinite set of knots:(1) whose Alexander polynomials are strongly coprime, AND INFECTION. 3 (2) whose Tristram-Levine signatures have vanishing integrals,(3) whose prime factors have square-free Alexander polynomials and(4) whose ρ -invariants do not vanish, that is ρ ( K i ) (cid:54) = 0 .For i = 1 , , . . . and j = 1 , , . . . , n let R i,j be a slice knot and η i,j be anunknotted curve in the complement of R i,j such that the pair ( R i,j , η i,j ) isdoubly anisotropic. Let K i = K i and K ji = R i,j ( η i,j , K j − i ) .Then { K ni } ∞ i =1 is linearly independent in C modulo ( n +1 . ) solvable knots. A noteworthy difference between this and previous results producing infi-nite rank subgroups via iterated infection is the condition on the Tristram-Levine signature. Previous constructions assume that the deepest infectingknots, K i , be complicated in some sense. In [9, Theorem 7.5] their inte-grals are required to be rationally linearly independent. In [3, Lemma 4.11and Proposition 4.12] they are required to take large values at a specificset of points. By contrast, Theorem 1.1 is designed to apply even whenLevine-Tristram signature functions vanish.Another key difference between this result and such previous results is theease of verifying that the assumptions of the theorem are satisfied. Specifi-cally, notice that in Theorem 1.1 there is no assumption on the ρ -invariantsof the slice knots R i,j . The techniques of [9] require a condition on first ordervon Neumann ρ -invariants of the knots along which infection is performed.Without a means of computation, they cannot verify that any fixed knot sat-isfies this condition. The techniques of [3] require that the Tristram-Levinesignature of the infecting knots exceed all of the von Neumann ρ -invariantsof the knots along which infection is done. Without any means of gettingconcrete bounds, these techniques will not give any explicit linearly inde-pendent sets.In Section 3, as an application of Theorem 1.1 we generate the followingfamily of linearly independent knots deep in the filtration using the baseknot R , infecting curve η (see Figure 1) and deepest infecting knot given bytwist knots T n of finite algebraic order (see Figure 2). Theorem 3.3.
For the slice knot R and infecting curve η , if T n is the n -twistknot then { ( R η ) m ( T n ) = ( R η ◦ · · · ◦ R η )( T n ) | n = − x − x − , x ≥ } is linearly independent in F m − . / F m +1 . , where F − . is taken to be the wholeconcordance group. This family of knots appears to be the first linearly independent set deepin the solvable filtration of C constructed by an iterated infection procedurewith deepest infecting knots whose Tristram-Levine signature functions van-ish.1.1. Outline of the paper.
In Section 2 we provide what in this paperis taken as the definition of the von Neumann ρ -invariant, as well as some CHRISTOPHER DAVIS +1 -1+2 η Figure 1.
A slice knot R with a doubly anisotropic curve, η . The depicted derivative is the unlink. n Figure 2. T n , the n -twist knot.properties of the L -signature. We go on to provide definitions of the ρ and ρ -invariants which are employed in this paper.In Section 3, we find a family of knots with nonzero ρ -invariant whoseAlexander polynomials are strongly coprime and square-free, as well as aset of slice knots whose rational Alexander modules have doubly anisotropicelements, that is, a set of knots satisfying the conditions of Theorem 1.1.This provides an explicit linearly independent set of knots sitting arbitrarilydeep in the solvable filtration of the concordance group. The remainingsections are devoted to the development of the machinery used in the proofof Theorem 1.1In Section 4 we discuss a localization of the Alexander module, (cid:102) A p ( K ). Inorder to capture information involving this localization we define a new classof von Neumann ρ -invariant. It enjoys additivity properties over connectedsum and infection and in some cases agrees with ρ .In Section 5 we study the Blanchfield linking form on (cid:102) A p ( K ) and findsufficient conditions for this localized Blanchfield form to have no nontrivialisotropic submodules. The significance of this result appears in Section 6 AND INFECTION. 5 in which we show that isotropic submodules of this localization are wellbehaved with respect to the operation of extension of scalars.Finally, in Section 7 we give the proof of Theorem 1.1.2.
Background: von Neumann ρ -invariants and L signatures In this section we state the properties of von Neumann ρ -invariants and L signatures needed in this paper.In [15, Section 3] the following property is proven of the von Neumann ρ -invariant. It serves here as the definition. Definition . Consider oriented 3-manifolds M , . . . , M n , with homomor-phisms φ i : π ( M i ) → Γ i . Suppose that M (cid:116) M (cid:116) · · · (cid:116) M n is the orientedboundary of a compact oriented 4-manifold W and ψ : π ( W ) → Λ is ahomomorphism such that, for each i , there is a monomorphism α i : Γ i → Λmaking the following diagram commute: π ( M i ) Γ i π ( W ) Λ (cid:47) (cid:47) φ i (cid:15) (cid:15) (cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31) i ∗ (cid:127) (cid:95) (cid:15) (cid:15) (cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31) α i (cid:47) (cid:47) ψ Then n (cid:88) i =1 ρ ( M i , φ i ) = σ (2) ( W, ψ ) − σ ( W ) where σ ( W ) is the regular signa-ture of W and σ (2) ( W, ψ ) is the L signature of W twisted by the coefficientsystem ψ . The expression σ (2) ( W, ψ ) − σ ( W ) is called the signature defectof W with respect to ψ .For a compact oriented 4-manifold W with coefficient system φ : π ( W ) → Γ, σ (2) ( W, Γ) ∈ R is defined. The properties of the L signature which areused in this paper are Novikov additivity and a bound in terms of ranksof twisted second homology in the case that Γ is PTFA (Poly Torsion FreeAbelian, see [10, Definition 2.1]). Proposition (Novikov additivity, [10, Lemma 5.9 (3)]) . If compact oriented4-manifolds W and W intersect in a single common boundary component, W = W ∪ W , and i : W → W and i : W → W are the inclusion maps,then for every homomorphism φ : π ( W ) → Γ , σ (2) ( W, φ ) = σ (2) ( W , φ ◦ i ∗ ) + σ (2) ( W , φ ◦ i ∗ ) . The second property is that when Γ is PTFA and more generally whenever Q [Γ] is an Ore domain,(2.1) (cid:12)(cid:12)(cid:12) σ (2) ( W, φ ) (cid:12)(cid:12)(cid:12) ≤ Rank Q [Γ] (cid:18) H ( W ; Q [Γ]) i ∗ [ H ( ∂W ; Q [Γ])] (cid:19) where i ∗ : H ( ∂W ; Q [Γ]) → H ( W ; Q [Γ]) is the inclusion induced map.This follows from the monotonicity of von Neumann dimension (see [19, CHRISTOPHER DAVIS
Lemma 1.4]) and the fact that the L Betti number agrees with Q [Γ] rankwhen Q [Γ] is an Ore Domain (see [5, Lemma 2.4] or [14, Proposition 2.4]).2.1. The ρ and ρ -invariants. Definition . For a knot K , Let φ : π ( M ( K )) → Z be the Abelianizationmap. Let ρ ( K ) := ρ ( M ( K ) , φ ) be the corresponding ρ -invariant.It is shown in [11, Proposition 5.1] that ρ ( K ) is given by the integral ofthe Tristram-Levine signature function.Recall that the rational derived series of a group G is given by setting G (0) r = G and inductively defining G ( n +1) r to be the set of all g ∈ G ( n ) r whichare torsion in the Abelianization of G ( n ) r . This series is the most quicklydescending series with the property that each of the successive quotients G ( n ) r /G ( n +1) r is TFA (Torsion Free Abelian). Definition . For a knot K , let φ : π ( M ( K )) → π ( M ( K )) π ( M ( K )) (2) r be the quotient by the second term in the rational derived series. Let ρ ( K ) := ρ ( M ( K ) , φ ) be the corresponding ρ -invariant.In [12] the ρ -invariant is shown to provide a sliceness obstruction and isused to find an infinite collection of twist knots of algebraic order 2 whichis linearly independent in C .3. generating explicit linearly independent families of knotsarbitrarily deep in the solvable filtration In this section we verify the assumptions of Theorem 1.1 for an explicitset of knots.We first address the restriction we put on the slice knots we infect andcurves along which we infect them.
Definition . For a knot, R , an element of A ( R ), η , is called doublyanisotropic if η cannot be written as a sum of isotropic elements, thatis there do not exist any α, β ∈ A ( R ) with Bl ( α, α ) = Bl ( β, β ) = 0 and α + β = η .In this paper we concern ourselves with the operator R η : C → C givenby sending a knot J to the result of infection R η ( J ) where R is a slice knotand η is an unknotted curve representing a doubly anisotropic element of A ( R ).The following proposition serves to illustrate that there are many sliceknots whose Alexander modules have doubly anisotropic elements. Proposition 3.2.
Let K be a slice knot with cyclic Alexander module iso-morphic to Q [ t ± ]( δ ( t ) ) where δ is any symmetric polynomial with δ (1) = ± . If AND INFECTION. 7 δ has a prime symmetric factor then η , the generator of A ( R ) , is doublyanisotropic.Proof. Let q be the assumed prime symmetric factor of δ . Consider anyelement, f η , in the Alexander module ( f ∈ Q [ t ± ]). If f η were an isotropicelement then 0 = Bl ( f η, f η ) = f ( t ) f ( t − ) r ( t ) δ ( t ) ∈ Q ( t ) Q [ t ± ]where ( r, δ ) = 1. Then q , being a factor of δ , must divide f ( t ) f ( t − ) r ( t ).Since q is prime and ( r, q ) = 1 q must divide f ( t ) or f ( t − ). Since q issymmetric, it divides both. Thus, any isotropic element of the A ( R ) (andso any sum of two isotropic elements) sits in the proper submodule P = (cid:104) qη (cid:105) and any element of A ( R ) − P (for example, η , the generator of A ( R )) isdoubly anisotropic. (cid:3) Cha ([4, Theorem 5.18]) shows that for every symmetric polynomial δ there is a ribbon knot with Alexander module of the form Q [ t ± ]( δ ( t ) ) , so that sliceknots with doubly anisotropic curves abound. For the sake of concreteness,let R be the slice knot depicted in Figure 1 and Let η be the curve in S − R , also depicted in Figure 1. The Alexander module of R is cyclicgenerated by η and has Alexander polynomial of the form δ ( t ) where δ ( t ) = t − t + 1 is a symmetric prime polynomial. By Proposition 3.2 ( R, η ) isdoubly anisotropic.What remains is to find infinitely many infecting knots whose Tristram-Levine signatures have vanishing integrals, whose ρ -invariants are nonzeroand whose Alexander polynomials are strongly coprime and square-free. For n <
0, the twist knot T n (see Figure 2) is algebraically of finite order, sothat the Tristram-Levine signature vanishes. It is shown in [12, Theorem6.1] that for n ( x ) = − x − x −
1, and x ≥ ρ ( T n ( x ) ) (cid:54) = 0. Their Alexanderpolynomials are prime and so square-free.Theorem 3.1 of [1] gives us that the strong coprimality condition is satis-fied. A note on conventions, the knot which is called T n in this paper is thereverse of the mirror image of the knot called T − n in [1]. Theorem (Theorem 3.1, [1]) . For positive integers m (cid:54) = n the Alexanderpolynomials ∆ T − n and ∆ T − m are strongly coprime. Thus, the slice knot R and infecting curve η , together with the deepestinfecting knot T n ( x ) for x ≥ Theorem 3.3.
The set { ( R η ) m ( T n ) | n = − x − x − , x ≥ } is linearly inde-pendent in F m − . / F m +1 . , where F − . is taken to be all of the concordancegroup. Starting with a different choice of R η , we construct families of knots thatare linearly independent in the concordance group but which many previousinvariants fail to detect. CHRISTOPHER DAVIS
Theorem 3.4.
Let p m ( t ) denote the m th cyclotomic polynomial where m isdivisible by three distinct prime numbers.Let R be a ribbon knot with cyclic Alexander module A ( R ) ∼ = Q [ t ± ]( p m ) . Let η be an unknotted curve representing a generator of A ( R ) . Let T n be the n -twist knot.Then { R η ( T n ) | n = − x − x − with x ≥ } is linearly independent in F . / F . (and so in C ); however, the Casson-Gordon sliceness obstructionof [2] , the metabelian η -invariant obstruction of [13] and the (1 . -solvabilityobstructions of [16] , [10] and [3] all vanish for each element of this set.Proof. The fact that this set is linearly independent is an immediate conse-quence of Theorem 1.1.For every n the Alexander polynomial of R η ( T n ) is the same as the Alexan-der polynomial of R , which is p m . By [18, Theorem 1.2], every prime powercyclic branched cover of R η ( T n ) is a homology sphere. Thus, the metabelian η -invariants of [13] and the Casson-Gordon obstructions vanish.In order to compute the obstructions of [10] (using the specialization of theobstruction to (1 . A ( R η ( T n )) ∼ = A ( R ),namely P = (cid:104) p m (cid:105) . We first compute the obstruction of [8]. By [7, Lemma2.3] ρ P ( R η ( T n )) = ρ P ( R ) + ρ ( T n ) . As T n is of finite algebraic order, ρ ( T n ) = 0. Since R is slice and P isthe only Lagrangian submodule of A ( R ), [8, Theorem 4.2] implies that ρ P ( R ) = 0. This completes the proof that this invariant vanishes.In order to compute the obstruction of [16], let x ∈ P and consider themap φ x defined in that paper. By [7, Lemma 2.3], ρ ( R η ( T n ) , φ x ) = ρ ( R, φ x ) + ρ ( T n ) , similarly to before, ρ ( T n ) vanishes and since R is slice and has only oneLagrangian submodule, P , [16, Theorem 1.1] implies that ρ ( R, φ x ) = 0 forall x ∈ P .Finally, we check that the (1 . R is slice, it follows that ρ ( M ( R ) , φ ) = 0 for somecoefficient system φ : π ( M ( R )) → Γ where Γ (2) = 0, Γ is amenable and Γis in Strebel’s class D ( S ) (See [3] for a definition) where S is either Q or afinite cyclic group. By [7, Lemma 2.3], then ρ ( M ( R η ( T n )) , φ ) = ρ ( M ( R ) , φ ) + ρ ( M ( T n ) , φ ) . As we observed, ρ ( M ( R ) , φ ) = 0. By [3, Lemma 4.5], ρ ( M ( T n ) , φ ) is a sumor integral of the Tristram-Levine signature of T n and so is zero. (cid:3) We do not know if it is possible to use the (2 . . AND INFECTION. 9 strongly localized ρ -invariants For a knot K and a polynomial p , the strongly localized ρ -invariant of K , (cid:101) ρ p ( K ) is defined in terms of a localization of the Alexander module of K .We begin by describing this localization.For polynomials p, q ∈ Q [ t ± ] we say that p and q are strongly coprime ([9, Definition 4.4]) denoted (cid:93) ( p, q ) = 1 if there is no nonzero complex number z and integers m, n such that p ( z m ) = q ( z n ) = 0. Let (cid:102) S p = { q ∈ Q [ t ± ] | (cid:93) ( p, q ) = 1 } be the multiplicative set consisting of polynomials strongly coprime to p .Let (cid:102) R p = Q [ t ± ] (cid:102) S p − = (cid:26) fg ∈ Q ( t ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:93) ( g, p ) = 1 (cid:27) be the strong localization of Q [ t ± ] at p . By [20, Theorem 10.30] (cid:102) R p isflat as a Q [ t ± ] module so that for a knot K , the first homology of M ( K )with coefficients in (cid:102) R p is given by H ( M ( K ); (cid:102) R p ) ∼ = H ( M ( K ); Q [ t ± ]) ⊗ Q [ t ± ] (cid:102) R p . We call this module the strongly localized Alexander module anddenote it by (cid:102) A p ( K ).Let π ( M ( K )) (2) (cid:101) p be the kernel of the composition(4.1) π ( M ( K )) (1) → π ( M ( K )) (1) π ( M ( K )) (2) (cid:44) → A ( K ) → (cid:102) A p ( K ) . Let (cid:102) φ p : π ( M ( K )) → π ( M ( K )) π ( M ( K )) (2)˜ p be the quotient map. Definition . Let (cid:101) ρ p ( K ) = ρ ( M ( K ) , (cid:102) φ p ) be the strongly localized ρ -invariant of K at p .We will be flexible with notation. For any CW-complex X with infinitecyclic first homology generated by t we can similarly define A ( X ), (cid:102) A p ( X )and π ( X ) (2) (cid:101) p .The strongly localized ρ -invariant shares many properties with the simi-larly defined localized ρ -invariant of [12]. The proofs of the following propo-sitions are identical to proofs in [12] and so are omitted. Proposition 4.2 ([12, Proposition 3.4]) . If ∆ is the Alexander polynomialof a knot K , then(1) (cid:102) ρ ( K ) = ρ ( K ) (2) If p and ∆ are strongly coprime, then (cid:101) ρ p ( K ) = ρ ( K ) . Proposition 4.3 ([12, Proposition 3.5]) . Let J and K be knots and η be anunknot in the complement of J such than J and η have zero linking number.(1) (cid:101) ρ p ( J K ) = (cid:101) ρ p ( J ) + (cid:101) ρ p ( K ) (2) (cid:101) ρ p ( J η ( K )) = (cid:40) (cid:101) ρ p ( J ) if η = 0 in (cid:102) A p ( J ) (cid:101) ρ p ( J ) + ρ ( K ) if η (cid:54) = 0 in (cid:102) A p ( J )5. Strongly p -anisotropic knots For a knot K the classical rational Blanchfield form Bl is sesquilinear withrespect to the involution q ( t ) = q ( t − ). For a symmetric polynomial p , thisinvolution extends over (cid:102) R p , so the Blanchfield form extends to a sesquilinearform which we call the strongly localized Blanchfield form , (cid:102) Bl p : (cid:102) A p ( K ) × (cid:102) A p ( K ) → Q ( t ) (cid:102) R p . A submodule P of (cid:102) A p ( K ) is called isotropic if P ⊆ P ⊥ with respect to (cid:102) Bl p and it is called Lagrangian or self-annihilating if P = P ⊥ . A knot K is called strongly p -anisotropic if (cid:102) A p ( K ) has no nontrivial isotropicsubmodules.We now provide examples of strongly p -anisotropic knots. Proposition 5.1.
Let ∆ be the Alexander polynomial of a knot K . Let p be a symmetric polynomial and h = (∆ , p ) be the greatest common divisorof ∆ and p . Suppose that h has no non-symmetric factors and no roots ofmultiplicity greater than . If (cid:94) (cid:0) ∆ h , p (cid:1) = 1 then K is strongly p -anisotropic.Proof. As a first step, we show that (cid:102) A p ( K ) is cyclic. Since the unlocalizedAlexander module A ( K ) is torsion over the PID Q [ t ± ], it has a decompo-sition into elementary factors: A ( K ) = k ⊕ i =1 Q [ t ± ]( q i )where q i divides q i +1 for each 0 < i < k and k (cid:89) i =1 q i = ∆. Thus, the localizedAlexander module has the decomposition (cid:102) A p ( K ) = A ( K ) ⊗ Q [ t ± ] (cid:102) R p = k ⊕ i =1 (cid:102) R p ( q i )If for some i < k there exist some z ∈ C , m, n ∈ Z such that q i ( z n ) = p ( z m ) =0 then q i +1 ( z n ) is also zero since q i divides q i +1 . Thus, ∆ = k (cid:89) i =1 q i has a rootof multiplicity at least two at z n . Since h has no roots of multiplicity greaterthan 1, ∆ h ( z n ) = 0 contradicting that (cid:94) ( ∆ h , p ) = 1. Thus, q i ∈ (cid:102) S p is a unit in (cid:102) R p for every i < k and (cid:102) A p ( K ) = (cid:102) R p ( q k ) . In particular, (cid:102) A p ( K ) is cyclic. AND INFECTION. 11
Let ˆ h = ( q k , h ). Since q k ˆ h divides ∆ h , which is a polynomial stronglycoprime to p , it follows that q k ˆ h ∈ (cid:102) S p is a unit in (cid:102) R p so that the idealsgenerated by ˆ h and q k in (cid:102) R p are the same and (cid:102) A p ( K ) ∼ = (cid:102) R p / (ˆ h ) . Let η be a generator of (cid:102) A p ( K ). If (cid:103) Bl p ( η, xy η ) = 0 for some xy ∈ (cid:102) R p , then forany other rq η ∈ (cid:102) A p ( K ), (cid:102) Bl p ( rq η, xy η ) = rq (cid:102) Bl p ( η, xy η ) = 0. Thus, (cid:102) Bl p ( − , xy η ) isidentically zero. By the non-singularity of the Blanchfield form, this impliesthat xy η is zero, so that xy is zero in (cid:102) R p / (ˆ h ) and xy ∈ (ˆ h )If xy η is an isotropic element of (cid:102) A p ( K ), then0 = (cid:102) Bl p (cid:18) xy η, xy η (cid:19) = (cid:102) Bl p (cid:18) η, xxyy η (cid:19) so that xxyy ∈ (ˆ h ). Thus, there is some uv ∈ (cid:102) R p such that xxyy = ˆ h uv in (cid:102) R p .Cross-multiplying gives the equality in Q [ t ± ](5.1) xxv = ˆ huyy. Thus, ˆ h divides xxv . The fact that v ∈ (cid:102) S p implies (cid:93) ( v, p ) = 1 and in particular( v, p ) = 1. Since ˆ h divides p it follows that ( v, ˆ h )=1. Therefore (5.1) impliesthat ˆ h divides xx . Since ˆ h divides h , which has neither any non-symmetricfactors nor any repeated factors, it must be that ˆ h has neither any non-symmetric factors nor any repeated factors. Thus, ˆ h dividing xx impliesthat ˆ h divides x so that xy is in the ideal of (cid:102) R p generated by ˆ h and xy η = 0.Thus, (cid:102) A p ( K ) has no nonzero isotropic submodules with respect to (cid:102) Bl p . (cid:3) We restrict Proposition 5.1 to the setting from which we draw examples:
Corollary 5.2.
Let ∆ be the Alexander polynomial of a knot K .(1) If (cid:94) ( p, ∆) = 1 then (cid:102) A p ( K ) = 0 and K is strongly p -anisotropic.(2) If p = ∆ has no repeated roots and has no non-symmetric factorsthen K is strongly p -anisotropic. Isotropy and extension of coefficient systems
In this section we discuss the affect that the conditions of strong p -anisotropy and double anisotropy have on higher order Alexander modulesof a knot. More precisely, if a knot is strongly p -anisotropic we discover a re-striction on the structure of isotropic submodules of certain localizations ofhigher order Alexander modules. We go on to show that doubly anisotropicelements of the Alexander module produce elements of unlocalized higherorder Alexander modules which are almost doubly anisotropic.We begin by describing what we mean by higher order Alexander modules. For a knot, K . Let ψ : π ( M ( K )) → Γ be a homomorphism to a PTFAgroup, Γ, which factors as π ( M ( K )) → (cid:104) t (cid:105) (cid:44) → A (cid:69) Γwhere t is the generator of the Abelianization of π ( M ( K )) and A is a TFAnormal subgroup of Γ. Let S p ( A ) = { q ( a ) . . . q n ( a n ) | (cid:94) ( q i , p ) = 1 , a i ∈ A } ⊆ Q [ A ] . Since A is normal, S p ( A ) is a Γ-invariant divisor set for Q [ A ], [9, Proposition4.1] shows that S p ( A ) ⊆ Q [Γ] satisfies the right (as well as the left) Ore con-dition and the localization R := Q [Γ] S p ( A ) − is defined. For the definitionof the Ore condition and a treatment of localization for noncommutativerings, see [21, Chapter 2]. Let K (Γ) = Q [Γ]( Q [Γ] − { } ) − be the skew fieldof fractions of Q [Γ].We are interested in the localized higher order Alexander module of K , H ( M ( K ); R ).According to [10, Theorem 2.13], there exists a sesquilinear form Bl Γ : H ( M ( K ); R ) × H ( M ( K ); R ) → K (Γ) R . We summarize the construction. Consider the Bockstien exact sequence oncohomology, H ( M ( K ); K (Γ)) → H (cid:16) M ( K ); K (Γ) R (cid:17) Bo → H ( M ( K ); R ) → H ( M ( K ); K (Γ)) . By Poincar´e duality and [6, Lemma 3.9], H ( M ( K ); K (Γ)) ∼ = H ( M ( K ); K (Γ)) = 0By [10, Remark 2.8.1] there is a universal coefficient theorem for skew fieldcoefficients and H ( M ( K ); K (Γ)) ∼ = Hom K (Γ) ( H ( M ( K ); K (Γ)) , K (Γ)) = 0 . What remains of the Bockstien exact sequence is that the Bockstien homo-morphism Bo : H ( M ( K ); K (Γ) / R ) → H ( M ( K ); R ) is an isomorphism.The Blanchfield form, Bl Γ , is defined by the composition H ( M ( K ); R ) P.D. → H ( M ( K ); R ) Bo − → H ( M ( K ); K (Γ) / R ) κ → Hom R ( H ( M ( K ); R ) , K (Γ) / R )where P.D. denotes Poincar´e duality and κ is the Kronecker map, that is, Bl Γ ( a, b ) = (cid:0) ( κ ◦ Bo − ◦ P.D. )( a ) (cid:1) ( b )By [17, Lemma 3.2 and Proposition 3.6], since π ( M ( K )) → Γ factorsnontrivially through Abelianization and has image in the normal TFA sub-group A , H ( M ( K ); R ) ∼ = H ( M ; (cid:102) R p ) ⊗ (cid:102) R p R AND INFECTION. 13 and for any a ⊗ α and b ⊗ β in H ( M ( K ); Q [Γ] S p ( A ) − ), Bl Γ ( a ⊗ α, b ⊗ β ) = α Ψ( (cid:103) Bl p ( a, b )) β, where Ψ : Q ( t ) (cid:102) R p → K (Γ) R is the map induced by ψ .Thus, in this section we start with a torsion (cid:102) R p module, M with a bilinearform B : M × M → Q ( t ) (cid:102) R p and study the bilinear form on M Γ = M ⊗ (cid:102) R p R , B Γ : M Γ × M Γ → K (Γ) R given by B Γ ( a ⊗ α, b ⊗ β ) = α Ψ( B ( a, b )) β. The inheritance of anisotropy under extension of coefficients.
The following theorem reveals an aspect of the behavior of isotropic sub-modules under this extension of coefficients.
Theorem 6.1.
Consider the infinite cyclic group (cid:104) t (cid:105) and a torsion (cid:102) R p mod-ule M , with bilinear form B : M × M → Q ( t ) (cid:102) R p .Suppose (cid:104) t (cid:105) injects into a TFA group A which is a normal subgroup of aPTFA group Γ . If P is an isotropic submodule of M ⊗ R with respect to B Γ ,then { m ∈ M | m ⊗ ∈ P } is isotropic with respect to B . Theorem 6.1 is a consequence of Lemma 6.2 below.
Lemma 6.2.
Suppose t (cid:55)→ T ∈ A defines a monomorphism from (cid:104) t (cid:105) to A where A is a TFA group and a normal subgroup of a PTFA group Γ . Thenthe induced map Ψ : Q ( t ) (cid:102) R p (cid:44) → K (Γ) R . is a monomorphism.Proof. Suppose that f ( t ) g ( t ) is in the kernel of this map. Then f ( T ) g ( T ) is con-tained in R and there exist some r ∈ Q [Γ] and q ∈ S p ( A ) such that f ( T ) g ( T ) = rq . By the definition of equality in K (Γ) (see [21, Chapter 2 Propo-sition 1.4]) This implies that there exist nonzero c, d ∈ Q [Γ] such that f ( T ) c = rd, (6.1) g ( T ) c = qd. (6.2)Considering (6.1) as an equation in Q [Γ]( Q [ A ] − { } ) − (into which Q [Γ]injects), it reduces to c = ( f ( T )) − rd . This substitution reduces (6.2) to g ( T )( f ( T )) − rd = qd . Cancelation gives that g ( T )( f ( T )) − r = q . Now, g ( T ) and ( f ( T )) − sit in the image of the field Q ( t ) and so commute witheach other. Thus, ( f ( T )) − g ( T ) r = q . Multiplying by f ( T ) on the left givesus that(6.3) g ( T ) r = f ( T ) q. Let X be a transversal for Γ /A (that is, a subset of Γ containing the iden-tity, 1, such that every equivalence class in Γ /A has a unique representativein X ). The group ring Q [Γ] is free as a Q [ A ] module and has basis X , so that r can be uniquely realized as r = (cid:88) x ∈ X r x x where each r x is in Q [ A ]. Sincethe right hand side of (6.3) is in Q [ A ], that is, the span of { } , it reduces to g ( T ) r = f ( T ) q and(6.4) r x = 0 for x ∈ X − { } . (6.5)Now suppose that f ( t ) g ( t ) ∈ Q ( t ) is in reduced terms and that g in not in (cid:102) S p .Since q is assumed to be in S p ( A ), there exist polynomials q , . . . , q n ∈ (cid:102) S p and a , . . . , a n ∈ A so that q = n (cid:89) i =1 q i ( a i ). Equation (6.4) as an equality in Q [ A ] involves only finitely many elements of A , namely, T, a , . . . , a n and b , . . . , b k where b , . . . , b k are whatever terms appear in r . The span of { T, a , . . . , a n , b , . . . , b k } is a finitely generated subgroup of the TFA group, A , so it is free Abelian. Pick a basis { s, c , . . . , c m } such that s l = T for some l . The equality (6.4) can then be realized as an equality in the multivariableLaurent polynomial ring Q [ s ± , c ± , . . . , c ± m ]:(6.6) g ( s k ) r ( s, c , . . . , c m ) = f ( s k ) n (cid:89) i =1 q i ( s k i c k i, . . . c k i,m m ) . Since g is not an element of (cid:102) S p , (i.e. it is not strongly coprime to p )there exists some z ∈ C and α, β ∈ Z − { } such that g ( z β ) = p ( z α ) = 0.Evaluating (6.6) at s = z β/k and c = · · · = c n = 1 gives an equality in C (6.7) 0 = f ( z β ) n (cid:89) i =1 q i ( z βk i /k ) . Since z β is a root of g and f is assumed to be relatively prime to g , f ( z β ) (cid:54) = 0.Therefore q i ( z βk i /k ) = 0 for some i , contradicting that q i ∈ (cid:102) S p is stronglycoprime to p . Thus, it must be that g ∈ (cid:102) S p and fg = 0 in Q ( t ) / (cid:102) R p . (cid:3) We now prove Theorem 6.1.
Proof of Theorem 6.1.
Consider any m, n ∈ M such that m ⊗ , n ⊗ ∈ P .Since P is isotropic, 0 = B Γ ( m ⊗ , n ⊗
1) = Ψ( B ( m, n )). Since Ψ is injectiveby Lemma 6.2 this implies that B ( m, n ) = 0 completing the proof. (cid:3) The inheritance of double anisotropy under extension of co-efficients.
The following Proposition reveals that the double anisotropy ispartially inherited under the extension of coefficients. A similar claim couldbe proven after localization. For our purposes the unlocalized claim is suf-ficient. AND INFECTION. 15
Proposition 6.3.
Let A be a TFA group which is a normal subgroup of aPTFA group Γ . Let M be a torsion Q [ A ] module with Q [ A ] -sesquilinear form B A : M × M → K ( A ) Q [ A ] . This sesquilinear form extends to a Q [Γ] -sesquilinearform B Γ : M ⊗ Q [Γ] × M ⊗ Q [Γ] → K (Γ) Q [Γ] . Let Q be an isotropic submodule of M with respect to B A and P be anisotropic submodule of M ⊗ Q [Γ] with respect to B Γ . If η ⊗ ∈ M ⊗ Q [Γ] sits in P + Q ⊗ Q [Γ] , then η = p + q for some q ∈ Q and B A ( p, p ) = 0 .Proof. Let X be a transversal for Γ /A . Suppose that(6.8) η ⊗ (cid:88) x ∈ X p x ⊗ x + (cid:88) x ∈ X q x ⊗ x with q x ∈ Q , p x ∈ M for all x and p := (cid:88) x ∈ X p x ⊗ x ∈ P . Thinking of Q [Γ]as the free Q [ A ] module generated by X , (6.8) implies that p x + q x = 0 for x (cid:54) = 1 and η = p + q . Since P is isotropic,(6.9) 0 = B Γ ( p, p ) = B Γ (cid:88) x ∈ X p x ⊗ x, (cid:88) y ∈ X p y ⊗ y . Appealing to the Γ-sesquilinearity of B Γ , this implies0 = (cid:80) x ∈ X (cid:80) y ∈ X x − B Γ ( p x ⊗ , p y ⊗ y = (cid:80) x ∈ X (cid:80) y ∈ X ( x − B A ( p x , p y ) x ) x − y. (6.10)Since A is normal in Γ, x − B A ( p x , p y ) x is in K ( A ) Q [ A ] for each x, y ∈ X .Since X is a choice of coset representatives for Γ /A , each x − y is equiv-alent modulo A to some z in X , that is, there is some a x,y ∈ A such that x − y = a x,y z . We use this to rearrange (6.10),(6.11) 0 = (cid:88) z ∈ X (cid:88) x − y ≡ z ( x − B A ( p x , p y ) x ) a x,y z The map (cid:18) K ( A ) Q [ A ] (cid:19) | X | → K (Γ) Q [Γ] defined by sending (cid:104) r x (cid:105) x ∈ X to (cid:88) x ∈ X r x ⊗ x isinjective. Indeed, if (cid:104) a x b x (cid:105) x ∈ X is in the kernel of this map then there existssome c = (cid:88) x ∈ X c x x with c x ∈ Q [ A ] such that (cid:88) x ∈ X a x b − x x = (cid:88) x ∈ X c x x Since A is Abelian, a x b − x = b − x a x . Left multiplying by b = (cid:89) x ∈ X b x we seethat (cid:88) x ∈ X ( b − x b ) a x x = (cid:88) x ∈ X bc x x This is an equation in Q [Γ]. The set X is a basis for Q [Γ] as a free Q [ A ]module. Thus, for all x ∈ X , ( b − x b ) a x = bc x so a x b x = c x ∈ Q [ A ] holds in Q ( A ) and (cid:104) a x b x (cid:105) x ∈ X is zero in (cid:18) K ( A ) Q [ A ] (cid:19) | X | .This injectivity together with (6.11) implies that for each z ∈ X ,(6.12) 0 = (cid:88) x − y ≡ z ( x − B A ( p x , p y ) x ) a x,y . Taking z = 1 ∈ X we see that(6.13) 0 = (cid:88) x ∈ X ( x − B A ( p x , p x ) x ) a x,x . As we observed previously, for x (cid:54) = 1, p x = − q x is in Q , so B A ( p x , p x ) = 0.Thus, all but one of the terms in (6.13) vanishes. Dropping them, we seethat 0 = B A ( p , p ) . Since, η = p + q , this completes the proof. (cid:3) Proposition 6.4.
Let M be a torsion Q [ t ± ] -module with bilinear form B : M × M → Q ( t ) Q [ t ± ] . Let Q ⊆ M be isotropic.Suppose (cid:104) t (cid:105) injects into a TFA group A which is a normal subgroup of aPTFA group Γ and P ⊆ M ⊗ Q [Γ] is isotropic. If η ⊗ is in P + Q ⊗ Q [Γ] ,then η is not doubly anisotropic with respect to B .Proof. Consider η A = η ⊗ ∈ M ⊗ Q [ A ] and the isotropic module Q A = Q ⊗ Q [ A ]. Notice that η A ⊗ ∈ ( M ⊗ Q [ A ]) ⊗ Q [Γ] is in Q A ⊗ Q [Γ] + P by assumption. Applying Proposition 6.3 gives that there is some q A ∈ Q A and p A ∈ M ⊗ Q [ A ] with B A ( p A , p A ) = 0 and η A = p A + q A .But then η ⊗ ∈ M ⊗ Q [ A ] sits in the sum of Q ⊗ Q [ A ] with the isotropicsubmodule (cid:104) p A (cid:105) . Applying Proposition 6.3 again gives that η = p + q where B ( p, p ) = 0 and q ∈ Q . In this case, η sits in the sum of the isotropicsubmodules Q and (cid:104) p (cid:105) and so is not doubly anisotropic. (cid:3) The proof of Theorem 1.1
In this section we set out to prove the main result of this paper, Theo-rem 1.1:
Theorem 1.1.
Let { K i } be a possibly infinite set of knots:(1) whose Alexander polynomials are strongly coprime,(2) whose Tristram-Levine signatures have vanishing integrals,(3) whose prime factors have square-free Alexander polynomials and(4) whose ρ -invariants do not vanish, that is ρ ( K i ) (cid:54) = 0 . AND INFECTION. 17
For i = 1 , , . . . and j = 1 , , . . . , n let R i,j be a slice knot and η i,j be anunknotted curve in the complement of R i,j such that the pair ( R i,j , η i,j ) isdoubly anisotropic.Let K i = K i and K ji = R i,j ( η i,j , K j − i ) .Then { K ni } ∞ i =1 is linearly independent modulo ( n + 1 . )-solvable knots. In order to prove the theorem we explore the interaction between aniterated infection procedure, strongly localized ρ -invariants and the followingtechnical condition on a bounded 4-manifold. Definition . Let K , . . . , K m be knots in S . Consider a 4-manifold W with ∂W = (cid:116) M ( K i ) and an epimorphism φ : π ( W ) (cid:16) Γ. The pair ( W, Γ)is said to satisfy condition C with respect to integers n, h (which we ab-breviate by saying that ( W, Γ) is C ( n , h )) if the following conditions hold:(C1) Γ ( n +1) r = 0 (This condition implies that Γ is PTFA).(C2) There is a normal Abelian subgroup A (cid:67) Γ such that for each i ,there is a monomorphism α i : π ( M ( K i )) π ( M ( K i )) (1) r ∼ = Z → Γ making thefollowing diagram commute, π ( M ( K i )) π ( M ( K i )) π ( M ( K i )) (1) r π ( W ) Γ (cid:47) (cid:47) (cid:15) (cid:15) (cid:127) (cid:95) (cid:15) (cid:15) α i (cid:47) (cid:47) φ , and Im( α i ) sits in A . The subgroup A does not depend on i .(C3) For any coefficient system ψ : π ( W ) → Λ with Λ ( h +1) r = 1 and anepimorphism, β , making the following diagram commute π ( W ) ΓΛ (cid:47) (cid:47) φ (cid:31) (cid:31) (cid:63)(cid:63)(cid:63)(cid:63)(cid:63)(cid:63)(cid:63)(cid:63)(cid:63)(cid:63)(cid:63) ψ (cid:79) (cid:79) (cid:79) (cid:79) (cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31) β and for any Ore localization Q [Λ] S − of Q [Λ],Ker (cid:0) H ( ∂W ; Q [Λ] S − ) → H ( W ; Q [Λ] S − ) (cid:1) is isotropic with respect to the Blanchfield form Bl Λ : H ( ∂W ; Q [Λ] S − ) × H ( ∂W ; Q [Λ] S − ) → K (Λ) Q [Λ] S − . (C4) For any PTFA coefficient system ψ : π ( W ) → Θ on W , withΘ ( h +2) r = 0, σ (2) ( W ; ψ ) − σ ( W ) = 0 If such a pair ( W, Γ) exists then we say that (cid:116) M ( K i ) bounds a C ( n, h ). Remark . Notice that if n ≤ h , then taking Λ = Γ, β to be the identitymap and A to be the Abelian group of condition (C2), then condition ( C p ,Ker (cid:0) H ( ∂W ; Q [Γ] S p ( A ) − ) → H ( W ; Q [Γ] S p ( A ) − ) (cid:1) is isotropic with respect to the Blanchfield form on H ( ∂W ; Q [Γ] S p ( A ) − ).Notice that ( h. Lemma 7.3. If W is an h. solution for K and φ : π ( W ) → Z is theAbelianization map then ( W, Z ) is C (0 , h − .Proof. Condition (C1) holds since Z is torsion free Abelian. Condition (C2)holds since by definition of h. W being an ( h. h )-solution and so is an ( h )-null-bordism. Condition (C4) follows from [10,Theorem 4.2]. (cid:3) The action of connected sum and infection on Condition C are providedby the following lemmas. Their proofs are delayed until subsection 7.1. Lemma 7.4.
Let K i = K i, . . . K i,m i for i = 1 , . . . , p . If p (cid:116) i =1 M ( K i ) bounds a C ( n, h ) , ( W, Γ) , then p (cid:116) i =1 m i (cid:116) j =1 M ( K i,j ) bounds a C ( n, h ) . Lemma 7.5.
For ≤ i ≤ m , let K i be a knot, R i be a slice knot and η i be an unknotted curve representing a doubly anisotropic element of A ( R i ) .If m (cid:116) i =1 M ( R i ( η i , K i )) bounds a C ( n, h ) with n ≤ h , then m (cid:116) i =1 M ( K i ) bounds a C ( n + 1 , h ) . We now combine the three lemmas above to discover a relationship be-tween the result of iterated infection being solvable and zero surgery on thedeepest infecting knots cobounding a 4-manifold satisfying condition C . Lemma 7.6.
For integers ≤ i ≤ m and ≤ j ≤ n let R i,j be a slice knotwith doubly anisotropic curve η i,j . For ≤ i ≤ m let K i be a knot. Let K i,j be recursively defined by K i, = K i and K i,j = R i,jη i,j ( K i,j − ) . If m i =1 K i,n is( h. )-solvable for h ≥ n , then m (cid:116) i =1 M ( K i ) bounds a 4-manifold with coefficientsystem Γ such that ( W, Γ) is C ( n, h − .Proof. If m i =1 K i,n is ( h. M (cid:18) m i =1 K i,n (cid:19) bounds a C (0 , h − m (cid:116) i =1 M (cid:0) K i,n (cid:1) bounds a C (0 , h − AND INFECTION. 19
Now, applying Lemma 7.5 (if h − ≥
0) implies that m (cid:116) i =1 M (cid:0) K i,n − (cid:1) bounds a C (1 , h − h − ≥
1) implies that m (cid:116) i =1 M (cid:0) K i,n − (cid:1) bounds a C (2 , h − n times (provided that h − ≥ n −
1) gives that m (cid:116) i =1 M (cid:0) K i,n − n (cid:1) = m (cid:116) i =1 M ( K i ) bounds a C ( n, h − (cid:3) Next, we prove that the (cid:101) ρ p -invariant is an obstruction to a disjoint unionof zero surgeries on knots bounding a 4-manifold satisfying condition C . Lemma 7.7.
Let p ∈ Q [ t ± ] be a polynomial. If { K i } mi =1 is a set of knotssuch that for each i , K i decomposes as a connected sum of p -anisotropicknots and m (cid:116) i =1 M ( K i ) bounds a 4-manifold W with coefficient system Γ suchthat ( W, Γ) is C ( n, h ) with n ≤ h , then m (cid:88) i =1 (cid:101) ρ p ( K i ) = 0 .Proof. We first prove the lemma in the more restrictive setting that each K i is strongly p -anisotropic. By (C2), the coefficient system φ : π ( W ) → Γrestricted to the M ( K i )-boundary component factors nontrivially throughthe Abelianization of π ( M ( K i )) and by (C4) the associated signature defectis zero. Thus, (cid:88) i ρ ( K i ) = 0 . While noteworthy, this is not the desired conclusion. We find another coeffi-cient system on W which, when restricted to each M ( K i ) boundary compo-nent, factors injectively through the quotient of π ( M ( K i )) by π ( M ( K i )) (2) (cid:101) p .By Remark 7.2 P = Ker( H ( M ( K i ); Q [Γ] S p ( A ) − ) → H ( W ; Q [Γ] S p ( A ) − ))is isotropic.Since π ( M ( K )) → Γ factors nontrivially through the Abelianization, [17,Lemma 3.2 and Proposition 3.6] reveals that H ( M ( K i ); Q [Γ] S p ( A ) − ) ∼ = H ( M ( K i ); (cid:102) R p ) ⊗ (cid:102) R p Q [Γ] S p ( A ) − , and for any a ⊗ α and b ⊗ β in H ( M ( K i ); (cid:102) R p ) ⊗ (cid:102) R p Q [Γ] S p ( A ) − , Bl Γ ( a ⊗ α, b ⊗ β ) = α Ψ( (cid:103) Bl p ( a, b )) β, where Ψ : Q ( t ) (cid:102) R p → K (Γ) (cid:102) R p is induced by the map α i of condition (C2). Thus,we can think of P as an isotropic submodule of (cid:102) A p ( K ) ⊗ Q [Γ] S p ( A ) − withrespect to the bilinear form induced by (cid:103) Bl p By applying Theorem 6.1, we see that the kernel of the map(7.1) (cid:102) A p ( K i ) → (cid:102) A p ( K i ) ⊗ (cid:102) R p Q [Γ] S p ( A ) − = H ( W ; Q [Γ] S p ( A ) − ) , which is equal to (cid:110) m ∈ (cid:102) A p ( K i ) | m ⊗ ∈ P (cid:111) is isotropic. We assume that (cid:102) A p ( K i ) has no nontrivial isotropy so (7.1) is injective.Now we build a new coefficient system on W . Let G := Ker ( φ : π ( W ) → Γ),so that G is isomorphic to the fundamental group of (cid:102) W Γ , the Γ-cover of W .Let G (1) p ⊆ G be given by the kernel of the following composition: F : G ∼ = → π ( (cid:103) W Γ ) → H ( (cid:103) W Γ ; Z ) Z -torsion (cid:44) → H ( W ; Q [Γ]) → H ( W ; Q [Γ] S p ( A ) − )Observe that G (1) p is normal in π ( W ). In order to see this, let g ∈ G (1) p and γ ∈ π ( W ). Then F ( γ − gγ ) is given by letting γ ∗ , the deck translation on (cid:103) W Γ corresponding to γ , act on F ( g ) ∈ H ( W ; Q [Γ] S p ( A ) − ). Since F ( g ) = 0,it follows that F ( γ − gγ ) = γ ∗ ( F ( g )) = 0, so γ − gγ is in G (1) p .Since (C1) gives us that π ( M ( K i )) → Γ factors through Abelianization,it follows that φ is trivial on π ( M ( K i )) (1) and the map induced by inclusionsends π ( M ( K i )) (1) to G . Consider the following commutative diagram:(7.2) π ( M ( K i )) (1) π ( M ( K i )) (1) π ( M ( K i )) (2) (cid:101) p (cid:102) A p ( K ) G GG (1) p H ( W ; Q [Γ] S p ( A ) − ) (cid:47) (cid:47) a (cid:15) (cid:15) i ∗ (cid:31) (cid:127) (cid:47) (cid:47) b (cid:15) (cid:15) β (cid:127) (cid:95) (cid:15) (cid:15) c (cid:47) (cid:47) e (cid:31) (cid:127) (cid:47) (cid:47) d The dotted map, β , is induced by i ∗ . In order to see that it is welldefined, one must check that i ∗ maps π ( M ( K i )) (2) (cid:101) p to G (1) p . In orderto see this, take x ∈ π ( M ( K i )) (2) (cid:101) p . It follows that c ( b ( a ( x ))) is zero in H ( W ; Q [Γ] S p ( A ) − ). By the commutativity of the diagram, d ( e ( i ∗ ( x ))) iszero in H ( W ; Q [Γ] S p ( A ) − ) and so i ∗ ( x ) ∈ Ker( e ) = G (1) p . Thus, the map β is well defined.The maps b and d in (7.2) are injections by the definition of π ( M ( K )) (2) (cid:101) p and G p . The map c is the monomorphism in (7.1). Thus, β is injectiveIf x is in the kernel of the composition(7.3) π ( M ( K i )) i ∗ → π ( W ) → π ( W ) G (1) p , AND INFECTION. 21 then i ∗ ( x ) is in G (1) p ⊆ G := Ker( π ( W ) → Γ). Since the map from π ( M ( K i )) to Γ factors nontrivially through Abelianization, it must be that x is in π ( M ( K )) (1) . This means that x ∈ Ker (cid:32) π ( M ( K i )) (1) → i ∗ G → GG (1) p (cid:33) .By the commutativity of (7.2) and the injectivity of β , the kernel of this mapis π ( M ( K i )) (2) (cid:101) p .If we set Θ := π ( W ) G (1) p then the following commutative diagram holds foreach M ( K i )-boundary component: π ( M ( K i )) π ( M ( K i )) π ( M ( K i )) (2) (cid:101) p π ( W ) Θ . (cid:15) (cid:15) (cid:47) (cid:47) (cid:127) (cid:95) (cid:15) (cid:15) (cid:47) (cid:47) This implies that m (cid:88) i =1 (cid:101) ρ p ( K ) = σ ( W, Θ) − σ ( W ). It remains only to checkthat this signature defect is zero.In order to apply condition (C4) to get this conclusion, consider the fol-lowing short exact sequence:0 → GG (1) p → π ( W ) G (1) p → π ( W ) G → GG (1) p , is TFA. The rightmost term, π ( W ) G , injects intoΓ, so that (cid:18) π ( W ) G (cid:19) ( n +1) r = 0 since Γ ( n +1) r = 1. This implies thatΘ ( n +2) r = (cid:32) π ( W ) G (1) p (cid:33) ( n +2) r = 0and condition (C4) applies to give that σ ( W, Θ) − σ ( W ) = 0. This completesthe proof in the case that each K i is p -anisotropic.In order to see it under the weaker assumption that each K i has onlystrongly p -anisotropic factors, suppose that K i = B i b =1 J i,b with each J i,b strongly p -anisotropic. An application of Lemma 7.4 gives that m (cid:116) i =1 B i (cid:116) b =1 M ( J i,b ) . bounds a C ( n, h ). Now we can apply the theorem in the case already provento see that m (cid:88) i =1 (cid:32) B i (cid:88) b =1 (cid:101) ρ p ( J i,b ) (cid:33) = 0By Proposition 4.3, B i (cid:88) b =1 (cid:101) ρ p ( J i,b ) = (cid:101) ρ p ( B i b =1 J i,b ) = (cid:101) ρ p ( K i ). Making this substi-tution completes the proof. (cid:3) We are now ready to prove Theorem 1.1. We prove a stronger theoremfrom which we get it as a corollary.
Theorem 7.8.
Let p be a polynomial. Let { K i } be a possibly infinite setof knots each of which decomposes into a connected sum of strongly p -anisotropic knots. For i = 1 , , . . . and j = 1 , , . . . , n let R i,j be a sliceknot and η i,j be an unknotted curve in the complement of R i,j representinga doubly anisotropic element of A ( R i,j ) .Let K i = K i and K ji = R i,j ( η i,j , K j − i ) .If m i =1 a i K ni is ( n + 1 . )-solvable then m (cid:88) i =1 a i (cid:101) ρ p ( K i ) = 0 .Proof. Suppose that m i =1 a i K ni were ( n +1 . m (cid:116) i =1 (cid:18) a i (cid:116) k =1 M ( K i ) (cid:19) bounds a C ( n, n ) so by Lemma 7.7 (cid:88) i a i (cid:101) ρ p ( K i ) = 0. (cid:3) Proof of Theorem 1.1.
Suppose that some linear combination m j =1 a i K i is ( n +1 . p be the Alexander polynomial of K i . For j (cid:54) = i K j has Alexander polynomial strongly coprime to p . By Corollary 5.2, K j isstrongly p -anisotropic for all j . Theorem 7.8 gives that(7.4) (cid:88) j a j (cid:101) ρ p ( K j ) = 0Proposition 4.2 applies to give that (cid:101) ρ p ( K i ) = ρ ( K i ) (cid:54) = 0 and that for j (cid:54) = i , (cid:101) ρ p ( K j ) = ρ ( K j ) = 0. plugging these into (7.4) yields a i ρ ( K i ) = 0 so a i = 0.Since the choice of i was arbitrary, a i = 0 for all i and there are nonontrivial linear relationships amongst these knots modulo n +1 . (cid:3) Proofs of Lemmas 7.4 and 7.5.
Before we prove these two importantlemmas we discuss the cobordisms used to prove them AND INFECTION. 23
Definition . For knots K , . . . , K n , let V = V ( K , . . . , K n ) be thecobordism between (cid:116) M ( K i ) and M ( K i ) constructed by starting with n (cid:116) i =1 M ( K i ) × [0 ,
1] and connecting it by gluing together neighborhoods ofcurves in M ( K i − ) × { } and M ( K i ) × { } representing the meridians of K i − and K i . Definition . Consider knots K and J and an unknotted curve, η , in S − K which has zero linking with K . Let V inf = V inf ( K, η, J ) be thecobordism between M ( K ) (cid:116) M ( J ) and M ( K η ( J )) given by starting with M ( K ) × I (cid:116) M ( J ) × I and gluing a neighborhood of η in M ( K ) × { } to aneighborhood of the meridian of J in M ( J ) × { } .By virtue of the fact that the inclusion induced maps H ( M ( K )) ⊕ H ( M ( J )) → H ( V ) and H ( M ( R )) ⊕ H ( M ( K )) → H ( V inf ) are epi-morphisms, each of V and V inf are rational ( k )-null-bordisms for everynonnegative integer, k (see [7, Definition 5.1]).The following is a key result about rational ( k )-null-bordisms. (For an R module M , T ( M ) denotes the R -torsion part of M .) Theorem ( [7, Theorem 6.3] ) . Suppose W is a rational ( k )-null-bordismand φ : π ( W ) → Γ is a nontrivial coefficient system where Γ is a PTFAgroup with Γ ( k ) = 1 . Let R be an Ore localization of Z [Γ] so Z [Γ] ⊆ R ⊆K (Γ) . Suppose that for each component M i of ∂W on which φ is nontrivial, Rank Z [Γ] ( H ( M i ; Z [Γ])) = β ( M i ) − . Then if P is the kernel of the inclu-sion induced map T ( H ( ∂W ; R )) → T ( H ( W ; R )) then P is isotropic withrespect to the Blanchfield form on T ( H ( ∂W ; R )) . Now, for any PTFA group, Γ, Γ ( k ) = 1 for some k . Since the cobordismsin which we are interested are ( k )-null bordisms for every k , this condi-tion imposes only the restriction that Γ be PTFA in our setting. Sincethe components of V inf and V are all given by zero surgery along knotsThe condition that Rank Z [Γ] ( H ( M i ; Z [Γ])) = β ( M i ) − M i by [6, Proposition 3.10].In the case of V , the meridian of any one of the components normallygenerates π ( V ) so that if φ is nontrivial on V then it is nontrivial onevery boundary component. In the case of V inf , the meridians of R and R η ( K ) each normally generate π ( V inf ) so that φ is nontrivial on the M ( R )and M ( R η ( K )) boundary components Proposition 7.11.
Consider knots K and J . Let η be an unknotted curvein the complement of K . Let V be either V ( K, J ) or V inf ( K, η, J ) Given anontrivial PTFA coefficient system φ : π ( V ) → Γ and S ⊆ Q [Γ] a rightdivisor set which is closed under the involution on Q [Γ] . Let R = Q [Γ] S − .Then(1) σ (2) ( V, Γ) = σ ( V ) = 0 ,(2) Ker ( H ( ∂V ; R ) → H ( V R )) is isotropic with respect to Bl Γ . (3) If V = V inf and φ is nontrivial on M ( J ) , then Ker ( H ( ∂V inf ; R ) → H ( V ; R )) is isotropic with respect to Bl Γ .(4) If φ is trivial on M ( J ) , then Ker ( H ( M ( K ( η, J )); R ) ⊕ H ( M ( K ); R ) → H ( V inf ; R )) is isotropic with respect to Bl Γ .Proof. By [7, Theorem 5.9], σ (2) ( W, Γ) = σ ( W ) = 0.The remaining conclusions are immediate consequences of [7, Theorem6.3]. Conclusions (2) and (3) follow since H ( ∂V ; R ) is torsion in thesecases. Conclusion (4) holds since if φ is trivial on M ( J ), then the torsionpart of H ( ∂V inf ; R ) is H ( M ( K ( η, J )); R ) ⊕ H ( M ( K ); R ). (cid:3) Finally, we prove the technical lemmas needed in the proof of Lemma 7.6.For convenience we restate them as we prove them.
Lemma 7.4.
Let K i = K i, . . . K i,m i for i = 1 , . . . , p . If p (cid:116) i =1 M ( K i ) bounds a C ( n, h ) , ( W, Γ) , then p (cid:116) i =1 m i (cid:116) j =1 M ( K i,j ) bounds a C ( n, h ) .Proof. Construct a new 4-manifold, (cid:99) W , by gluing to the M ( K i ) = M ( K i, . . . K i,m i )boundary component of W a copy of V ( K i, , . . . , K i,m i ) which we call V i .Do this for each i . The resulting 4-manifold has boundary given by p (cid:116) i =1 m i (cid:116) j =1 M ( K i,j ). Since the map i ∗ : H ( M ( K i )) → H ( V i ) is an isomorphism, onecan use Condition (C2) to conclude that φ : π ( W ) → Γ extends over π ( V i )for each i . Specifically, if α i is the monomorphism which exists since ( W, Γ)satisfies (C2), one can define the extension of φ to π ( V i ) by the composition: π ( V i ) → H ( V i ) i − ∗ → H ( M ( K i )) α i (cid:44) → ΓWe claim that ( (cid:99)
W ,
Γ) is a C ( k, n ). Since the underlying group, Γ, did notchange, Condition (C1) still holds. In order to see Condition (C2) consider AND INFECTION. 25 the following diagram: π ( M ( K i,j )) H ( M ( K i,j )) π ( V ) H ( V ) π ( M ( K i )) H ( M ( K i )) π ( W ) Γ (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) ∼ = (cid:47) (cid:47) (cid:15) (cid:15) i − ∗ ∼ = (cid:79) (cid:79) (cid:47) (cid:47) (cid:15) (cid:15) (cid:127) (cid:95) (cid:15) (cid:15) α i (cid:47) (cid:47) The composition of the maps on the right hand column is the requiredmonomorphism. Its image is contained in the Abelian subgroup, A , givenby the fact that ( W, Γ) satisfies (C2).We now use Proposition 7.11 (2) to show Condition (C3). Let ψ : π ( (cid:99) W ) → Λ and S be as in the statement of Condition (C3). If x and y are in P := Ker (cid:18) ⊕ i,j H ( M ( K i,j ); Q [Λ] S − ) → H ( (cid:99) W ; Q [Λ] S − ) (cid:19) then there exist x (cid:48) and y (cid:48) in P (cid:48) := Ker (cid:18) ⊕ i H ( M ( K i ); Q [Λ] S − ) → H ( W ; Q [Λ] S − ) (cid:19) such that x − x (cid:48) and y − y (cid:48) are in Q = ⊕ i Q i where Q i := Ker (cid:0) H ( ∂V i ; Q [Λ] S − ) → H ( V i ; Q [Λ] S − ) (cid:1) . Consider the equality from the sesquilinearity of the Blanchfield form,(7.5) Bl Λ ( x − x (cid:48) , y − y (cid:48) ) = Bl Λ ( x, y ) − Bl Λ ( x, y (cid:48) ) − Bl Λ ( x (cid:48) , y ) + Bl Λ ( x (cid:48) , y (cid:48) ) . Since Q is isotropic by Proposition 7.11 (2), Bl Λ ( x − x (cid:48) , y − y (cid:48) ) = 0. By as-sumption, Condition (C3) holds for ( W, Γ), so P (cid:48) is isotropic and Bl Λ ( x (cid:48) , y (cid:48) ) =0. Since x and y (cid:48) are carried by different components of ∂V , as are x (cid:48) and y , Bl Λ ( x, y (cid:48) ) = Bl Λ ( y (cid:48) , x ) = 0. Thus, (7.5) reduces to 0 = Bl Λ ( x, y ) so that P is isotropic and Condition (C3) holds.Condition (C4) holds because of Novikov additivity, since by Proposi-tion 7.11 (1), σ ( V i ) = σ (2) ( V i , Θ) = 0. This completes the proof. (cid:3)
Lemma 7.5.
For i = 1 , . . . m , let K i be a knot, R i be slice and η i represent adoubly anisotropic element of A ( R i ) . If m (cid:116) i =1 M ( R i ( η i , K i )) bounds a C ( n, h ) with n ≤ h , ( W, Γ) , then m (cid:116) i =1 M ( K i ) bounds a C ( n + 1 , h ) .Proof. To each M ( R i ( η i , K i ))-boundary component of W glue a copy of V inf ( R i , K i , η i ) which we abbreviate as V i . Do this for each i . Call theresulting 4-manifold W . It has boundary ∂W = (cid:116) i ( M ( K i ) (cid:116) M ( R i )). Foreach i , let E i be the complement of a slice disk for the slice knot R i . Let (cid:99) W be given by gluing E i to W along the M ( R i ) boundary component foreach i .Similar to the proof of Lemma 7.4, the coefficient system extends over V i ∪ E and on V i ∪ E it factors through Abelianization. Since µ i (the meridianof K i ) is isotopic in V i to η i which is nullhomologous, φ ( µ i ) is trivial. Thus, µ i ∼ = η i lifts to a curve in the Γ-cover of (cid:99) W and can be regarded as anelement of H ( (cid:99) W ; Q [Γ]).If x and y are elements of P := Ker (cid:18) ⊕ i H ( M ( R i ); Q [Γ]) → H ( W ; Q [Γ]) (cid:19) then there must exists x (cid:48) and y (cid:48) in P (cid:48) := Ker (cid:18) ⊕ i H ( M ( R i ( η i , K i )); Q [Γ]) → H ( W ; Q [Γ]) (cid:19) such that x − x (cid:48) and y − y (cid:48) are in S = ⊕ i S i where S i := Ker ( H ( M ( R i ); Q [Γ]) ⊕ H ( M ( R i ( η i , K i )); Q [Γ]) → H ( V i ; Q [Γ])) . By Proposition 7.11 (4) S is isotropic so that0 = Bl ( x − x (cid:48) , y − y (cid:48) ) = Bl ( x, y ) − Bl ( x, y (cid:48) ) − Bl ( x (cid:48) , y ) + Bl ( x (cid:48) , y (cid:48) ) . By remark 7.2, P (cid:48) is isotropic so that Bl ( x (cid:48) , y (cid:48) ) = 0. Since x and y (cid:48) as wellas x (cid:48) and y sit in different components, Bl ( x, y (cid:48) ) = Bl ( x (cid:48) , y ) = 0. Thus, Bl ( x, y ) = 0 and P is isotropic.Since the inclusion induced map π ( M ( R i )) π ( M ( R i )) (1) r → π ( M ( E i )) π ( M ( E i )) (1) r ∼ = Z isan isomorphism, it follows that if Q i = Ker( A ( R i ) → A ( E i )), thenKer ( H ( M ( R i ); Q [Γ]) → H ( E i ; Q [Γ])) = Q i ⊗ Q [Γ] . By a Mayer-Vietoris argument,Ker (cid:18) ⊕ i H ( M ( R i ); Q [Γ]) → H ( W ; Q [Γ]) (cid:19) = P + ⊕ i ( Q i ⊗ Q [Γ]) . If (cid:104) , . . . , , η j ⊗ , , . . . , (cid:105) ∈ ⊕ i H ( M ( R i ); Q [Γ]) were in P + ⊕ i ( Q i ⊗ Q [Γ]),then there would exist some p = (cid:104) p i (cid:105) ∈ P and (cid:104) q i (cid:105) ∈ ⊕ i ( Q i ⊗ Q [Γ]) with AND INFECTION. 27 η j ⊗ p j + q j and 0 = p i + q i when i (cid:54) = j . Since q i is in the isotropicsubmodule Q i ⊗ Q [Γ] for each i and P is isotropic, this implies0 = Bl Γ ( p, p ) = (cid:88) i ( Bl Γ ( p i , p i )) = Bl Γ ( p j , p j ) + (cid:88) i (cid:54) = j Bl Γ ( q i , q i )= Bl Γ ( p j , p j )So that η j ⊗ p j + q j sits in the sum of the isotropic submodule Q j ⊗ Q [Γ] together with the isotropic submodule generated by p j . Corollary 6.4then contradicts the assumption that η j be doubly anisotorpic. Thus, µ j ∼ = η j must be nonzero in H ( (cid:99) W ; Q [Γ]) and H ( M ( K i )) maps injectively to H ( (cid:99) W ; Q [Γ]).Letting G = Ker( π ( (cid:99) W ) → Γ), define (cid:98)
Γ to be the quotient π ( W ) G (1) r where G (1) r is the first term in the rational derived series of G . Consider the fol-lowing short exact sequence,0 → GG (1) r → (cid:98) Γ → Γ → . It reveals first that (cid:98)
Γ is PTFA, since Γ is PTFA and GG (1) r is TFA. Secondly,since G is the fundamental group of the Γ cover of (cid:99) W , GG (1) r is the quotientof H ( (cid:99) W ; Z [Γ]) by its Z -torsion, into which H ( M ( K i )) was shown to inject.Thus, this choice of ( (cid:99) W , (cid:98)
Γ) satisfies (C2). It satisfies (C1) for n + 1 since (cid:98) Γ ( n +1) r sits in the TFA group GG (1) r , so that (cid:98) Γ ( k +2) r = 0.The argument that ( (cid:99) W , (cid:98)
Γ) satisfies conditions (C3) and (C4) is just as inthe proof of 7.4 with part (3) of 7.11 replacing part (2). (cid:3)
References [1] Evan Bullock and Christopher William Davis. Strong coprimality and strong irre-ducibility of Alexander polynomials.
Topology and its Applications , 2011. to appear.[2] A. J. Casson and C. McA. Gordon. Cobordism of classical knots. In `A la recherchede la topologie perdue , volume 62 of
Progr. Math. , pages 181–199. Birkh¨auser Boston,Boston, MA, 1986. With an appendix by P. M. Gilmer.[3] Jae Choon Cha. Amenable L -theoretic methods and knot concordance. Preprintavailable at http://arxiv.org/abs/1010.1058.[4] Jae Choon Cha. The structure of the rational concordance group of knots. Mem.Amer. Math. Soc. , 189(885):x+95, 2007.[5] Jae Choon Cha. Topological minimal genus and L -signatures. Algebr. Geom. Topol. ,8(2):885–909, 2008.[6] Tim D. Cochran. Noncommutative knot theory.
Algebr. Geom. Topol. , 4:347–398,2004.[7] Tim D. Cochran, Shelly Harvey, and Constance Leidy. Knot concordance and higher-order Blanchfield duality.
Geom. Topol. , 13(3):1419–1482, 2009. [8] Tim D. Cochran, Shelly Harvey, and Constance Leidy. Derivatives of knots andsecond-order signatures.
Algebr. Geom. Topol. , 10(2):739–787, 2010.[9] Tim D. Cochran, Shelly Harvey, and Constance Leidy. Primary decomposition andthe fractal nature of knot concordance.
Math Annalen , 2010. DOI: 10.1007/s00208-010-0604-5.[10] Tim D. Cochran, Kent E. Orr, and Peter Teichner. Knot concordance, Whitneytowers and L -signatures. Ann. of Math. (2) , 157(2):433–519, 2003.[11] Tim D. Cochran, Kent E. Orr, and Peter Teichner. Structure in the classical knotconcordance group.
Comment. Math. Helv. , 79(1):105–123, 2004.[12] Christopher William Davis. Von Neumann rho invariants as obstructions to tor-sion in the topological knot concordance group. 2010. Preprint available athttp://arxiv.org/abs/1010.5020.[13] Stefan Friedl. L -eta-invariants and their approximation by unitary eta-invariants. Math. Proc. Cambridge Philos. Soc. , 138(2):327–338, 2005.[14] Stefan Friedl, Constance Leidy, and Laurentiu Maxim. L -Betti numbers of planealgebraic curves. Michigan Math. J. , 58(2):411–421, 2009.[15] Shelly L. Harvey. Homology cobordism invariants and the Cochran-Orr-Teichner fil-tration of the link concordance group.
Geom. Topol. , 12(1):387–430, 2008.[16] Se-Goo Kim and Taehee Kim. Polynomial splittings of metabelian von Neumannrho-invariants of knots.
Proc. Amer. Math. Soc. , 136(11):4079–4087, 2008.[17] Constance Leidy. Higher-order linking forms for 3-manifolds. Preprint.[18] Charles Livingston. Seifert forms and concordance.
Geom. Topol. , 6:403–408 (elec-tronic), 2002.[19] Wolfgang L¨uck. L invariants of regular coverings of compact manifolds and CW-complexes. In Handbook of Geometric Topology , pages 735–817. North-Holland, Am-sterdam, 2002.[20] Joseph J. Rotman.
Advanced modern algebra , volume 114 of
Graduate Studies inMathematics . American Mathematical Society, Providence, RI, 2010. Second edition[of MR2043445].[21] Bo Stenstr¨om.
Rings of quotients . Springer-Verlag, New York, 1975. Die Grundlehrender Mathematischen Wissenschaften, Band 217, An introduction to methods of ringtheory.
Department of Mathematics, Rice UNIVERSITY
E-mail address ::