2-torsion in the n-solvable filtration of the knot concordance group
22-TORSION IN THE n -SOLVABLE FILTRATION OF THE KNOTCONCORDANCE GROUP TIM D. COCHRAN † , SHELLY HARVEY †† , AND CONSTANCE LEIDY ††† Abstract.
Cochran-Orr-Teichner introduced in [11] a natural filtration of the smooth knotconcordance group
C· · · ⊂ F n +1 ⊂ F n. ⊂ F n ⊂ · · · ⊂ F ⊂ F . ⊂ F ⊂ C , called the ( n )-solvable filtration. We show that each associated graded abelian group { G n = F n / F n. | n ∈ N } , n ≥ n = 0 , s -invariant, τ -invariant, δ -invariants and Casson-Gordon invariants. Moreovereach is slice in a rational homology 4-ball. In fact we show that there are many distinct suchclasses in G n , distinguished by their Alexander polynomials and, more generally, by the torsionin their higher-order Alexander modules. Introduction
A (classical) knot K is the image of a smooth embedding of an oriented circle in S . Twoknots, K (cid:44) → S × { } and K (cid:44) → S × { } , are concordant if there exists a proper smoothembedding of an annulus into S × [0 ,
1] that restricts to the knots on S × { , } . Let C denotethe set of (smooth) concordance classes of knots. The equivalence relation of concordancefirst arose in the early 1960’s in work of Fox, Kervaire and Milnor in their study of isolatedsingularities of 2-spheres in 4-manifolds and, indeed, certain concordance problems are knownto be equivalent to whether higher-dimensional surgery techniques “work” for topological 4-manifolds [15][28][3]. It is well-known that C can be endowed with the structure of an abeliangroup (under the operation of connected-sum), called the smooth knot concordance group. Theidentity element is the class of the trivial knot. Any knot in this class is concordant to a trivialknot and is called a slice knot. Equivalently, a slice knot is one that is the boundary of asmooth embedding of a 2-disk in B . In general, the abelian group structure of C is still poorlyunderstood. But much work has been done on the subject of knot concordance (for excellentsurveys see [19] and [37]). In particular, [11] introduced a natural filtration of C by subgroups · · · ⊂ F n +1 ⊂ F n. ⊂ F n ⊂ · · · ⊂ F ⊂ F . ⊂ F ⊂ C . Mathematics Subject Classification.
Primary 57M25; Secondary 20J. † Partially supported by NSF DMS-0706929. †† Partially supported by NSF CAREER DMS-0748458. †††
Partially supported by NSF DMS-0805867. a r X i v : . [ m a t h . G T ] A p r TIM D. COCHRAN † , SHELLY HARVEY †† , AND CONSTANCE LEIDY ††† called the ( n )- solvable filtration of C and denoted {F n } (defined in Section 3). The non-trivialityof C can be measured in terms of the associated graded abelian groups { G n = F n / F n. | n ∈ N } (here we ignore the other “half” of the filtration, F n. / F n +1 , where almost nothing is known).This paper is concerned with elements of order two in C and, more generally, with elements oforder two in G n .We will review some of the history of 2-torsion phenomena in C in the context of the n -solvable filtration. One of the earliest results concerning C was an epimorphism constructed byFox and Milnor [15] F M : C (cid:16) Z ∞ . Soon thereafter, Levine constructed an epimorphism(1.1) C (cid:16) AC ∼ = Z ∞ ⊕ Z ∞ ⊕ Z ∞ , to a group, AC , that became known as the algebraic knot concordance group . Any knot inthe kernel of (1.1) is called an algebraically slice knot. In terms of the n -solvable filtration,Levine’s result is [11, Remark 1.3.2, Thm. 1.1]: G ∼ = Z ∞ ⊕ Z ∞ ⊕ Z ∞ . It is known that there exist elements of order two in C that realize some of the above 2-torsioninvariants. Let K denote the mirror image of the oriented knot K , obtained as the imageof K under an orientation reversing homeomorphism of S ; and let r ( K ) denote the reverse of K , which is obtained by merely changing the orientation of the circle. Then it is knownthat K r ( K ) is a slice knot, so the inverse of [ K ] in C , denoted − [ K ], is represented by r ( K ),denoted − K . A knot K is called negative amphichiral if K is isotopic to r ( K ). It followsthat, for any negative amphichiral knot K , K K is a slice knot, since it is isotopic to K − K .Hence negative amphichiral knots represent elements of order either 1 or 2 in C . It is a conjectureof Gordon that every class of order two in C can be represented by a negative amphichiral knot[19].In fact the work of Milnor and Levine in the 1960’s resulted in a more precise statement: G ∼ = (cid:77) p ( t ) (cid:0) Z r p ⊕ Z m p ⊕ Z n p (cid:1) where the sum is over all primes p ( t ) ∈ Z [ t ] where p ( t ) . = p ( t − ) and p (1) = ± G ) admits a certain p ( t ) -primary decomposition , wherein a knot has a nontrivial p ( t )-primary part only if p ( t ) is afactor of its Alexander polynomial. (Indeed, Levine and Stoltzfus classified G by first splittingthe Witt class of the Alexander module (with its Blanchfield form) into its p ( t )-primary parts).In the 1970’s the introduction of Casson-Gordon invariants in [1][2] led to the discovery thatthe subgroup of algebraically slice knots was of infinite rank and contained infinite linearlyindependent sets of elements of order two [27][36]. In terms of the n -solvable filtration thisimplies the existence of Z ∞ ⊕ Z ∞ ⊂ G . -TORSION IN THE n -SOLVABLE FILTRATION OF THE KNOT CONCORDANCE GROUP 3 Different Z ∞ -summands were exhibited in [31][16]. More recent work of Se-Goo Kim [29] on the“polynomial splitting” properties of Casson-Gordon invariants led to a generalization analogousto the result of Milnor-Levine: (cid:77) p ( t ) Z ∞ ⊂ G . Thus there is evidence that G also exhibits a p ( t )-primary decomposition. Further strongevidence is given in [30]. Although a similar statement for the 2-torsion in G has not appeared,it is expected from combining the work of [29] and Livingston [36]. Several authors have shownthat certain knots that projected to classes of order 2 and 4 in AC are in fact of infiniteorder in C [38][39][26][20][35]. A number of papers have addressed the non-triviality of { G n } ,[18][17][31][16][11][12][13], culminating in [10] where it was shown that, for any integer n , thereexists Z ∞ ⊂ G n . Moreover the recent work [8] of the authors resulted in a generalization of the latter fact,along the lines of the Levine-Milnor primary decomposition and [30]: for each “distinct” n -tuple P = ( p ( t ) , ..., p n ( t )) of prime polynomials with p i (1) = ±
1, there is a distinct subgroup Z ∞ ∼ = I ( P ) ⊂ G n , yielding a subgroup(1.2) (cid:77) P n Z ∞ ∼ = (cid:77) P∈ P n I ( P ) ⊂ G n . Given a knot K , such an n -tuple encodes the orders of certain submodules of the sequence ofhigher-order Alexander modules of K . Thus one can distinguish concordance classes of knotsnot only by their classical Alexander polynomials, but also, loosely speaking, by their higher-order Alexander polynomials . This result indicates that G n decomposes not just according tothe prime factors of the classical Alexander polynomial, but also according to types of torsionin the higher-order Alexander polynomials.Here we show corresponding results for 2-torsion. That is, for any n ≥
2, not only will weexhibit(1.3) Z ∞ ⊂ G n , but we also will exhibit many distinct such subgroups(1.4) (cid:77) P n − Z ∞ ⊂ G n , parametrized by their Alexander polynomials and the types of torsion in the higher-orderAlexander polynomials. The representative knots are distinguished by families of von Neu-mann signature defects associated to their classical Alexander polynomials and “higher-orderAlexander polynomials”. The precise statement is given in Theorem 5.8. Each of these con-cordance classes has a negative amphichiral representative that is smoothly slice in a rationalhomology 4-ball. Thus the classical signatures and the Casson-Gordon signature-defect ob-structions [1] (indeed all metabelian obstructions) vanish for these knots [11, Theorem 9.11].In addition, the s -invariant of Rasmussen [45], the τ -invariant of Ozsv´ath-Szab´o [43], and the δ p n invariants of Manolescu-Owens and Jabuka [41][25][42] vanish on these concordance classes, TIM D. COCHRAN † , SHELLY HARVEY †† , AND CONSTANCE LEIDY ††† since each of these invariants induces a homomorphism C → Z and so must have value zeroon classes representing torsion in C . Our examples are inspired by those of Livingston, whoprovided examples that can be used to establish (1.3) in the case n = 1 [36]. His examplesare distinguished by their Casson-Gordon signature defects. Our examples are distinguished byhigher-order L (2) -signature defects. It is striking that elements of finite order can sometimesbe detected by signatures. The key observation is that, unlike invariants such as the classicalknot signatures, the s invariant, the τ -invariant, or the δ -invariants, the invariants arising fromhigher-order signature defects (including Casson-Gordon invariants) are not additive under con-nected sum. Therefore there is no reason to expect that they would vanish on elements of finiteorder.Our work is further evidence that G n exhibits some sort of primary decomposition, butwherein not only the classical Alexander polynomial, but also some higher-order Alexanderpolynomials are involved.We remark that [11] also defined a filtration, {F topn } , of the topological concordance group, C top . Since it is known, by work of Freedman and Quinn, that a knot lies in F topn if and only ifit lies in F n , all of the results of this paper apply equally well, without change, to the filtration {F topn } . Therefore, for simplicity, in this paper we will always work in the smooth category.2. The examples
Our examples are inspired by those of Livingston [36], who exhibited an infinite “linearlyindependent” set of negative amphichiral algebraically slice knots. His examples can be usedto establish the existence of the aforementioned Z ∞ ⊂ G . The Building Blocks.
Consider the knot shown on the left-hand side of Figure 2.1.Here J is an arbitrary pure two component string link [32][21]. The disk containing the letter Jf ( J ) J f ( J ) J Figure 2.1.
Families of Negative Amphichiral Knots K symbolizes replacing the trivial 2-string link by the 2-string link J . Viewing the knot diagramas being in the xy -plane ( y being vertical), the mirror image can be defined as the image -TORSION IN THE n -SOLVABLE FILTRATION OF THE KNOT CONCORDANCE GROUP 5 under the reflection ( x, y, z ) (cid:55)→ ( x, y, − z ), which alters a knot diagram by replacing all positivecrossings by negative crossings and vice-versa. Recall that the image of J under this reflectionis denoted J . We also consider a “flip” homeomorphism of S which flips over a diagram,given by rotation of 180 degrees about the y -axis or f ( x, y, z ) = ( − x, y, − z ). Note that thesehomeomorphisms commute. Special cases of the following elementary observation appearedin [36, Lemma 2.1] [37, p.326] and [4, p. 60]. Lemma 2.1.
Suppose J is an arbitrary pure two component string link. Then the knot K onthe left-hand side of Figure 2.1 is negative amphichiral. f ( J ) J f ( J ) J Figure 2.2
Proof.
The knot on the right-hand side of Figure 2.1 is a diagram for r ( K ), since it is obtainedby a reflection, in the plane of the paper, of the diagram for K , followed by a reversal of thestring orientation. Here we use that f commutes with the reflection. We claim that the resultis isotopic to K . Flipping the diagram (rotating by 180 degrees about the vertical axis in theplane of the paper), we arrive at the diagram shown on the left-hand side of Figure 2.2. Thisis identical to the original diagram of K except that the left-hand band passes under the right-hand band instead of over. But the left-hand band can be “swung” around by an isotopy assuggested in the right-hand side of Figure 2.2, bringing it on top of the other band, at whichpoint one arrives at the original diagram of K . (cid:3) The following result was shown for the figure-eight knot (the case that the string link J is asingle twist) by the first author (inspired by [14]). It was extended, by Cha, to the case that J is an arbitrary number of twists in [4, p.63]. Our contribution here is just to note that Cha’sproof suffices to prove this more general result. Lemma 2.2.
Each knot K in the family shown in Figure 2.1 is slice in a rational homology -ball. TIM D. COCHRAN † , SHELLY HARVEY †† , AND CONSTANCE LEIDY ††† Proof.
We follow the argument of [4], only indicating where our more general argument deviates.It suffices to show that the zero-framed surgery, M K , as shown on the left-hand side of Figure 2.3,is rational homology cobordant to S × S . After adding, to M K × [0 , M (cid:48) given by surgery on the 3-componentlink drawn as the solid lines on the right-hand side of Figure 2.3. Therefore M K is rationallyhomology cobordant to M (cid:48) . J J − J J
Figure 2.3
Next one shows, as follows, that this underlying 3-component link, L , is concordant to thesimple 3-component link, L shown on the right-hand side of Figure 2.4. Ignoring the framingson L , add a band as shown by the dashed lines on the right-hand side of Figure 2.3, resulting inthe 4-component link, L , shown on the left-hand side of Figure 2.4. Here − J is the image of the(unoriented) string link under the map ( x, y, z ) (cid:55)→ ( x, − y, z ). One must be careful here since J ,which is reflection in the plane of the paper, is not the correct notion of mirror image for a stringlink. Since our y -axis is the true axis of the string link (the [0 ,
1] factor in D × [0 , − J is theconcordance inverse of J in the string link concordance group [32][22], so J +( − J ) is concordantto the trivial 2-string link. Hence the link L is concordant to the 4-component link, L , thatwould result from taking J to be trivial. Capping off the right-most unknotted component of L , we arrive at the 3-component link, L , shown on the right-hand side of Figure 2.4. Thisdescribes the desired concordance from L to L . Consequently, M (cid:48) is homology cobordantto the 3-manifold described by the framed link on the right-hand side of Figure 2.4, which isknown to homeomorphic to S × S . (cid:3) In this paper we will only need the special case of these lemmas wherein the string link J consists of two twisted parallels of a single knotted arc as indicated by the examples inFigures 2.5 and 2.6. Here an m inside the rectangle indicates m full positive twists between -TORSION IN THE n -SOLVABLE FILTRATION OF THE KNOT CONCORDANCE GROUP 7 − JJ − Figure 2.4 the two strands, and the J inside the rectangle indicates that the trivial two component stringlink has been replaced by two parallel zero-twisted copies of a single knotted arc J . This isexplained more fully in Subsection 2.2. m − m Figure 2.5.
Negative Amphichiral Knots E m Proposition 2.3. If m and n are distinct positive integers then the Alexander polynomials ∆ m ( t ) of E m and ∆ n ( t ) of E n are distinct and irreducible, hence coprime.Proof. A Seifert matrix for E m with respect to the obvious basis is (cid:18) m − − m (cid:19) . TIM D. COCHRAN † , SHELLY HARVEY †† , AND CONSTANCE LEIDY ††† mJ − mJ Figure 2.6.
Families of Negative Amphichiral Knots E m ( J )Thus the Alexander polynomial of E m is ∆ m ( t ) = m t − (2 m + 1) t + m . The discriminant4 m + 1 is easily seen, for m (cid:54) = 0, to never be the square of an integer, so the roots of ∆ m ( t ) arereal and irrational. Hence ∆ m ( t ) is irreducible over Q [ t, t − ]. It follows that if ∆ m ( t ) and ∆ n ( t )had a common factor then they would be identical up to a unit. But the equations m = qn and 2 m + 1 = q (2 n + 1) imply q = 1 so m = ± n . (cid:3) Doubling Operators.
To construct knots that lie deep in the n -solvable filtration, weuse iterated generalized satellite operations.Suppose R is a knot in S and (cid:126)α = ( α , α , . . . , α m ) be an ordered, oriented, trivial link in S , that misses R , bounding a collection of oriented disks that meet R transversely as shownon the left-hand side of Figure 2.7. Suppose ( K , K , . . . , K m ) is an m -tuple of auxiliary knots.Let R (cid:126)α ( K , . . . , K m ) denote the result of the operation pictured in Figure 2.7, that is, for each α j , take the embedded disk in S bounded by α j ; cut off R along the disk; grab the cut strands,tie them into the knot K j (such that the strands have linking number zero pairwise) and reglueas shown schematically on the right-hand side of Figure 2.7. α α m . . . . . .K K m R (cid:126)α ( K , . . . , K m ) R R
Figure 2.7. R (cid:126)α ( K , . . . , K m ): Infection of R by K j along α j We will call this the result of infection performed on the knot R using the infectionknots K j along the curves α j [12]. In the case that m = 1 this is the same as the classicalsatellite construction. This construction has an alternative description. For each α j , removea tubular neighborhood of α j in S and glue in the exterior of a tubular neighborhood of K j along their common boundary, which is a torus, in such a way that (the longitude of) α j is -TORSION IN THE n -SOLVABLE FILTRATION OF THE KNOT CONCORDANCE GROUP 9 identified with the meridian, µ j , of K j and the meridian of α j is identified with the reverseof the longitude, (cid:96) j , of K j as suggested by Figure 2.8. The resulting space can be seen to behomeomorphic to S and the image of R is the new knot. (cid:96) j µ j α j RS \ K j Figure 2.8.
Infection as replacing a solid torus by a knot exteriorIt is well known that if the input knots K and K are concordant, then the output knots R α ( K ) and R α ( K ) are concordant. Thus the functions R (cid:126)α descend to C . Definition 2.4 ([9, 8]) . A doubling operator , R (cid:126)α : C × · · · × C → C is a function, as inFigure 2.7, that is given by infection on a ribbon knot R wherein, for each i , lk ( R, α i ) = 0.Often we suppress α i from the notation.These are called doubling operators because they generalize untwisted Whitehead doubling.In particular we will consider the family of doubling operators R mη ,η ( − , − ) shown in Fig-ure 2.9. Note that, since E m is negative amphichiral by Lemma 2.1, m m − m − mη η Figure 2.9.
Negative Amphichiral Doubling Operators R m ≡ E m E m R m ≡ E m E m ∼ = E m − E m , which is well known to be a ribbon knot [46, Exercise 8E.30]. Thus R m is a negative amphichiralribbon knot. For the case m = 1, this was already noted in [36]. † , SHELLY HARVEY †† , AND CONSTANCE LEIDY ††† We will also consider the family of doubling operators, R mα , shown in Figure 2.10 (wherehere the − m inside a box symbolizes m full negative twists between the bands but where theindividual bands remain untwisted), equipped with a specified circle α that can be shown togenerate its Alexander module. R m α − m Figure 2.10.
Doubling operators R mα Elements of order 2 in F n . Now we describe large families of examples of negativeamphichiral knots that lie in F n . Let K be any knot with Arf( K )= 0. Let K n − be theimage of K under the composition of any n − K n − ≡ R n − ◦ · · · ◦ R ( K ) . Then, for any integer m we define K n as in Figure 2.11, that is, K n ≡ R mη ,η ( K n − , K n − ). mK n − mK n − − m − m Figure 2.11.
The examples K n Proposition 2.5.
For any n ≥ , any m , any composition of n − doubling operators and anyArf invariant zero input knot K , the knot K n of Figure 2.11 satisfies -TORSION IN THE n -SOLVABLE FILTRATION OF THE KNOT CONCORDANCE GROUP 11 • K n is negative amphichiral; • K n ∈ F n ; • K n is (smoothly) slice in a smooth rational homology -ball; and • K n K n is a slice knot.Proof. It was shown in [10, Theorem 7.1] that, for any any doubling operator R , R ( F i , . . . , F i ) ⊂ F i +1 . Since any knot of Arf invariant zero is known to lie in F [11, Remark 8.14,Thm. 8.11], and since K n is the image of K under a composition of n doubling operators, it follows that K n ∈ F n .Note that K n is the connected sum of two knots each of which is of the form shown inFigure 2.6 (hence of the form of Figure 2.1). Thus, by Lemma 2.2, each such K n is slice in arational homology 4-ball. Moreover, by Lemma 2.1, K n is negative amphichiral so K n K n isisotopic to K n r ( K n ). But the latter is a ribbon knot and hence a slice knot. (cid:3) For specificity we define the following infinite families:
Definition 2.6.
Given an n -tuple ( m , ..., m n ) of integers and an Arf invariant zero knot K ,we define K n ( m , . . . , m n , K ) to be the image of K under the following composition of n doubling operators. Specifically let K n ≡ K n ( m , . . . , m n , K ) ≡ R m n η ,η ( K n − , K n − ) , as shown in Figure 2.11, where K n − is R m n − ◦ · · · ◦ R m ( K ) , where the R m i are the operators of Figure 2.10. In other words, we recursively set: K = R m α ( K ); K = R m α ◦ R m α ( K );... K n − = R m n − α ◦ · · · ◦ R m α ( K ); K n = R m n η ,η ( K n − , K n − ) . Even though K n depends on ( m , ..., m n , K ), we will often suppress the latter from the notation.3. Commutator Series and Filtrations of the knot concordance groups
To accomplish our goals, we must establish that many of the knots in the families given byFigure 2.11, and specifically those in Definition 2.6, are not in F n. and, indeed, are distinctfrom each other in F n / F n. . To this end we review recent work of the authors that introducedrefinements of the n -solvable filtration parameterized by certain classes of group series thatgeneralized the derived series. In particular the authors defined specific filtrations of C thatdepend on a sequence of polynomials. These filtrations can then be used to distinguish betweenknots with different Alexander modules or different higher-order Alexander modules. All of thematerial in this section is a review of the relevant terminology of [8, Sections 2,3]. † , SHELLY HARVEY †† , AND CONSTANCE LEIDY ††† Recall that the derived series , { G ( n ) | n ≥ } , of a group G is defined recursively by G (0) ≡ G and G ( n +1) ≡ [ G ( n ) , G ( n ) ]. The rational derived series [23], { G ( n ) r | n ≥ } , isdefined by G (0) r ≡ G and G ( n +1) r = ker (cid:32) G ( n ) r → G ( n ) r [ G ( n ) r , G ( n ) r ] → G ( n ) r [ G ( n ) r , G ( n ) r ] ⊗ Z Q (cid:33) . More generally,
Definition 3.1 ([8, Definition 2.1]) . A commutator series defined on a class of groups is afunction, ∗ , that assigns to each group G in the class a nested sequence of normal subgroups · · · (cid:67) G ( n +1) ∗ (cid:67) G ( n ) ∗ (cid:67) · · · (cid:67) G (0) ∗ ≡ G, such that G ( n ) ∗ /G ( n +1) ∗ is a torsion-free abelian group. Proposition 3.2 ([8, Proposition 2.2]) . For any commutator series { G ( n ) ∗ } , { x ∈ G ( n ) ∗ | ∃ k > , x k ∈ [ G ( n ) ∗ , G ( n ) ∗ ] } ⊂ G ( n +1) ∗ (and in particular [ G ( n ) ∗ , G ( n ) ∗ ] ⊂ G ( n +1) ∗ ,whence the name commutator series); G ( n ) r ⊂ G ( n ) ∗ , that is, every commutator series contains the rational derived series; G/G ( n ) ∗ is a poly-(torsion-free abelian) group (abbreviated PTFA); Z [ G/G ( n ) ∗ ] and Q [ G/G ( n ) ∗ ] are right (and left) Ore domains. Any commutator series that satisfies a weak functoriality condition induces a filtration, {F ∗ n } ,of C by subgroups. These filtrations generalize and refine the ( n )-solvable filtration {F n } of [11].Let M K denote the closed 3-manifold obtained by zero-framed surgery on S along K . Definition 3.3 ([8, Definition 2.3]) . A knot K is an element of F ∗ n if the zero-framed surgery M K bounds a compact, connected, oriented, smooth 4-manifold W such that1. H ( M K ; Z ) → H ( W ; Z ) is an isomorphism;2. H ( W ; Z ) has a basis consisting of connected, compact, oriented surfaces, { L i , D i | ≤ i ≤ r } , embedded in W with trivial normal bundles, wherein the surfaces are pairwisedisjoint except that, for each i , L i intersects D i transversely once with positive sign;3. for each i , π ( L i ) ⊂ π ( W ) ( n ) ∗ and π ( D i ) ⊂ π ( W ) ( n ) ∗ .A knot K ∈ F ∗ n. if in addition,4. for each i , π ( L i ) ⊂ π ( W ) ( n +1) ∗ .Such a 4-manifold is called an ( n, ∗ )-solution (respectively an ( n. , ∗ )-solution) for K and itis said that K is ( n, ∗ ) - solvable (respectively ( n. , ∗ )-solvable) via W . The case where thecommutator series is the derived series (without the torsion-free abelian restriction) is denoted F n and we speak of W being an ( n )-solution, and K or M K being ( n )-solvable via W [11,Section 8]. Definition 3.4.
A commutator series { G ( n ) ∗ } is weakly functorial (on a class of { groups,maps } ) if f ( G ( n ) ∗ ) ⊂ π ( n ) ∗ for each n and for any homomorphism f : G → π (in the class) thatinduces an isomorphism G/G (1) r ∼ = π/π (1) r (i.e. induces an isomorphism on H ( − ; Q )). -TORSION IN THE n -SOLVABLE FILTRATION OF THE KNOT CONCORDANCE GROUP 13 Proposition 3.5 ([8, Prop. 2.5]) . Suppose ∗ is a weakly functorial commutator series definedon the class of groups with β = 1 . Then {F ∗ n } n ≥ is a filtration by subgroups of the classical(smooth) knot concordance group C : · · · ⊂ F ∗ n +1 ⊂ F ∗ n. ⊂ F ∗ n ⊂ · · · ⊂ F ∗ ⊂ F ∗ . ⊂ F ∗ ⊂ C . Moreover, for any n ∈ Z F n ⊂ F ∗ n . The case where the commutator series is the derived series (without the torsion-free abelianrestriction) is the ( n )-solvable filtration [11], denoted {F n } .3.1. The Derived Series Localized at P . Fix an n -tuple P = ( p ( t ) , ..., p n ( t )) of non-zero elements of Q [ t, t − ], such that p ( t ) . = p ( t − ). For each such P we now recall from [8] the definition of a partial commutator seriesthat we call the (polarized) derived series localized at P , that is defined on the class of groupswith β = 1.Suppose G is a group such that G/G (1) r ∼ = Z = (cid:104) µ (cid:105) . Then we define the derived series localizedat P recursively in terms of certain right divisor sets S p n ⊂ Q [ G/G ( n ) P ]. Definition 3.6.
For n ≥
0, let G (0) P ≡ G ; G (1) P ≡ G (1) r ;and for n ≥ G ( n +1) P ≡ ker (cid:32) G ( n ) P → G ( n ) P [ G ( n ) P , G ( n ) P ] ⊗ Z [ G/G ( n ) P ] Q [ G/G ( n ) P ] S − p n (cid:33) . (3.1)To make sense of (3.1) one must realize that, for any H (cid:67) G , H/ [ H, H ] is a right Z [ G/H ]-module where g acts on h by h (cid:55)→ g − hg . One must also verify, at each stage, that G ( n ) P hasbeen defined in such a way that G ( k ) P /G ( k +1) P is a torsion-free abelian group for each k < n , so G/G ( n ) P is a poly-(torsion-free-abelian) group (PTFA), from which it follows that Q [ G/G ( n ) P ] isa right Ore domain. Therefore, for any right divisor set S p n ⊂ Q [ G/G ( n ) P ] we may define theOre localization Q [ G/G ( n ) P ] S − p n as in (3.1) (see [8, Sections 3,4]).For the (polarized) derived series localized at P we use the following right divisor sets: Definition 3.7.
The (polarized) derived series localized at P is defined as in Definition 3.6by setting S p = S p ( G ) = { q ( µ ) ...q r ( µ ) | ( p ( t ) , q j ( t )) = 1; G/G (1) r ∼ = (cid:104) µ (cid:105)} ⊂ Q [ G/G (1) r ];(3.2)and for n ≥ S p n = S p n ( G ) = { q ( a ) ...q r ( a r ) | (cid:94) ( p n , q j ) = 1; q j (1) (cid:54) = 0; a j ∈ G ( n − P /G ( n ) P } , (3.3)so S p n ⊂ Q [ G ( n − P /G ( n ) P ]. † , SHELLY HARVEY †† , AND CONSTANCE LEIDY ††† Here p i ( t ) and q j ( t ) are in Q [ t, t − ]. By ( p , q j ) = 1 we mean that p is coprime to q j in Q [ t, t − ], as usual. But by (cid:94) ( p n , q j ) = 1 we mean something slightly stronger. Definition 3.8 ([8, Defn. 4.4]) . Two non-zero polynomials p ( t ) , q ( t ) ∈ Q [ t, t − ] are said tobe strongly coprime , denoted (cid:93) ( p, q ) = 1 if, for every pair of non-zero integers, n, k , p ( t n )is relatively prime to q ( t k ). Alternatively, (cid:93) ( p, q ) (cid:54) = 1 if and only if there exist non-zero roots, r p , r q ∈ C *, of p ( t ) and q ( t ) respectively, and non-zero integers k, n , such that r kp = r nq . Clearly, (cid:93) ( p, q ) = 1 if and only if for each prime factor p i ( t ) of p ( t ) and q j ( t ) of q ( t ), (cid:94) ( p i , q j ) = 1.Note that Q − { } ⊂ S p n (take q j to be a non-zero constant). It is easy to see (and wasproved in [8, Section 4]) that S p n is closed (up to units) under the involution on Q [ G/G ( n ) P ].Here we need p ( t ) . = p ( t − ). Example 3.9.
Consider the family of quadratic polynomials { q m ( t ) = ( mt − ( m + 1))(( m + 1) t − m ) | m ∈ Z + } , whose roots are { m/ ( m + 1) , ( m + 1) /m } . The polynomial q m is the Alexander polynomialof the ribbon knot R m shown in in Figure 2.10. It can easily be seen (and was proved in [8,Example 4.10]) that (cid:94) ( q m , q n ) = 1 if m (cid:54) = n . Theorem 3.10 ([8, Thm. 4.16]) . The (polarized) derived series localized at P is a weaklyfunctorial commutator series on the class of groups with β = 1 . von Neumann signature defects as obstructions to ( n. , ∗ ) -solvability To each commutator series there exist signature defects that offer obstructions to a givenknot lying in a term of F ∗ . Given a closed, oriented 3-manifold M , a discrete group Γ, anda representation φ : π ( M ) → Γ, the von Neumann ρ -invariant , ρ ( M, φ ) ∈ R , was definedby Cheeger and Gromov [5]. If ( M, φ ) = ∂ ( W, ψ ) for some compact, oriented 4-manifold W and ψ : π ( W ) → Γ, then it is known that ρ ( M, φ ) = σ (2)Γ ( W, ψ ) − σ ( W ) where σ (2)Γ ( W, ψ )is the L (2) -signature (von Neumann signature) of the equivariant intersection form defined on H ( W ; Z Γ) twisted by ψ , and σ ( W ) is the ordinary signature of W [40][13, Section 2]. Thus the ρ -invariants should be thought of as signature defects . They were first used to detect non-sliceknots in [11]. For a more thorough discussion see [11, Section 5][13, Section 2][12, Section 2]. Allof the coefficient systems Γ in this paper will be of the form π/π ( n ) ∗ where π is the fundamentalgroup of a space. Hence all such Γ will be PTFA. Aside from the definition, the properties thatwe use in this paper are: Proposition 4.1. If φ factors through φ (cid:48) : π ( M ) → Γ (cid:48) where Γ (cid:48) is a subgroup of Γ , then ρ ( M, φ (cid:48) ) = ρ ( M, φ ) . If φ is trivial (the zero map), then ρ ( M, φ ) = 0 . -TORSION IN THE n -SOLVABLE FILTRATION OF THE KNOT CONCORDANCE GROUP 15 If M = M K is the zero-surgery on a knot K and φ : π ( M ) → Z is the abelianization,then ρ ( M, φ ) is denoted ρ ( K ) and is equal to the integral over the circle of the Levine-Tristram signature function of K [12, Prop. 5.1] . Thus ρ ( K ) is the average of theclassical signatures of K . If K is a slice knot or link and φ : M K → Γ ( Γ PTFA) extends over π of a slice diskexterior then ρ ( M K , φ ) = 0 by [11, Theorem 4.2] . The von Neumann signature satisfies Novikov additivity, i.e. if W and W intersectalong a common boundary component then σ (2)Γ ( W ∪ W ) = σ (2)Γ ( W ) + σ (2)Γ ( W ) [11,Lemma 5.9] . For any -manifold M , there is a positive real number C M , called the Cheeger-Gromovconstant [5][13, Section 2] of M such that, for any φ | ρ ( M, φ ) | < C M . We will also need the following generalization of property (4).
Theorem 4.2 ([8, Theorem 5.2]) . Suppose ∗ is a commutator series (no functoriality is re-quired). Suppose K ∈ F ∗ n. , so the zero-framed surgery M K is ( n. , ∗ ) -solvable via W as inDefinition 3.3. Let G = π ( W ) and consider φ : π ( M K ) → G → G/G ( n +1) ∗ → Γ , where Γ is an arbitrary PTFA group. Then σ (2)Γ ( W, φ ) − σ ( W ) = 0 = ρ ( M K , φ ) . Statements of Main Results and the outline of the proof
We will show that for any n ≥
2, not only does there exist Z ∞ ⊂ G n ≡ F n / F n. , but there are also many distinct such classes (cid:77) P n − Z ∞ ⊂ G n , distinguished by the sequence of orders of certain higher-order Alexander modules of the knots.Given the sequence P = ( p ( t ) , ..., p n ( t )), we have defined (in Definitions 3.6 and 3.7) anassociated commutator series called the derived series localized at P . Definition 5.1.
Let {F P n } denote the filtration of C associated, by Definition 3.3, to the derivedseries localized at P .Since for any group G and integer n (or half-integer), G ( n ) ⊂ G ( n ) P , one sees that F n ⊂ F P n . Inparticular F n. ⊂ F P n. , so there is a surjection F n F n. π (cid:16) F n F P n. . † , SHELLY HARVEY †† , AND CONSTANCE LEIDY ††† The point of the filtration {F P n } , is that any knot K ∈ F n , whose classical Alexander polynomialis coprime to p ( t ), will lie in the kernel of π . Moreover, the idea of Theorem 5.3 below isthat a knot will of necessity lie in the kernel of π , unless p ( t ) divides its classical Alexanderpolynomial and , loosely speaking, the higher p i ( t ) are related to torsion in its i th higher-orderAlexander module. Definition 5.2.
Given P = ( p ( t ) , ..., p n ( t )) and Q = ( q ( t ) , ..., q n ( t )), we say that P is strongly coprime to Q if either ( p , q ) = 1, or for some k > (cid:94) ( p k , q k ) = 1. Theorem 5.3 ([8, Theorem 6.5]) . For any n ≥ , let R n − α n − , . . . , R α be any doubling operatorsand K be any Arf invariant zero input knot. Consider the knot K n ≡ R mη ,η ( K n − , K n − ) ,where K n − = R n − α n − ◦ · · · ◦ R α ( K ) . Then K n ∈ F P n +1 for each P = ( p ( t ) , p ( t ) , ..., p n ( t )) , with p ( t ) . = p ( t − ) , that is strongly coprime to (∆ m ( t ) , q n − ( t ) , . . . , q ( t )) , where ∆ m is the Alexander polynomial of E m and q i is the Alexanderpolynomial of R i . This applies, in particular, to the families of Definition 2.6, constructed using the ribbonknots of Figures 2.9 and 2.10.
Corollary 5.4.
For any ( m , . . . , m n ) and any input knot K with Arf invariant zero, K n ( m , . . . , m n , K ) ∈ F P n +1 for each P = ( p ( t ) , p ( t ) , ..., p n ( t )) that is strongly coprime to (∆ m n ( t ) , q n − ( t ) . . . , q ( t )) where ∆ m n is the Alexander polynomial of E m n and q i is the Alexander polynomial of R m i . Now we need a non-triviality theorem to complement Theorem 5.3.
Theorem 5.5.
Suppose K n ≡ R mη ,η ( K n − , K n − ) , where K n − is the result of applying any sequence of n − doubling operators, R n − α n − ◦ · · · ◦ R α to an Arf invariant zero “input” knot K . Suppose additionally that n ≥ and m (cid:54) = 0 ; for each i , α i generates the rational Alexander module of R i , and this module is non-trivial; | ρ ( K ) | , the average Levine-Tristram signature of K , is greater than twice the sum ofthe Cheeger-Gromov constants of the ribbon knots R m , R , . . . , R n − (see Section 4).If P is the sequence of classical Alexander polynomials of the knots ( E m , R n − , . . . , R ) , then K n / ∈ F P n. . This can be applied to the specific families of Definition 2.6. -TORSION IN THE n -SOLVABLE FILTRATION OF THE KNOT CONCORDANCE GROUP 17 Corollary 5.6.
Fix n ≥ and an n -tuple of positive integers ( m , . . . , m n ) . Suppose K ischosen so that | ρ ( K ) | is greater than twice the sum of the Cheeger-Gromov constants of theribbon knots R m n , R m n − , . . . , R m . If P is the n -tuple of Alexander polynomials of the knots ( E m n , R m n − , . . . , R m ) , then K n / ∈ F P n. . The proofs of Theorems 5.3 and 5.5 will constitute Sections 6 and 7. Assuming these theo-rems, we now derive our main results.
Theorem 5.7.
Fix n ≥ . For any n -tuple of positive integers ( m , ..., m n ) choose an Arfinvariant zero knot K ( m , ..., m n ) such that | ρ ( K ) | is greater than twice the sum of theCheeger-Gromov constants of R m n , R m n − , . . . , R m . Then the resulting set of knots {K n ( m , ..., m n , K ) | m i ∈ Z + } , as in Definition 2.6, represent linearly independent, order two elements of F n / F n. . They alsorepresent linearly independent order two elements in C . In particular this gives Z ∞ ⊂ G n ≡ F n F n. , where each class is represented by a negative amphichiral knot that is slice in a rational homology -ball.Proof of Theorem 5.7 assuming Theorems 5.3 and 5.5. By Proposition 2.5, K n is negative am-phichiral, K n ∈ F n and K n K n is a slice knot. Thus 2[ K n ] = 0 in F n / F n. . By Corollary 5.6,for a certain P , K n / ∈ F P n. , so in particular K n / ∈ F n. by Proposition 3.5. Therefore each [ K n ]has order precisely two in G n .Suppose there exists a nontrivial relation J = K n ( m , ..., m n , K ) ... K n ( m k , ..., m kn , K k ) ∈ F n. . Set P = ( p , ..., p n ) = (∆ n , q n − , ..., q ), the reverse of the sequence of Alexander polynomialsof the operators corresponding to the first summand of J . For each of the other summands of J , the corresponding n -tuple ( m i , ..., m in ) is assumed distinct from ( m , ..., m n ). Therefore,the (reversed) sequence of Alexander polynomials of the operators corresponding to this othersummand is strongly coprime to P by Proposition 2.3 and Example 3.9. Thus, by Theorem 5.3,each summand of J , aside from the first, lies in F P n +1 and hence in F P n. . Since J ∈ F n. , J ∈ F P n. , by Proposition 3.5. Since F P n. is a subgroup, it would follow that the first summandof J also lay in F P n. , contradicting Corollary 5.6. (cid:3) More generally,
Theorem 5.8.
Suppose n ≥ . Let P n be any set of n -tuples P = ( δ ( t ) , δ ( t ) , . . . , δ n ( t )) ofprime polynomials δ i ( t ) ∈ Z [ t, t − ] such that δ i (1) = ± , δ ( t ) = ∆ m = m t − (2 m + 1) t + m and with the property that, for any distinct P , P (cid:48) ∈ P n , P and P (cid:48) are strongly coprime. † , SHELLY HARVEY †† , AND CONSTANCE LEIDY ††† Then there exists a set of negative amphichiral n -solvable knots indexed by P n that is linearlyindependent modulo F n. , that is, that spans (cid:77) P n Z ⊂ G n , where the knot corresponding to the sequence ( δ ( t ) , δ ( t ) , . . . , δ n ( t )) admits a sequence of higher-order Alexander modules containing submodules whose orders are determined by the sequence ( δ ( t ) δ ( t − ) , . . . , δ n ( t ) δ n ( t − )) with the classical Alexander polynomial being δ ( t ) δ ( t − ) . More-over each class is represented by a negative amphichiral knot that is slice in a rational homology -ball.Proof of Theorem 5.8 assuming Theorems 5.3 and 5.5. By [49], for any prime δ ( t ) with δ (1) = ± δ ( t ) δ ( t − ). Hence,given P = ( δ ( t ) , δ ( t ) , . . . , δ n ( t )), choose such ribbon knots R n − , . . . , R whose Alexanderpolynomials are δ ( t ) δ ( t − ) , . . . , δ n ( t ) δ n ( t − ) respectively, and choose curves α i (unknotted in S ), that generate the Alexander modules of the R i . Thus doubling operators R iα i , 1 ≤ i ≤ n − δ ( t ) = ∆ m = m t − (2 m + 1) t + m , there is a ribbon knot, namely R m = E m E m of Figure 2.9, whose Alexander polynomial is δ ( t ) δ ( t − ). The hypothesesimply m (cid:54) = 0. Choose any Arf invariant zero knot K such that | ρ ( K ) | is greater than twicethe sum of the Cheeger-Gromov constants of R m , R n − , . . . , R . Then set(5.1) K n P ≡ R mη ,η ( K n − , K n − ) , where K n − ≡ R n − α n − ◦ · · · ◦ R α ( K ). To each P there is an associated n -tuple, P ∗ =( δ , δ ( t ) δ ( t − ) , . . . , δ n ( t ) δ n ( t − )), that gives the sequence of Alexander polynomials of theknots E m , R n − , . . . , R that define K n P .By Lemma 2.1 and Proposition 2.5, each K n P is negative amphichiral and n -solvable. ByTheorem 5.5,(5.2) K n P / ∈ F P ∗ n. , so K n P / ∈ F n. . Thus [ K n P ] has order precisely two in G n . Suppose there were a non-trivialrelation J = k (cid:88) i =1 K n P i ∈ F n. . By hypothesis, if i (cid:54) = 1 then P i is strongly coprime to P . It follows that P ∗ i is strongly coprimeto P ∗ . Thus, by Theorem 5.3, if i (cid:54) = 1 then K n P i ∈ F P ∗ n +1 ⊂ F P ∗ n. . Since J ∈ F n. , J ∈ F P ∗ n. . Since the latter is a subgroup, K n P ∈ F P ∗ n. , contradicting (5.2).It remains only to relate the sequence P to the higher-order Alexander modules of the knots K n P . Since this is not central to our results, we sketch the proof. Recall: -TORSION IN THE n -SOLVABLE FILTRATION OF THE KNOT CONCORDANCE GROUP 19 Definition 5.9 ([6, Def. 2.8][23, Def. 5.3]) . The i th , i ≥ higher-order (integral) Alexandermodule of a knot K is A Z i ( K ) ≡ H ( M K ; Z [ G/G ( i +1) r ]) ∼ = G ( i +1) r [ G ( i +1) r , G ( i +1) r ] , where G ≡ π ( M K ). Note: The case i = 0 would give the classical Alexander module.Thus A Z i ( K n P ) is a module over Γ i ≡ G/G ( i +1) r , where G ≡ π ( M K n P ). The following lemmashows that the (two) images of the classical Alexander polynomial, δ i +1 ( t ) δ i +1 ( t − ), of theconstituent operator R n − i under certain maps Z [ t, t − ] → Z [ G ( i ) /G ( i +1) r ] ⊂ Z Γ i , wherein t (cid:55)→ x and t (cid:55)→ x , appear as the orders of cyclic submodules of A Z i ( K n P ). Lemma 5.10.
Fixing P = ( δ ( t ) , δ ( t ) , . . . , δ n ( t )) , for each ≤ i ≤ n − , the i th higher-orderAlexander module of K n P (the knot defined in (5.1) ) contains two non-trivial summands Z Γ i δ i +1 ( x ) δ i +1 ( x − ) Z Γ i ⊕ Z Γ i δ i +1 ( x ) δ i +1 ( x − ) Z Γ i for certain x , x ∈ G ( i ) /G ( i +1) r .Proof. Recall that K n P is defined as the image of K under a composition of n doubling operators.In particular K n − ≡ R n − α n − ◦ · · · ◦ R α ( K ). Sequences of satellite operations have a certainassociativity property yielding, for each i ≥
2, an alternative description of K n − as a single infection on single ribbon knot, ˜ R i , along a curve lying in π ( S − ˜ R i ) ( i − , using the knot K n − i [7, Prop. 4.7][9, Prop. 5.10]. Specifically, K n − = R n − α n − ◦ · · · ◦ R n − i +1 α n − i +1 (cid:16) R n − iα n − i · · · ◦ R α ( K ) (cid:17) K n − = R n − α n − ◦ · · · ◦ R n − i +1 α n − i +1 ( K n − i ) K n − = (cid:16) R n − α n − ◦ · · · ◦ R n − i +2 α n − i +2 ( R n − i +1 α n − i +1 ) (cid:17) β i (cid:0) K n − i (cid:1) K n − = ˜ R iβ i ( K n − i ) , where ˜ R iβ i ≡ R n − α n − ◦ · · · ◦ R n − i +2 α n − i +2 ( R n − i +1 α n − i +1 )and β i is the image of α n − i +1 . The specific nature of ˜ R i is not important to our presentconsiderations. If i = 1, let ˜ R iβ i be the identity operator. Then, for any i ≥
1, it follows that K n = R mη ,η (cid:16) ˜ R iβ i ( K n − i ) , ˜ R iβ i ( K n − i ) (cid:17) . This can be reformulated, by the same considerations as above, to yield K n = ˜ R γ ,γ (cid:16) K n − i , K n − i (cid:17) † , SHELLY HARVEY †† , AND CONSTANCE LEIDY ††† where ˜ R = R mη ,η ( ˜ R i , ˜ R i ) and { γ , γ } are the images of the two copies of β i . These curves caninductively shown to lie in π ( S − ˜ R ) ( i ) [7, Prop. 4.7][9, Prop. 5.10]. The latter computationis very similar to the computation we will perform in (7.15).Now we can apply known results about the effect of single infection on the higher-orderAlexander modules [33, Theorem 3.5][6, Theorem 8.2]: A Z i ( K n ) = A Z i ( ˜ R ) ⊕ (cid:16) A Z ( K n − i ) ⊗ Z [ t,t − ] Z Γ i (cid:17) ⊕ (cid:16) A Z ( K n − i ) ⊗ Z [ t,t − ] Z Γ i (cid:17) . where A Z denotes the classical Alexander module and the first tensor product is given by t (cid:55)→ x = γ and the second by t (cid:55)→ x = γ . But A Z ( K n − i ) ∼ = A Z ( R n − i ) ∼ = Z [ t, t − ] δ i +1 ( t ) δ − i +1 ( t ) Z [ t, t − ] . where t (cid:55)→ x . The Alexander modules of R n − i and R n − i are isomorphic. Thus A Z i ( K n ) containstwo cyclic summands as claimed. By [33, Theorem 3.5][6, Theorem 8.2] these summands arenon-zero precisely when x and x are not zero in Γ i . The verification of the latter requiresfurther computation as in [7, Theorem 4.11][9, Propoposition 5.14]. These calculations areentirely similar to and easier than the ones we will do to verify our Proposition 7.4. They arenot included.This concludes what we will say about the connections between P and the orders of thehigher-order Alexander modules of K n P . (cid:3) This concludes the proof of Theorem 5.8. (cid:3) Sketch of Proof of Theorem 5.3
Theorem 5.3 is a consequence of [8, Theorem 6.5]. However, we shall sketch the proof sincethe basic idea is elementary and it also shows that K n ∈ F n . Theorem 5.3 ([8, Theorem 6.5]) . For any n ≥ m ∈ Z , let R n − α n − , . . . , R α be anydoubling operators and K be any Arf invariant zero input knot. Consider the knot K n ≡ R mη ,η ( K n − , K n − ), where K n − = R n − α n − ◦ · · · ◦ R α ( K ). Then K n ∈ F P n +1 for each P = ( p ( t ) , p ( t ) , ..., p n ( t )), with p ( t ) . = p ( t − ), that is strongly coprime to(∆ m ( t ) , q n − ( t ) , . . . , q ( t )), where ∆ m is the Alexander polynomial of E m and q i is the Alexanderpolynomial of R i . Proof of Theorem 5.3.
We set K = R ( K ) , . . . , K i = R i ( K i − ) for i = 1 , . . . , n − K n = R m ( K n − , K n − ). Recall from [10, Lemma 2.3, Figure 2.1] that, whenever a knot L is obtained from a knot R by infection using knots K , K , . . . there is a cobordism E whoseboundary is the disjoint union of the zero surgeries M L , − M R and − M K , − M K et cetera,as shown on the left-hand side of Figure 6.1. Therefore, since K n = R m ( K n − , K n − ), thereis a cobordism E n whose boundary is the disjoint union of the zero surgeries on K n , K n − , -TORSION IN THE n -SOLVABLE FILTRATION OF THE KNOT CONCORDANCE GROUP 21 M R M L M K M K M K n − M K n − M K n M R m E n ≡ E ≡ Figure 6.1.
The cobordism K n − and R m as shown on the right-hand side of Figure 6.1 and schematically in Figure 6.2.Similarly there is a cobordism E i , for 1 ≤ i < n whose boundary is the disjoint union of the zerosurgeries on K i , K i − and R i . Consider X = E n ∪ E n − ∪ E n − ∪ ... ∪ E ∪ E , gluing E i to E i − along their common boundary component M K i − , and gluing E i to E i − along their commonboundary component M K i − , as shown schematically in Figure 6.2. The boundary of X is adisjoint union of M K n , − M R m , − M K , − M K and two copies each of ± M R n − , ..., ± M R . For1 ≤ i < n , let S i denote the exterior of any ribbon disk in B for the ribbon knot R i . Let S n denote the exterior of any ribbon disk in B for the ribbon knot R m . Since Arf( K )= 0, K ∈ F via some V [12, Section 5]. Gluing V , V = − V and all the S i and S i to X , we obtaina 4-manifold, Z as shown in Figure 6.2. Note ∂Z = M K n . We claim that,(6.1) K n ∈ F n via Z, and if P is strongly coprime to (∆ m ( t ) , q n − ( t ) , . . . , q ( t )), then(6.2) K n ∈ F P n +1 via Z. First, simple Mayer-Vietoris sequences together with an analysis of the homology of the E i (as given by Lemma 6.1 below) imply that H ( Z ) ∼ = H ( V ) ⊕ H ( V ) since H ( S i ) = 0. Since V is a 0-solution, H ( V ) has a basis of connected compact oriented surfaces, { L j , D j | ≤ j ≤ r } ,satisfying the conditions of Definition 3.3. Similarly for H ( V ). We claim that,(6.3) π ( V ) ⊂ π ( Z ) ( n ) and if P is strongly coprime to (∆ m ( t ) , q n − ( t ) , . . . , q ( t )) then(6.4) π ( V ) ⊂ π ( Z ) ( n +1) P . † , SHELLY HARVEY †† , AND CONSTANCE LEIDY ††† ...... M R m S n E n M K n S n − S n − M R n − M R n − M K n − M K n − E n − E n − S S → VV E E M K n − M K n − Figure 6.2. Z Indeed equations (6.3) and (6.4) were shown inductively in the proof of [8, Theorem 6.2, The-orem 6.5] using the fact that, for each i , the doubling operator R iα i satisfies (cid:96)k ( α i , R i ) = 0leading to the fact that π ( M K i − ) ⊂ π ( E i ) (1) . Then, π ( L j ) ⊂ π ( V ) ⊂ π ( Z ) ( n ) , and if P is strongly coprime to (∆ m ( t ) , q n − ( t ) , . . . , q ( t )), π ( L j ) ⊂ π ( V ) ⊂ π ( Z ) ( n +1) P , and similarly for π ( D j ). The same holds for V . This would complete the verification ofclaims (6.1) and (6.2) since { L j , D j } (together with their counterparts in V would then satisfythe criteria of Definition 3.3.This concludes our sketch of the proof as given in [8, Theorem 6.5]. We include the rele-vant result about the elementary topology of the cobordism E . We will need several of theseproperties in later proofs. Lemma 6.1 ([10, Lemma 2.5]) . With regard to E on the left-hand side of Figure 6.1, theinclusion maps induce -TORSION IN THE n -SOLVABLE FILTRATION OF THE KNOT CONCORDANCE GROUP 23 (1) an epimorphism π ( M L ) → π ( E ) whose kernel is the normal closure of the longitudesof the infecting knots K i viewed as curves (cid:96) i ⊂ S − K i ⊂ M L ; (2) isomorphisms H ( M L ) → H ( E ) and H ( M R ) → H ( E ) ; (3) and isomorphisms H ( E ) ∼ = H ( M L ) ⊕ i H ( M K i ) ∼ = H ( M R ) ⊕ i H ( M K i ) . (4) The meridian of K , µ K ⊂ M K is isotopic in E to both α ⊂ M R and to the longitudinalpush-off of α , often called α ⊂ M L by abuse of notation. (5) The longitude of K , (cid:96) K ⊂ M K is isotopic in E to the reverse of the meridian of α , ( µ α ) − ⊂ M L and to the longitude of K in S − K ⊂ M L and to the reverse of themeridian of α , ( µ α ) − ⊂ M R (the latter bounds a disk in M R ). (cid:3) Proof of Theorem 5.5
The proof of Theorem 5.5 will occupy the remainder of the paper.
Theorem 5.5.
Consider knots K n , n ≥ K n ≡ R mη ,η ( K n − , K n − ) , where K n − is the result of applying a composition of n − R n − α n − ◦ · · · ◦ R α to some Arf invariant zero input knot K . Suppose additionally that1. m (cid:54) = 0;2. for each i , α i generates the rational Alexander module of R i , and this module is non-trivial;3. | ρ ( K ) | , the average Levine-Tristram signature of K , is greater than twice the sum ofthe Cheeger-Gromov constants of the ribbon knots R m , R , . . . , R n − (see Section 4).If P is the n -tuple of Alexander polynomials of the knots ( E m , R n − , . . . , R ), then K n / ∈ F P n. . Proof of Theorem 5.5.
We assume that P = ( p ( t ) , . . . , p n ( t )) = (∆ m , q n − ( t ) , . . . , q ( t ))is the n -tuple of Alexander polynomials of the knots ( E m , R n − , . . . , R ). Suppose that K n ∈F P n. . Let V be the putative ( n. , P )-solution. We will derive a contradiction.Let W be the 4-manifold (refer to Figure 7.1) obtained from V by adjoining the cobordisms E n , E n − , E n − , . . . E , E as defined in the proof of Theorem 5.3. For specificity, set W n = V,W n − = W n ∪ E n ,W n − = W n − ∪ E n − ∪ E n − , ... W = W ∪ E ∪ E . † , SHELLY HARVEY †† , AND CONSTANCE LEIDY ††† Note that, unlike in the manifold Z of Figure 6.2, we do not cap off the zero surgeries on thevarious ribbon knots. Thus the boundary of W is the disjoint union of the zero surgeries onthe ribbon knots R m , R n − , . . . , R , R n − , . . . , R , together with the zero surgeries on K , K . ...... M R m E n V M R M R M K M K M K n M R n − M R n − M K n − M K n − E n − E n − E E M K n − M K n − Figure 7.1. W Below we will define a commutator series { π ( n ) S } that is slightly larger than the derived serieslocalized at P . In particular,(7.1) π ( W ) ( n +1) P ⊂ π ( W ) ( n +1) S . Then we consider the coefficient system on W given by the projection φ : π ( W ) → π ( W ) /π ( W ) ( n +1) P → π ( W ) /π ( W ) ( n +1) S . The bulk of the proof (14 pages!) will be to show that:(7.2) the restriction of φ to π ( M K ⊂ ∂W ) factors non-trivially through Z ; and(7.3) the restriction of φ to π ( M K ⊂ ∂W ) is zero. -TORSION IN THE n -SOLVABLE FILTRATION OF THE KNOT CONCORDANCE GROUP 25 We now show that (7.1), (7.2) and (7.3) imply Theorem 5.5. Consider the von Neumannsignature defect of ( W , φ ): σ (2) ( W , φ ) − σ ( W ) . By the additivity of these signatures (property (5) of Proposition 4.1), this quantity is the sumof the signature defects for V and those of the E i and E i . Note that the coefficient system on π ( V ) factors π ( V ) → π ( V ) π ( V ) ( n +1) P → π ( W ) π ( W ) ( n +1) P → π ( W ) π ( W ) ( n +1) S , where we used Theorem 3.10 to establish the second map and we use (7.1) for the third map.Thus, since V is an ( n. , P )-solution, the signature defect of V vanishes by Theorem 4.2. All ofthe signature defects of the E i vanish by [10, Lemma 2.4] (essentially because H ( E ) comes from H ( ∂E )). Therefore the signature defect vanishes for W . On the other hand, by Section 4, σ (2) ( W , φ ) − σ ( W ) = ρ ( ∂W , φ ) . Hence 0 = ρ ( M R m , φ ) + · · · + ρ ( M R , φ ) + ρ ( M R , φ ) + ρ ( M K , φ ) + ρ ( M K , φ ) . By (7.2) and properties (1) and (3) of Proposition 4.1, ρ ( M K , φ ) = ρ ( K );while by (7.3) and properties (1) and (2) of Proposition 4.1 ρ ( M K , φ ) = 0 . But, by choice, | ρ ( K ) | is greater than twice the sum of the Cheeger-Gromov constants of the3-manifolds M R m , . . . , M R , which is a contradiction (see property (6) of Proposition 4.1).Therefore the proof of Theorem 5.5 is reduced to defining a commutator series { π ( n ) S } suchthat (7.1), (7.2) and (7.3) hold.The commutator series π ( j ) S will be defined only for the groups π = π ( W i ), because weneed not be concerned with any other groups. It will be defined exactly as in Definition 3.6except that the sequence of right divisor sets S , ..., S n will be slightly different than those ofDefinition 3.7. We now define S , ..., S n . In these definitions π is the fundamental group of oneof the W i .We define S = S ( π ) = S p = S p ( π ) = { q ( µ ) ...q r ( µ ) | ( p ( t ) , q j ( t )) = 1; π/π (1) ∼ = (cid:104) µ (cid:105)} . (Note that π (1) = π (1) r = π (1) P = π (1) S .) Before defining the other S i we make a few remarks.Since p ( t ) is a knot polynomial, p ( t ) . = p ( t − ), so S is closed (up to units) under the naturalinvolution. In fact, since p ( t ) = ∆ m ( t ) is the Alexander polynomial of E m , p ( t ) is prime.Hence one sees that S = Q [ µ, µ − ] − (cid:104) ∆ m ( µ ) (cid:105) . Therefore for any Q [ µ ± ]-module M , M S − = M (cid:104) ∆ m (cid:105) , the classical localization of M atthe prime ideal (cid:104) ∆ m (cid:105) . † , SHELLY HARVEY †† , AND CONSTANCE LEIDY ††† Therefore, by (3.1),(7.4) π (2) S = π (2) P ≡ ker (cid:32) π (1) → π (1) [ π (1) , π (1) ] ⊗ Q [ µ, µ − ] S − ≡ A ( W ) S − ≡ A ( W ) (∆) (cid:33) , where A ( W ) (∆) is the classical localization of A ( W ) at the prime (cid:104) ∆ m (cid:105) . (If W is any spacewith π ( W ) = π and H ( W ) ∼ = Z then by its integral Alexander module, denoted A Z ( W )we mean H ( W ; Z [ µ, µ − ]) ∼ = π (1) /π (2) . By its rational Alexander module, denoted A ( W ), wemean H ( W ; Q [ µ, µ − ]).)Now let Γ ≡ π/π (2) S ≡ π/π (2) P and A = π (1) /π (2) S ≡ π (1) /π (2) P ⊂ Γ. Thus Γ is the semidirectproduct of the abelian group A with π/π (1) ∼ = Z . Note that the circle η (see Figure 2.9)represents an element of π ( M K n ) (1) and hence, under inclusion, an element of π (1) for eachof the groups π = π ( W i ) under consideration. Hence, for any π , η has an unambiguousinterpretation as an element of A . By abuse of notation we allow η to stand for its image inany of the appropriate groups. Recall that a set S ⊂ Γ is
Γ-invariant if gsg − ∈ S for all s ∈ S and g ∈ Γ. Note that the set { µ i η µ − i | i ∈ Z } is Γ-invariant where µ ∈ Γ generates π/π (1) .Then we define the other S n as follows: Definition 7.1.
Let S = S ( π ) ⊂ Q [ π (1) /π (2) S ] ⊂ Q [ π/π (2) S ] be the multiplicative set generatedby { q ( a ) | (cid:94) ( q, p ) = 1 , q (1) (cid:54) = 0 , a ∈ A } ∪ { p ( µ i η µ − i ) | i ∈ Z } ;and for 2 < i ≤ n let S n = S n ( π ) = { q ( a ) ...q r ( a r ) | (cid:94) ( p n , q j ) = 1; q j (1) (cid:54) = 0; a j ∈ π ( n − S /π ( n ) S } . (7.5)Since S is a multiplicative subset of Q A that is Γ-invariant, it is a right divisor set of Q Γby [8, Proposition 4.1]. Therefore Definition 3.6 applies to give a partially defined commutatorseries { π ( i ) S } . Since p ( t ) = q n − ( t ) is a knot polynomial, p ( t ) . = p ( t − ). Thus S is closed (upto units) under the natural involution. Lemma 7.2.
For each π and each ≤ i ≤ n + 1(7.6) π ( i ) P ⊂ π ( i ) S . Proof.
The proof is by induction on i . By Definition 3.6, π (1) P = π (1) r = π (1) S , so the Lemma istrue for i = 0 ,
1. Suppose it is true for all values up to some fixed i ≥
1. Let j : π → π be theidentity map. By [8, Proposition 3.2], it suffices to show that the induced ring map j ∗ : Z [ π/π ( i ) P ] → Z [ π/π ( i ) S ]has the property that j ∗ ( S p i ( π )) ⊂ S i ( π ). For i = 1, j ∗ is the identity map and S ( π ) is, bydefinition, identical to S p ( π ). It follows that π (2) P = π (2) S as already observed in (7.4). Thus,for i = 2, j ∗ is again the identity map and, by Definitions 7.1 and 3.7, S ( π ) strictly contains S p ( π ). For i >
2, the map j ∗ , although induced by the identity, will be a surjection withnon-zero kernel. Nonetheless, by the inductive hypothesis, j induces a homomorphism j ∗ : π ( i − P /π ( i ) P → π ( i − S /π ( i ) S . -TORSION IN THE n -SOLVABLE FILTRATION OF THE KNOT CONCORDANCE GROUP 27 Recall from Definition 3.7 that S p i ( π ) = { q ( a ) ...q r ( a r ) | (cid:94) ( p i , q j ) = 1; q j (1) (cid:54) = 0; a j ∈ π ( i − P /π ( i ) P } , which is the multiplicative set generated by the described set of polynomials q ( a ). If q ( a ) isany such polynomial then j ∗ ( q ( a )) = q ( j ∗ ( a )) and since a ∈ π ( i − P π ( i ) P , j ∗ ( a ) ∈ π ( i − S π ( i ) S . Thus, upon examining (7.5), we see that q ( j ∗ ( a )) ∈ S i ( π ). Hence j ∗ ( S p i ( π )) ⊂ S i ( π ) as desired. (cid:3) In particular this establishes (7.1).
Lemma 7.3.
The commutator series { π ( i ) S } is functorial with respect to any inclusion, W i → W j , where i > j .Proof. Note that any such inclusion induces an isomorphism H ( W i ) ∼ = H ( W j ) ∼ = Z = (cid:104) µ (cid:105) .If π ( i ) S were actually the polarized derived series localized at P , then the functoriality wouldfollow directly from our Theorem 3.10 [8, Thm. 4.16]. But since π ( i ) S is slightly different, wemust actually repeat some of the proof of [8, Thm. 4.16]. Suppose A = π ( W i ) , B = π ( W j )and ψ : A → B is induced by inclusion. We show, by induction on i , that ψ ( A ( i ) S ) ⊂ B ( i ) S . Thisholds for i = 0 so suppose it holds for i = n . We will show that ψ ( A ( n +1) S ) ⊂ B ( n +1) S . Theinduction hypothesis guarantees that, for each 1 ≤ k ≤ n , ψ induces a homomorphism of pairs ψ : ( A/A ( k ) S , A ( k − S /A ( k ) S ) → ( B/B ( k ) S , B ( k − S /B ( k ) S ) . By [8, Prop.3.2] (or by examining (3.1)) it suffices to show that this map satisfies(7.7) ψ ( S k ( A )) ⊂ S k ( B )for each 1 ≤ k ≤ n . First consider k = 1. Recall that S ( A ) = { q ( µ ) ...q r ( µ ) | ( p ( t ) , q j ( t )) = 1; A/A (1) ∼ = (cid:104) µ (cid:105)} ⊂ Q [ A/A (1) ] . Since ψ induces an isomorphism ψ : A/A (1) → B/B (1) , ψ ( µ ) = ± µ . By choosing generatorsonce and for all, we may assume that ψ ( µ ) = µ . So, for any such q j ( t ), ψ ( q ( µ ) . . . q r ( µ )) = q ( ψ ( µ )) . . . q r ( ψ ( µ )) = q ( µ ) . . . q r ( µ ) ∈ S ( B ) . This verifies (7.7) for k = 1.Now suppose k >
1. Recall that S k ( A ) = { q ( a ) ...q r ( a r ) | (cid:94) ( p n , q j ) = 1; q j (1) (cid:54) = 0; a j ∈ A ( k − S /A ( k ) S } . So, for any such q j ( t ), ψ ( q ( a ) . . . q r ( a r )) = q ( ψ ( a )) . . . q r ( ψ ( a r )) ∈ S k ( B ) , since ψ ( a j ) ∈ B ( k − S /B ( k ) S .Thus ψ ( S k ( A )) ⊂ S k ( B ). (cid:3) † , SHELLY HARVEY †† , AND CONSTANCE LEIDY ††† Establishing (7.2) and (7.3) . Since π ( M K ) ⊂ π ( W ) is normally generated by its meridian, µ , and π ( M K ) is normallygenerated by its meridian, (that we denote) µ , the case i = 0 of the following Proposition willestablish (7.2) and (7.3). Therefore the rest of the paper will be spent establishing Proposi-tion 7.4. Proposition 7.4.
For any i , ≤ i ≤ n − , µ i = α i +1 is non-trivial, while µ i = α i +1 is trivialin π ( W i ) ( n − i ) π ( W i ) ( n − i +1) S . To clarify the notation of this proposition, recall that, for 0 ≤ i ≤ n − ∂W i contains thedisjoint union of the zero surgeries on the knots K i (refer to the schematic Figure 7.2), and K i .Let µ i and µ i denote the meridians of K i and K i in these copies of M K i and M K i respectively.Also recall that K i +1 = R i +1 α i +1 ( K i ) for some circle α i +1 that generates the Alexander moduleof R i +1 ; and K i +1 = R i +1 α i +1 ( K i ). Let α i +1 denote (a push-off of) this circle in M K i +1 ⊂ ∂W i +1 (referring to Figure 7.2); and let α i +1 denote (a push-off of) the other copy of α i +1 in M K i +1 ⊂ ∂W i +1 . Note that, by property (4) of Lemma 6.1, µ i is isotopic to α i +1 in E i +1 and µ i is isotopicto α i +1 in E i +1 . Hence µ i = α i +1 and µ i = α i +1 as elements of π ( W i ). µ i α i +1 E i +1 W i +1 M K i +1 α i +1 µ i +1 M K i M R i +1 Figure 7.2. W i Proof of Proposition 7.4.
The proof is by reverse induction on i , starting with i = n − -TORSION IN THE n -SOLVABLE FILTRATION OF THE KNOT CONCORDANCE GROUP 29 Before proving the base case i = n −
2, we need to work out the “pre-base-case”, i = n − α n and α n are what we have previouslycalled η and η respectively. Lemma 7.5. µ n − = η and µ n − = η are both non-trivial in π ( W n − ) (1) π ( W n − ) (2) S . Proof.
Throughout the proof of this lemma we abbreviate W = W n − , π = π ( W n − ) and∆ = ∆ n . We make use of the fact that the integral and rational Alexander modules of aknot agree with those of its zero-framed surgery. Specifically we use A ( K ) to denote both therational Alexander module of K and that of M K . The inclusion maps induce a commutativediagram of maps between integral and rational Alexander modules as shown: A Z ( E m ) A Z ( R m ) A Z ( V ) A Z ( W ) π (1) π (2) S A ( E m ) A ( E m ) (∆) A ( R m ) (∆) A ( V ) (∆) A ( W ) (∆) (cid:45) i (cid:63) i (cid:48) (cid:45) j ∗ (cid:63) i (cid:45) k ∗ (cid:63) i (cid:45)(cid:63) i (cid:19)(cid:19)(cid:19)(cid:19)(cid:19)(cid:19)(cid:19)(cid:19)(cid:19)(cid:47) i (cid:63) i (cid:48)(cid:48) (cid:45) i (cid:45) j ∗ (cid:45) k ∗ Notice that A Z ( R m ) ∼ = A Z ( K n ) ∼ = A Z ( M K n ) ∼ = A Z ( ∂V ). The maps j ∗ and k ∗ are induced byinclusion. The map i is induced by the connected sum decomposition, where here E m denotesthe “left-hand” copy in R m ≡ E m E m . The existence and injectivity of i is given by(7.4). Since the η i represent elements in the Alexander module of E m , it suffices to show thatthe composition in the top row is injective. For this it suffices to show that the composition k ∗ ◦ j ∗ ◦ i ◦ i (cid:48)(cid:48) ◦ i (cid:48) is a monomorphism. Since it is well known that the integral Alexander modules A Z ( E m ) ∼ = A Z ( S − E m ) are Z -torsion-free, i (cid:48) is injective. Since A ( E m ) is a ∆-torsion module, i (cid:48)(cid:48) is injective. Under the connected sum decomposition the localized Alexander module of R m decomposes as the direct sum of the localized Alexander modules of its summands E m . TheBlanchfield form decomposes similarly. Hence i is injective. Now consider the map j ∗ inducedby the inclusion ∂V (cid:44) → V . j ∗ : A ( ∂V ) (∆) ∼ = A ( R m ) (∆) ≡ H ( M R m ; Q [ t, t − ] S − p ) → H ( V ; Q [ t, t − ] S − p ) ≡ A ( V ) (∆) . Since V is an ( n. , P )-solution for ∂V , and π ( i ) P ⊂ π ( i ) S , V is an ( n. , S ) solution, so it iscertainly a (1 , S )-solution. Consider the coefficient system ψ : π ( V ) → π ( V ) /π ( V ) (1) S ∼ = Z (recall G (1) S = G (1) r for any group G ). Then [10, Theorem 7.15] applies to say that the kernel of j ∗ is isotropic with respect to the classical Blanchfield form on A ( R m ) (∆) . Hence the kernel, P , † , SHELLY HARVEY †† , AND CONSTANCE LEIDY ††† of i ◦ j ∗ is isotropic with respect to the classical Blanchfield form on A ( E m ) (∆) . But, since theAlexander polynomial of E m is irreducible by Proposition 2.3, the rational Alexander moduleof E m has no proper submodules. The case P = A ( E m ) (∆) is not possible since the localizedclassical Blanchfield form is non-singular and A ( E m ) (∆) (cid:54) = 0. Thus P = 0 so i ◦ j ∗ is injective.It only remains to show that the lower map k ∗ is injective (actually an isomorphism). Sincelocalization is an exact functor, this is equivalent to showing that the inclusion map inducesan isomorphism between the rational Alexander modules of V and W . Recall that W = W n = V ∪ E n . Recall from property (1) of Lemma 6.1 applied to E n , that the kernel on π of theinclusion M K n = ∂V → E n is normally generated by the longitudes of the infecting knots K n − and K n − as curves in π ( M K n ). These lie in the second derived subgroups of π ( S − K n − )and π ( S − K n − ) respectively and so lie in the third derived subgroup of π ( M K n ) (refer toFigure 2.8). Since the rational Alexander module of any space X with H ( X ) ∼ = Z may bedescribed as G (1) /G (2) ⊗ Q where G = π ( X ), this shows that the rational Alexander modulesof V and W are isomorphic. (cid:3) The crucial base case, i = n −
2, in the (reverse) inductive proof of Proposition 7.4 is:
Lemma 7.6. µ n − = α n − is non-trivial, while µ n − = α n − is trivial in π ( W n − ) (2) π ( W n − ) (3) S . Proof.
It might be helpful to refer to Figure 7.2 with i = n −
2. By property (1) of Lemma 6.1,the kernel of the map π ( W n − ) → π ( W n − ∪ E n − ∪ E n − ) = π ( W n − )is normally generated by the longitudes, (cid:96) n − , (cid:96) n − , of the infecting knots K n − and K n − viewed as curves in S \ K n − ⊂ M K n − ⊂ ∂W n − and S \ K n − ⊂ M K n − ⊂ ∂W n − . But ofcourse these lie in the second derived subgroups of π ( S \ K n − ) and π ( S \ K n − ) respectively,and so lie in the second derived subgroups of π ( M K n − ) and π ( M K n − ) respectively. But, asobserved in Lemma 7.5(7.8) π ( M K n − ) = (cid:104) µ n − (cid:105) ⊂ π ( W n − ) (1) , and similarly for π ( M K n − ). It follows that both (cid:96) n − and (cid:96) n − lie the third derived subgroupof π ( W n − ) and hence lie in π ( W n − ) (3) S . Thus the inclusion W n − → W n − induces anisomorphism π ( W n − ) π ( W n − ) (3) S ∼ = π ( W n − ) π ( W n − ) (3) S , by weak functoriality and by [8, Prop.4.7].Therefore, to prove Lemma 7.6, it suffices to let π = π ( W n − ), and show that α n − is non-trivial in π (2) /π (3) S and that α n − is trivial in π (2) /π (3) S . Throughout the rest of the proof of -TORSION IN THE n -SOLVABLE FILTRATION OF THE KNOT CONCORDANCE GROUP 31 Lemma 7.6, we will abbreviate W = W n − , π = π ( W n − ), J = K n − and J = K n − . Thus ∂W = M R m ∪ M J ∪ M J .Consider the following commutative diagram (which we justify below) where Γ = π/π (2) S and R = Q Γ S − . Since we may view α n − ∈ π ( M J ) (1) and α n − ∈ π ( M J ) (1) , we have reducedLemma 7.6 to showing that α n − is not in the kernel of the top row of the diagram while α n − does lie in this kernel. π ( M J ) (1) ⊕ π ( M J ) (1) π (2) π (2) S π (3) S ( A ( J ) ⊕ A ( J )) ⊗ R H ( M J ∪ M J ; R ) H ( W ; R ) π (2) S [ π (2) S , π (2) S ] ⊗ R (cid:45) j ∗ (cid:63) π (cid:45) φ (cid:63) (cid:63) j (cid:45) ∼ = (cid:45) j ∗ (cid:45) ∼ = The j ∗ in the upper row of the diagram is justified by our observation (7.8), which says that π ( M J ) ⊂ π (1) and π ( M J ) ⊂ π (1) . Now we consider the first map in the bottom row. ByLemma 7.5 the coefficient system π → Γ, when restricted to π ( M J ) is non-trivial: π ( M J ) = (cid:104) µ n − (cid:105) (cid:44) → π (1) π (2) S (cid:44) → ππ (2) S ≡ Γ , but also factors through π ( M J ) /π ( M J ) (1) ∼ = Z using (7.8). It follows that H ( M J ; Q Γ) ∼ = H ( M J ; Q [ t, t − ]) ⊗ Q Γ ≡ A ( J ) ⊗ Q [ t,t − ] Q Γ , where Q [ t, t − ] acts on Q Γ by t → µ n − (equivalently t → η ). Hence H ( M J ; R ) ∼ = A ( J ) ⊗ R ;and similarly for J , where t acts by µ n − = η . This explains the first map in the lowerrow of the diagram. To justify the last map in the lower row, recall that H ( W ; Z Γ) has aninterpretation as the first homology module of the Γ-covering space of W . The fundamentalgroup of this covering space is the kernel of π → Γ. Hence H ( W ; Z Γ) ∼ = π (2) S [ π (2) S , π (2) S ]Since the Ore localization R is a flat Z Γ-module, the ∼ = is justified. This completes the expla-nation of the diagram. Since, by Definitions 3.7 and 7.1, π (3) S = ker (cid:32) π (2) S → π (2) S [ π (2) S , π (2) S ] → π (2) S [ π (2) S , π (2) S ] ⊗ Q Γ S − (cid:33) , it follows that the vertical map j (in the diagram) is injective. Hence, to establish Lemma 7.6,it suffices to show that the class represented by α n − ⊗ not in the kernel of the bottom rowof the diagram while that represented by α n − ⊗ does lie in this kernel. † , SHELLY HARVEY †† , AND CONSTANCE LEIDY ††† Recall that J ≡ K n − ≡ R n − α n − ( K n − ) where α n − generates A ( R n − ) (note this implies thelatter module is cyclic). Therefore A ( J ) ∼ = A ( R n − ). By hypothesis, the Alexander polynomialof R n − is q n − ( t ) = p ( t ). Thus (cid:104) α n − (cid:105) ∼ = A ( J ) ∼ = Q [ t, t − ] p ( t ) Q [ t, t − ]and (cid:104) α n − ⊗ (cid:105) ∼ = A ( J ) ⊗ R ∼ = (cid:18) Q Γ p ( η ) Q Γ (cid:19) S − ∼ = 0 , where the last equality holds since p ( η ) ∈ S , by Definition 7.1 (see [8, Thm. 4.12] for moredetail). Therefore α n − ⊗ α n − ⊗ were in the kernel of the bottom row of the diagram. We shall reach acontradiction. Recall that W n − ≡ V ∪ E n . Recall that V is an ( n. , P )-solution. Since n ≥ V is a (2 , P )-solution. One easily checks that H ( W n − ) i ∗ ( H ( ∂W n − ) ∼ = H ( V ) . Hence this group has a basis consisting of surfaces that satisfy parts (2) and (3) of Definition 3.3(with n = 2). But W n − fails to satisfy part (1) of that definition and ∂W n − is disconnected.Such a manifold was named a (2 , P ) -bordism in [8, Definition 7.11]. By [8, Thms. 7.14, 7.15],if P is the kernel of the map j ∗ : H ( M J ; R ) → H ( W ; R ) , as in the bottom row of the diagram, then P is an isotropic submodule for the Blanchfieldlinking form on H ( M J ; R ). Since we have supposed that α n − ⊗ ∈ P and since this elementis a generator of H ( M J ; R ), it would follow that this Blanchfield form were identically zeroon H ( M J ; R ). But by [8, Lemma 7.16] this form is non-singular. This would imply that H ( M J ; R ) were the zero module. This is a contradiction once we show that(7.9) A ( J ) ⊗ R ∼ = (cid:18) Q Γ p ( η ) Q Γ (cid:19) S − (cid:54) = 0 . This is a non-trivial result since we are dealing with a noncommutative localization.Note that, by the hypotheses of Theorem 5.5, p ( t ) = q n − ( t ) is not a unit in Q [ t, t − ]. Themap Z → Γ given by t → η is not zero by Lemma 7.5. Since Γ is PTFA, it is torsion-free, so (cid:104) η (cid:105) ⊂ Γ. Hence Q Γ is a free left Q [ η , η − ]-module on the right cosets of (cid:104) η (cid:105) ∈ Γ [44, Chapter1, Lemma 1.3]. Thus, upon fixing a set of coset representatives, any x ∈ Q Γ has a uniquedecomposition x = Σ γ x γ γ, where x γ ∈ Q [ η , η − ] and the sum is over a set of coset representatives { γ ∈ Γ } . It follows that p ( η ) has no right inverse in Q Γ since if p ( η ) x = 1 then p ( η ) x = p ( η )Σ γ x γ γ = Σ γ p ( η ) x γ γ = 1 . -TORSION IN THE n -SOLVABLE FILTRATION OF THE KNOT CONCORDANCE GROUP 33 Looking at the coset γ = e , we have p ( η ) x e = 1 in Q [ η , η − ], contradicting the fact that p ( t ) is not a unit in Q [ t, t − ]. Therefore, since Q Γ is a domain, Q Γ p ( η ) Q Γ (cid:29) . Continuing, by [47, Corollary 3.3, p. 57], the kernel of Q Γ p ( η ) Q Γ → (cid:18) Q Γ p ( η ) Q Γ (cid:19) S − is precisely the S -torsion submodule. Hence to establish (7.9), it suffices to show that thegenerator of Q Γ /p ( η ) Q Γ is not S -torsion. Suppose [1] were S -torsion. We will show that[1] = 0, implying that Q Γ /p ( η ) Q Γ is S -torsion-free. If [1] were S -torsion then 1 s = p ( η ) y for some s ∈ S and for some y ∈ Q Γ. We examine this equation in Q Γ.Recall that Γ = π/π (2) S . Let A = π (1) /π (2) S (cid:67) Γ. Since A ⊂ Γ, Q Γ, viewed as a left Q A -module,is free on the right cosets of A in Γ. Thus any y ∈ Q Γ has a unique decomposition y = Σ γ y γ γ, where the sum is over a set of coset representatives { γ ∈ Γ } and y γ ∈ Q A . Therefore we have s = p ( η )Σ γ y γ γ. Recall from Definition 7.1 that s ∈ S ⊂ Q A . It follows that for each coset representative γ (cid:54) = e we have 0 = p ( η ) y γ so y γ = 0 (note that p ( η ) (cid:54) = 0 since Q [ η ± ] ⊂ Q Γ). Hence y ∈ Q A andwe have(7.10) s = p ( η ) y as an equation in Q A . Recall from Definition 7.1 that an arbitrary element of S is a productof terms of the form q ( a ) and terms of the form p ( µ i η µ − i ) for some a ∈ A , q ( t ) in Q [ t, t − ]where (cid:94) ( p , q ) = 1, q (1) (cid:54) = 0, and µ generates π/π (1) . Since A is a torsion-free abelian group,(7.10) may be viewed as an equation in Q F for some free abelian group F ⊂ A of finite rank r .Since Q F is a UFD and since (cid:94) ( p , q ) = 1 we can apply the following. Proposition 7.7 ([8, Prop.4.5]) . Suppose p ( t ) , q ( t ) ∈ Q [ t, t − ] are non-zero. Then p and q arestrongly coprime if and only if, for any finitely-generated free abelian group F and any nontrivial a, b ∈ F , p ( a ) is relatively prime to q ( b ) in Q F (a unique factorization domain). Thus the greatest common divisor, in Q F , of p ( η ) and q ( a ) is a unit (note that if a is trivialin F then q ( a ) = q (1) (cid:54) = 0 is itself a unit). Thus p ( η ) divides the product of the terms ofthe form p ( µ i η µ − i ). Choose a basis, { x, x , . . . , x r } , for F in which η = x r for some r > η (cid:54) = 0 by Lemma 7.5) and µ i η µ − i = x n i x n i, · · · x n i,r r . Then we may regard Q F as aLaurent polynomial ring in the variables { x, x , . . . , x r } . Since p is not zero and not a unit,there exists a non-zero complex root x = τ of p ( x r ). Suppose ˜ p ( x ) is an irreducible factor (in Q F ) of p ( x r ) of which τ is a root. Then, for some i , ˜ p ( x ) divides p ( x n i x n i, · · · x n i,r r ). Then τ must be a zero of p ( x n i x n i, · · · x n i,r r ) for every complex value of x , . . . , x r . This is impossible † , SHELLY HARVEY †† , AND CONSTANCE LEIDY ††† unless each n i,j = 0. Thus, for this value of i , µ i η µ − i = x n , in F , for some n . Note n (cid:54) = 0 since η is nontrivial by Lemma 7.5. Thus(7.11) µ i η r µ − i = ( µ i η µ − i ) r = x nr = η n , for some i and some non-zero integers n and r . This equation holds in A . However, the circles µ , η and η all live in M R m and in fact can be interpreted in A Z ( E m ) (the left-hand copy of E m ). But recall that in the proof of Lemma 7.5 we showed that the map A Z ( E m ) → A Z ( W ) → π (1) π (2) S ≡ A is injective. Hence if (7.11) holds in A then it holds as an equation in A Z ( E m ), and hence alsoin A ( E m ), where, in module notation, it has the form( t ∗ ) i ( rη ) = nη . But the simple computation in the following Lemma proves that this is impossible.
Lemma 7.8.
Let m be a non-zero integer, let E m be the knot of Figure 2.5 and let (cid:104) η i (cid:105) , i = 1 , be the subspace of A ( E m ) generated by the circle η i shown in Figure 2.9. Then, under theautomorphism t ∗ : A ( E m ) → A ( E m ) , for every integer k , ( t ∗ ) k ( (cid:104) η (cid:105) ) ∩ (cid:104) η (cid:105) = (cid:126) .Proof. We may assume that m >
0. If V is the Seifert matrix for E m as in the proof ofProposition 2.3, with respect to the basis { a i } consisting of the cores of the obvious bandswhere (cid:96)k ( a i , η i ) = 1, then the rational Alexander module is presented by V − tV T with respectto the basis { η , η } where the relations are given by the columns, that is, ( V − tV T ) (cid:126)v = (cid:126) (cid:126)v . Since V has non-zero determinant, upon left multiplying the latter equation by V − , onerecovers the fact that the automorphism t ∗ is given by left multiplication by ( V − ) T V . Hence t ∗ = 1 m (cid:18) m + 1 mm m (cid:19) = 1 m M, for M as indicated, with respect to the basis { η , η } . It then suffices to prove that, for any k ,there is no non-zero solution ( x , y ) to the equation M k (cid:18) y (cid:19) = (cid:18) x (cid:19) . If there were such a solution ( x , y ) then there would be one with y >
0. Let B = { ( x, y ) | x ≥ , y > } . Since (cid:18) m + 1 mm m (cid:19) (cid:18) xy (cid:19) = (cid:18) ( m + 1) x + mymx + m y (cid:19) we observe that M ( B ) ⊂ B . But then if k ≥ M k ( B ) ⊂ B . This is a contradiction since(0 , y ) ∈ B but ( x , / ∈ B . Therefore there is no non-zero solution if k ≥
0. If k < -TORSION IN THE n -SOLVABLE FILTRATION OF THE KNOT CONCORDANCE GROUP 35 have (cid:18) y (cid:19) = M − k (cid:18) x (cid:19) , where − k = s >
0. As above if there were a non-zero solution then there would be one with x >
0. Letting A = { ( x, y ) | x > , y ≥ } , we observe that M s ( A ) ⊂ A , ( x , ∈ A and(0 , y ) / ∈ A , which is a contradiction. (cid:3) This contradiction establishes (7.9), finally finishing the proof of Lemma 7.6. (cid:3)
We now complete the induction step in the proof of Proposition 7.4.Suppose, for some i , 1 ≤ i ≤ n −
2, Proposition 7.4 holds, that is, µ i = α i +1 is non-trivial,while µ i = α i +1 is trivial in(7.12) π ( W i ) ( n − i ) π ( W i ) ( n − i +1) S . To complete the inductive step we need to show that(7.13) µ i − = α i = 0 ∈ π ( W i − ) ( n − i +1) π ( W i − ) ( n − i +2) S . and show that(7.14) µ i − = α i (cid:54) = 0 ∈ π ( W i − ) ( n − i +1) π ( W i − ) ( n − i +2) S . By the inductive hypothesis and weak functoriality, µ i ∈ π ( W i ) ( n − i +1) S ⊂ π ( W i − ) ( n − i +1) S . But, by property (1) of Lemma 6.1, µ i ∈ π ( M K i ) normally generates π ( E i ) so π ( E i ) ⊂ π ( W i − ) ( n − i +1) S , and so by property (1) of Proposition 3.2,[ π ( E i ) , π ( E i )] ⊂ [ π ( W i − ) ( n − i +1) S , π ( W i − ) ( n − i +1) S ] ⊂ π ( W i − ) ( n − i +2) S . Since (cid:96)k ( α i , R i ) = 0, α i ∈ [ π ( M K i ) , π ( M K i )] ⊂ [ π ( E i ) , π ( E i )] ⊂ π ( W i − ) ( n − i +2) S . This proves (7.13).Now to we need to prove (7.14). By property (1) of Lemma 6.1, the kernel of the map π ( W i ) → π ( W i ∪ E i ∪ E i ) = π ( W i − )is normally generated by the longitudes, (cid:96) i − , (cid:96) i − , of the infecting knots K i − and K i − viewedas curves in S \ K i − ⊂ M K i ⊂ ∂W i and S \ K i − ⊂ M K i ⊂ ∂W i . But of course these liein the second derived subgroups of π ( S \ K i − ) and π ( S \ K i − ) respectively, and so lie † , SHELLY HARVEY †† , AND CONSTANCE LEIDY ††† in the second derived subgroups of π ( M K i ) and π ( M K i ) respectively. But, by the inductionhypothesis (7.12),(7.15) π ( M K i ) = (cid:104) µ i (cid:105) ⊂ π ( W i ) ( n − i ) , and similarly for π ( M K i ). It follows that both (cid:96) i − and (cid:96) i − lie in π ( W i ) ( n − i +2) ⊂ π ( W i ) ( n − i +2) S . Thus the inclusion W i → W i − induces an isomorphism π ( W i ) ( n − i +1) π ( W i ) ( n − i +2) S ∼ = π ( W i − ) ( n − i +1) π ( W i − ) ( n − i +2) S , by weak functoriality and by [8, Prop. 4.7].Consequently, to establish (7.14), it suffices to let π = π ( W i ), and show that α i is non-trivial in π ( n − i +1) /π ( n − i +2) S . Throughout the rest of the proof, we will abbreviate W = W i , π = π ( W i ), J = K i and J = K i . Thus M J ⊂ ∂W .Consider the following commutative diagram (which we justify below) where Γ = π/π ( n − i +1) S and R = Q Γ S − n − i +1 . Since α i ∈ π ( M J ) (1) we have reduced (7.14) to showing that α i is not inthe kernel of the top row of the diagram. π ( M J ) (1) π ( n − i +1) π ( n − i +1) S π ( n − i +2) S A ( J ) ⊗ R H ( M J ; R ) H ( W ; R ) π ( n − i +1) S [ π ( n − i +1) S , π ( n − i +1) S ] ⊗ R (cid:45) j ∗ (cid:63) π (cid:45) φ (cid:63) (cid:63) j (cid:45) ∼ = (cid:45) j ∗ (cid:45) ∼ = The j ∗ in the upper row of the diagram is justified by (7.15). Now we consider the first mapin the bottom row. By the inductive hypothesis (7.14) the coefficient system π → Γ, whenrestricted to π ( M J ) is non-trivial: π ( M J ) = (cid:104) µ i (cid:105) (cid:44) → π ( n − i ) π ( n − i +1) S (cid:44) → ππ ( n − i +1) S ≡ Γ , but also factors through π ( M J ) /π ( M J ) (1) ∼ = Z because of (7.15). It follows that H ( M J ; Q Γ) ∼ = H ( M J ; Q [ t, t − ]) ⊗ Q Γ ≡ A ( J ) ⊗ Q [ t,t − ] Q Γ , where Q [ t, t − ] acts on Q Γ by t → µ i . Hence H ( M J ; R ) ∼ = A ( J ) ⊗ R . -TORSION IN THE n -SOLVABLE FILTRATION OF THE KNOT CONCORDANCE GROUP 37 To justify the last map in the lower row, recall that H ( W ; Z Γ) has an interpretation as the firsthomology module of the Γ-covering space of W corresponding to the kernel of π → Γ. Hence H ( W ; Z Γ) ∼ = π ( n − i +1) S [ π ( n − i +1) S , π ( n − i +1) S ]This completes the explanation of the diagram. Since, by Definitions 3.7 and 7.1, π ( n − i +2) S = ker (cid:32) π ( n − i +1) S → π ( n − i +1) S [ π ( n − i +1) S , π ( n − i +1) S ] → π ( n − i +1) S [ π ( n − i +1) S , π ( n − i +1) S ] ⊗ Q Γ S − n − i +1 (cid:33) , it follows that the vertical map j (in the diagram) is injective. Hence, to establish (7.14), itsuffices to show that the class represented by α i ⊗ not in the kernel of the bottom row ofthe diagram.Recall that J ≡ K i ≡ R iα i ( K i − ) where α i generates A ( R i ). Therefore A ( J ) ∼ = A ( R i ). Bythe hypotheses of Theorem 5.5, the Alexander polynomial of R i is q i ( t ) = p n − i +1 ( t ). Thus(7.16) (cid:104) α i ⊗ (cid:105) ∼ = A ( J ) ⊗ R ∼ = (cid:18) Q Γ p n − i +1 ( µ i ) Q Γ (cid:19) S − p n − i +1 . where the last equality holds because, since 1 ≤ i ≤ n −
2, it follows that 3 ≤ n − i + 1 ≤ n , so S n − i +1 = S p n − i +1 , by Definition 7.1.Suppose that α i ⊗ were in the kernel of the bottom row of the diagram. We shall reach acontradiction. Recall that W = W i ≡ V ∪ E n ∪ E n − ∪ E n − ∪ · · · ∪ E i +1 ∪ E i +1 . Recall also that V is an ( n. , P )-solution. Thus, by (7.6), V is an ( n. , S )-solution and, since n − i + 1 ≤ n , V is also an ( n − i + 1 , S )-solution. One easily checks that H ( W i ) i ∗ ( H ( ∂W i )) ∼ = H ( V ) . Hence this group has a basis consisting of surfaces that satisfy parts (2) and (3) of Definition 3.3(with n − i + 1). Thus W i is an ( n − i + 1 , S )-bordism ([8, Definition 7.11]). By [8, Thms. 7.14,7.15], if P is the kernel of the map j ∗ : H ( M J ; R ) → H ( W ; R ) , then P is isotropic for the Blanchfield linking form on H ( M J ; R ). Therefore if the generator α i ⊗ P , it would follow that this Blanchfield form were identically zero on H ( M J ; R ).But by [8, Lemma 7.16] this form is non-singular. This would imply that H ( M J ; R ) = 0. Thisis a contradiction once we show that (7.16) is in fact a non-trivial module. It is shown in [8,Theorem 4.12] that(7.17) Q Γ p n − i +1 ( µ i ) Q Γ (cid:44) → (cid:18) Q Γ p n − i +1 ( µ i ) Q Γ (cid:19) S − p n − i +1 , † , SHELLY HARVEY †† , AND CONSTANCE LEIDY ††† is a monomorphism (using that p n − i +1 ( t ) (cid:54) = 0 and that µ i lies in the abelian normal subgroup A = π ( n − i ) /π ( n − i +1) S ⊂ Γ). This reduces us to showing that(7.18) Q Γ p n − i +1 ( µ i ) Q Γ (cid:54) = 0 . By the hypotheses of Theorem 5.5, p n − i +1 ( t ) = q i ( t ) is not a unit. The map Z → Γ givenby t (cid:55)→ µ i is not zero by the inductive hypothesis (7.12). Thus (cid:104) µ i (cid:105) ⊂ Γ and Q Γ is a free Q [ µ i , µ − i ]-module on the cosets of (cid:104) µ i (cid:105) ∈ Γ. In the same manner as we showed earlier in theproof, it follows that p n − i +1 ( µ i ) is not a unit in the domain Q Γ. Therefore (7.18) holds.This finishes, finally, the inductive step and hence the entire proof of Proposition 7.4, whichin turn completes the proofs of (7.2) and (7.3). (cid:3)
Having established (7.1), (7.2) and (7.3), the proof of Theorem 5.5 is complete. (cid:3)
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