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Polynomial splittings of Casson-Gordon invariants

In this paper we prove that the Casson-Gordon invariants of the connected sum of two knots split when the Alexander polynomials of the knots are coprime. As one application, for any knot K, all but finitely many algebraically slice twisted doubles of K are linearly independent in the knot concordance group.


Introduction
In his classification of the knot concordance groups, Levine [7] defined the algebraic concordance group, G, of Witt classes of Seifert matrices and a homomorphism from the knot concordance group, C, of knots in the 3-sphere S 3 to G. Casson and Gordon [1] proved that the kernel of Levine's homomorphism C → G, the concordance group of algebraically slice knots, is nontrivial. Gilmer [4] used the work of [1] to define a Witt type group Γ + and showed that there are homomorphisms C → Γ + → G. The group Γ + is roughly characterized by the property that a class of knots maps to zero in Γ + if and only if all of Levine's invariants and the Casson-Gordon invariants of a representative of the class vanish. However, the definition of Γ + used here is modified from the one used by Gilmer to correct for an error in [4], as described in Section 2.
It follows from Levine's work [8] that if the connected sum of two knots with relatively prime Alexander polynomials maps to zero in G, then so does each knot. We show a similar result for the Casson-Gordon invariants as follows.
Theorem 1.1. Let K 1 and K 2 be knots with relatively prime Alexander polynomials in Q[t ±1 ]. Suppose that either K 1 or K 2 has a non-singular Seifert form. Then if K 1 #K 2 is zero in Γ + then so are both K 1 and K 2 .
To demonstrate the strength of this result we study the family of k-twisted doubles of a given knot K, denoted D k (K). This family contains an infinite number of algebraically slice knots and these algebraically slice knots have been the subject of careful study. Casson and Gordon [1] found the first examples of nontrivial concordance classes in the kernel of Levine's homomorphism using the family D k (U ) where U is the unknot. Since then [5,10,11,12,16] have found infinite linearly independent families of algebraically slice knots among the knots D k (U ). In each case these families were very scarce: roughly one knot was chosen for each prime integer. Theorem 1.1 will yield that for every knot K (not just the unknot U ) the set of all algebraically slice knots in the family of knots D k (K) is (with finite exceptions) linearly independent. More precisely: Theorem 1.2. (a) For any knot K, all but finitely many algebraically slice twisted doubles of K are linearly independent in Γ + and so in C.
(b) If σ r (K) ≥ 0 for all r, then all algebraically slice twisted doubles of K except the untwisted and 2-twisted ones are linearly independent in Γ + and so in C, where σ r denotes the averaged Tristram-Levine signature. In addition, if σ r (K) > 0 for some r, then all algebraically slice twisted doubles of K except the untwisted one are linearly independent in Γ + and so in C. Corollary 1.3. (a) All algebraically slice twisted doubles of the unknot except the two known to be slice (untwisted and 2-twisted ones) are linearly independent in the knot concordance group.
(b) There are infinitely many knots K for each of which all algebraically slice twisted doubles of K except the untwisted one are linearly independent in the knot concordance group.
The full set of knots D k (K) contains knots which are not algebraically slice, representing elements of infinite order, order 2, and order 4 in G. It has not been possible to prove that this set is linearly independent in C, but we do have the following theorem, based on recent work of Friedl [2]: Theorem 1.4. (a) For any knot K, there is a set K containing all twisted doubles of K except a finite number of nonnegatively twisted ones such that no nontrivial linear combinations of elements in K are ribbon.
(b) If σ r (K) ≥ 0 for all r, then no nontrivial linear combinations of twisted doubles of K except those with 0, 1, 2 twists are ribbon. In addition, if σ r (K) > 0 for r = 2 9 , 1 3 , 2 5 , then no nontrivial linear combinations of twisted doubles of K except the untwisted one are ribbon.
In the past, the construction of independent knots depended on finding knots for which some branched covers had homology groups of order divisible by distinct primes. Such an approach could conceivably work with doubled knots by using high degree covers, but the argument would be far more burdensome than the one we give. A paper in preparation will address another application of Theorem 1.1 that there are examples of linearly independent algebraically slice knots having the same homology on all prime power fold branched covers, in which case no such approach could possibly work, and our approach using the splitting associated with the polynomial is definitely required.
The paper is organized as follows: In Section 2 we summarize the modified results of [4] about the Casson-Gordon invariants on slice knots and ribbon knots to correct for an error in [4]. In Section 3 we prove Theorem 1.1. In Section 4 we state that all algebraically slice twisted doubles of a knot (with finite exceptions) have infinite order in Γ + and similar results concerning their ribbonness. We also summarize some facts on the Casson-Gordon invariants of genus 1 knots and the Tristram-Levine signatures of satellite knots and torus knots. In Section 5 we estimate the Casson-Gordon invariants of algebraically slice D k (K) and prove Theorem 1.2 and Corollary 1.3. In Section 6 we estimate the Casson-Gordon invariants of all twisted doubles D k (K) for double branched covers and prove Theorem 1.4.

Gilmer's obstructions
In this section we state the modified results of [4] about the Casson-Gordon invariants on slice knots and ribbon knots to correct for an error in [4]. Conventions of [13] are followed here rather than those of [4] since we will use the formula for the Casson-Gordon invariants for genus 1 knots given in [13].
Let K be a knot in the 3-sphere S 3 with Seifert surface F having intersection pairing , . Let i + : H 1 (F ) → H 1 (S 3 − F ) denote the map which pushes a class off in the positive normal direction and let i − denote the map given by pushing off the other way.
Let θ denote the Seifert pairing, i.e. θ(x, y) = lk(x, i + y). Let A denote the Seifert matrix for θ with respect to some basis {a 1 , . . . , a 2g } for H 1 (F ). With respect to this basis the intersection form on F is then given by the matrix A t − A. Let {α 1 , . . . , α 2g } denote the basis for H 1 (S 3 − F ) such that lk(a i , α j ) = δ i j as in [14, page 209]. Then i + with respect to these bases is given by A. Define j : H 1 (S 3 −F ) → H 1 (F ) by jx, y = lk(x, y), so j is given by the matrix (A−A t ) −1 with respect to the above bases.
Following [6,15], we define the associated isometric structure s : H 1 (F ) → H 1 (F ) by the equation θ(x, y) = sx, y . We see that s is j • i + and is given by the matrix We are interested in H 1 (M q ; Q/Z), the set of characters on H 1 (M q ). Define ε q to be the endomorphism of H 1 (F ) given by s q − (s − 1) q and N q ⊂ H 1 (F ; Q/Z) to be the kernel of ε q ⊗ id Q/Z . Gilmer [4] proved that H 1 (M q ; Q/Z) is isomorphic to N q and the isomorphism can be uniquely constructed up to covering translations. So we may view the Casson-Gordon invariants τ (K, ) as a function on N q . See [1] for the definition of the Casson-Gordon invariants. From now on q will always denote a power of a prime. For a prime p, let N q p denote the p-primary component of N q .
A Seifert form θ on H 1 (F ) is said to be null-concordant if there is an s-invariant direct summand Z of H 1 (F ) such that Z = Z ⊥ with respect to the intersection pairing , . Such a direct summand Z is called a metabolizer for the isometric structure s associated to θ.
We say a knot K is slice if K bounds a smoothly embedded 2-disk D in the 4-ball B 4 with ∂(B 4 , D) = (S 3 , K). Knots K 1 and K 2 are called concordant if K 1 # − K 2 is slice, where −K denotes the mirror image of K with reversed orientation. The set of concordance classes of knots forms an abelian group under connected sum, called the knot concordance group and denoted C. We remark here that the result all apply in the category of locally flat oriented manifolds and pairs as well.
We say a knot K is ribbon if K bounds a smoothly immersed 2-disc f (D 2 ) in S 3 for a smooth map f : D 2 → S 3 which has the property that each component of self-intersection is an arc A ⊂ f (D 2 ) for which f −1 (A) is two arcs in D 2 , one of which lies in the interior of D 2 . It is easy to see that all ribbon knots are slice. There is no known concept of the knot ribbon concordance group more along the line of the knot concordance group as defined above. The difficulty is that it is unknown whether the following is true: If knots K and K#J are ribbon, then J is ribbon. In fact, this is an equivalent statement of Fox's conjecture: All slice knots are ribbon.
Gilmer [3,4] combined the slicing obstructions of [7] with those of [1] in a nontrivial way. Recently Gilmer has announced that there is an error in [3,4] (cf. [2]). However, the following two weaker statements are known to be valid. Note that the first statement has a weaker conclusion: the phrase "all primes p" is replaced by "all but finitely many primes p"; the second statement has a stronger hypothesis: "a slice knot K" is replaced by "a ribbon knot K." The first statement directly follows from Gilmer's original proof if primes p are chosen so that p do not divide the order of torsion of H 1 (R) (see [4] for the definition of R). The second statement is a corollary of Friedl's recent work [2, theorem 8.3 and corollary 8.5].
Theorem 2.1. (a) If F is a Seifert surface for a slice knot K then there is a metabolizer Z for the isometric structure on H 1 (F ) such that τ (K, N q p ∩ (Z ⊗ Q/Z)) vanishes for all prime powers q and all but finitely many primes p.
(b) If F is a Seifert surface for a ribbon knot K then there is a metabolizer Z for the isometric structure on H 1 (F ) such that τ (K, N q p ∩ (Z ⊗ Q/Z)) vanishes for all prime powers q and all primes p.
We say that a knot K has vanishing Gilmer slice (resp. ribbon) obstruction if the conclusion of Theorem 2.1(a) (resp. (b)) is satisfied for any Seifert surface F of K. Otherwise, we say that K has nonvanishing Gilmer slice (resp. ribbon) obstruction.
Gilmer [4] defined a Witt type group Γ + and a homomorphism from the knot concordance group C to Γ + such that the class of a knot maps to zero if and only if it satisfies the conclusion of Theorem 2.1(b). It became unknown whether the original Γ + is a group and whether there is a homomorphism C → Γ + since the proof of the cancellation lemma of [4, lemma 5] had a similar gap.
On the other hand, if the definition of Γ + is modified so that the class of a knot maps to zero if and only if it satisfies the conclusion of Theorem 2.1(a), then Γ + is a group and there are still homomorphisms C → Γ + → G. For instance, using the notation in [4, page 12], the definition of a metabolizer for (U, , , s, τ q p ) should be modified as done in the conclusion of Theorem 2.1(a), namely, "all primes p" should be replaced by "all but finitely many primes p." With this new definition of metabolizer we can eliminate the gap in the proof of [4, lemma 5] and hence this new Γ + becomes a group. Throughout this paper, Γ + is this modified one.

Alexander polynomial and proof of Theorem 1.1
Let K be a knot in the 3-sphere S 3 with Seifert surface F , Seifert pairing θ, and Seifert matrix A. We define ∆ K (t) = det(A − tA t ), called the Alexander polynomial of K corresponding to the Seifert surface F . As is well known, the Alexander polynomial of a knot is uniquely determined up to multiplication by ±t n in Q[t ±1 ]. Observe that the characteristic polynomial for the associated isometric To prove Theorem 1.1 we need a generalized notion of Seifert form. Consider integral valued bilinear forms θ on finitely generated free Z-modules H. Define the transpose of θ, denoted θ t , by θ t (x, y) = θ(y, x) for all x and y in H. We say that θ is a Seifert form if the form θ − θ t is unimodular, i.e. the associated map The notions of isometric structure, metabolizer, and Alexander polynomial extend to (algebraic) Seifert forms. Observe that if θ is a Seifert form on H, then the rank of H must be even. Polynomials in Q[t ±1 ] are said to be relatively prime if their greatest common divisor is a unit.
The following lemma is a refinement of [6, proposition 3]. Its geometrical origins are in the work of [8] on the knot concordance group. They did not need the splitting of a metabolizer as stated below, while we will need the splitting later.
Lemma 3.1. Let θ 1 and θ 2 be Seifert forms on H 1 and H 2 . Suppose that their Alexander polynomials are relatively prime in Q[t ±1 ] and that either θ 1 or θ 2 is nonsingular. Then if θ 1 ⊕ θ 2 is null-concordant with a metabolizer Z for the associated isometric structure, then θ 1 and θ 2 are null-concordant with metabolizers Z 1 and Z 2 for the associated isometric structures, respectively, such that Proof. Consider the associated isometric structures and ∆ θ2 are relatively prime in Q[t ±1 ] and since either θ 1 or θ 2 is non-singular, ∆ θ1 and ∆ θ2 are relatively prime in Q[t]. Then ϕ 1 and ϕ 2 are also relatively prime is a common factor of ∆ θ1 and ∆ θ2 . Thus there are polynomials u 1 and u 2 in Z[x] and a non-zero integer c such that u 1 ϕ 1 + u 2 ϕ 2 = c.
For z ∈ Z, there are z 1 ∈ H 1 and z 2 ∈ H 2 with z = z 1 + z 2 . As stated right after the definition of the Alexander polynomial, each ϕ i is the characteristic polynomial for s i , and hence ϕ i (s i ) = 0. Using this and s(z i ) = s i (z i ), we have Thus each Z i is a metabolizer for s i . Proof of Theorem 1.1. Let F 1 and F 2 be Seifert surfaces for K 1 and K 2 . Then a boundary connected sum F 1 ♮F 2 is a Seifert surface for K 1 #K 2 . Let Z be a metabolizer for the isometric structure on vanishes for all prime powers q and all primes p except p 1 , . . . , p n . Then by Lemma 3.1 there are metabolizers Z 1 and Z 2 for the isometric structures on H 1 (F 1 ) and H 1 (F 2 ), respectively, with Z = Z 1 ⊕ Z 2 . Let q be a power of a prime. Let N = ker ε q ⊗ id Q/Z and N i = ker ε q i ⊗ id Q/Z , where ε q and ε q i are the endomorphisms of H and H i , respectively, as denoted in Section 2. Then since Let N p and N i,p denote the p-primary components of N and N i , respectively. Let Also, by [10, corollary B2] τ is determined by the algebraic concordance class of the knot if the character is trivial. This implies that τ (K 2 , 0) = 0 and hence τ (K 1 , χ 1 ) = 0. Since χ 1 was chosen arbitrarily, we just have found a metabolizer Z 1 for the isometric structure on H 1 (F 1 ) such that τ (K 1 , N 1,p ∩ (Z 1 ⊗ Q/Z)) vanishes for all prime powers q and all primes p except p 1 , . . . , p n , i.e. K 1 is zero in Γ + . Similarly, K 2 is zero in Γ + . This completes the proof.
The proof given above also works for the Gilmer ribbon obstructions, in which case there are no exceptional primes p 1 , . . . , p n .

Twisted doubles of a knot
In this section we state that all but finitely many algebraically slice twisted doubles of a knot have infinite order in the knot concordance group C, in fact, in Γ + . We also state similar results concerning ribbonness in the line of Theorem 2.1(b). The proofs will be given in the next two sections. In preparations, we also summarize some facts on the Casson-Gordon invariants of genus 1 knots and the Tristram-Levine signatures of satellite knots and torus knots.
Let K be a knot in the 3-sphere S 3 . Let D k (K) denote the k-twisted double of K as illustrated in Figure 1. Here, k may be negative.
The following theorem is due to [8, corollary 23].
is not a perfect square, and every prime congruent to 3 mod 4 has even exponent in the prime power factorization of 4k + 1; (d) of order 4 in G if k > 0 and some prime congruent to 3 mod 4 has odd exponent in the prime power factorization of 4k + 1.
Immediate corollaries are that D k (K) is algebraically slice if and only if k = l(l + 1) for an integer l ≥ 0 and that D k (K) has infinite order in Γ + if k < 0.
We will further prove the following two theorems. For a nonnegative integer n let nD k (K) denote the connected sum of n copies of D k (K). The first of the following two theorems concerns the order of algebraically slice D k (K) in Γ + . The second concerns the Gilmer ribbon obstructions of nD k (K) for not only algebraically slice but all twisted doubles D k (K).
Theorem 4.2. (a) For any knot K, the algebraically slice k-twisted double D k (K) has infinite order in Γ + for all but finitely many k.
(b) If σ r (K) ≥ 0 for all r, the algebraically slice D k (K) has infinite order in Γ + for any k = 0, 2, where σ r (K) denotes the averaged Tristram-Levine signature of K (details will be given later). In addition, if σ r (K) > 0 for some r, then D 2 (K) has infinite order in Γ + as well. (b) If σ r (K) ≥ 0 for all r, for any integer n = 0 and any integer k = 0, 1, 2, nD k (K) has nonvanishing Gilmer ribbon obstruction. In addition, if σ r (K) > 0 for r = 2 9 , 1 3 , 2 5 , for any integer n = 0, nD 1 (K) and nD 2 (K) have nonvanishing Gilmer ribbon obstruction as well.
We devote the remaining two sections to proving these theorems. Before that, we summarize some useful facts in the rest of this section.

4.1.
Casson-Gordon invariants of a genus 1 knot. We state the work of [13, theorem 7] that gives a formula for τ for genus 1 knots in terms of the classical signatures. We remark that [3,4] first found the formula for the 2-fold branched cover case and algebraically slice case.
For a knot K and a character χ, τ (K, χ) is defined to be an element of the Witt group W (C(t), J ) ⊗ Z Q, where J denotes the involution on C(t) given by complex conjugation and by the map t → t −1 and W (C(t), J ) is the Witt group of finite dimensional hermitian inner product spaces. For details, see [1]. Let W (R) denote the Witt group of finite dimensional inner product spaces over R. The signature function σ : W (R) → Z is an isomorphism. Also there is a natural map W (R) → W (C(t), J ) given by tensoring with C(t) over R. Composing this map with σ −1 tensored with Q gives a homomorphism ρ : Q → W (C(t), J ) ⊗ Z Q. Note that for each complex number ζ with |ζ| = 1, there is a homomorphism σ ζ : W (C(t), J ) ⊗ Z Q → Q (see [1]). It is easy to see that σ 1 • ρ is the identity.
For any real number r, define A r (K) = 1 − e 2πir A + 1 − e −2πir A t , where A is a Seifert matrix of K, and define σ r (K) to be σ(A r ) if A r is non-singular and elsewhere to be the average of the one-sided limits of σ(A r ). This σ r (K) is a concordance invariant and is equal to the Tristram-Levine signature [7,17] of K except perhaps at finitely many r. It is an immediate consequence of the definition that σ r (K) = σ 1−r (K). Thus we only need to consider σ r (K) for 0 ≤ r ≤ 1 2 . The following is due to [13]. The case for q = 2 and the case for algebraically slice knots are due to [3,4].
where J x is a simple closed curve on F representing x and, for i = 0, . . . , q − 1, s i is an integer such that 0 < s i < d and s i ≡ (1 + m * ) i s mod d.
In particular, (a) if q = 2, then

Satellite knots and torus knots.
Let K be a knot in S 3 . By an axis for K of winding number w we mean an unknotted simple closed curve γ in S 3 − K having linking number w with K. Let V be a solid torus complementary to a tubular neighborhood of γ, with K contained in the interior of V . There is a preferred generator v for H 1 (V ), specified by the condition lk(v, γ) = +1. For any knot C in S 3 there is an untwisted orientation-preserving embedding h : V → S 3 taking V onto a tubular neighborhood of C such that C represents h * (v) in H 1 (hV ). We say that the knot h(K), denoted C(K), is a satellite of C with orbit K, axis γ, and winding number w.
Theorem 4.5. Let C(K) be a satellite of C with orbit K and winding number w. Then σ r (C(K)) = σ wr (C) + σ r (K).

Algebraically slice twisted doubles
In this section we will estimate the Casson-Gordon invariants of algebraically slice D k (K) and prove Theorem 4.2, Theorem 1.2, and Corollary 1. With rational coefficients, G is diagonalizable with respect to the basis {v + , v − }.
For a positive integer n, let nD l(l+1) (K) be the connected sum of n copies of D l(l+1) (K). We put a subscript n on objects corresponding to nD l(l+1) (K). For example, F n denotes the Seifert surface of nD l(l+1) (K) obtained by boundary connected summing n copies of the Seifert surface F of D l(l+1) (K).

5.1.
Metabolizer of nD l(l+1) (K). Let Z n be a metabolizer of the associated isometric structure of nD l(l+1) (K). (It should be remarked that unlike F n , s n , and θ n , Z n needs not be a direct sum of metabolizers of D l(l+1) (K).) Since G is diagonalizable with respect to the basis {v + , v − } with rational coefficients, G n associated with s n is diagonalizable with respect to the basis {v + j , v − j } j=1,...,n , where v ± j = 0 ⊕ · · · ⊕ v ± ⊕ · · · ⊕ 0 is an eigenvector of G n whose j-th coordinate is the only nonzero v ± under the identification of H 1 (F n ) with the direct sum of n copies of H 1 (F ). Note that {v + 1 , . . . , v + n } is a basis for the eigenspace of G n corresponding to the eigenvalue −l and {v − 1 , . . . , v − n } is a basis for the eigenspace of G n corresponding to l + 1.
The following lemma shows that Z n contains an eigenvector of G n for which the Casson-Gordon invariant can be easily estimated as will be shown later.
Lemma 5.1. There are integers e ≥ n 2 , a > 0, and a e+1 , . . . , a n ∈ Z such that Z n contains either a( Proof. From a basic result of linear algebra, Z n ⊗ Q has a basis consisting of eigenvectors of G n since Z n is invariant under G n and G n is diagonalizable over Q. In particular, Z n ⊗ Q = E + ⊕ E − , where E ± are the eigenspaces of G n restricted to Z n ⊗ Q. Since the rank of Z n is n, one of E ± has rank greater than or equal to n/2. Suppose that E + has rank e ≥ n/2. Using the Gauss-Jordan algorithm and rearranging basis elements, we may assume that a basis for E + consists of vectors of the form v + j + u j , 1 ≤ j ≤ e, where u j are linear combinations of v + e+1 , . . . , v + n . Adding these basis elements together we see that Z n ⊗ Q/Z contains a vector e j=1 v + j + n j=e+1 b j v + j for some b e+1 , . . . , b n ∈ Q. Multiplying the vector by a nonzero integer gives a desired vector. The same argument works if rank E − ≥ n/2.

Estimation of the Casson-Gordon invariants.
We use Theorem 4.4(b) to estimate τ of D l(l+1) (K) for the character corresponding to v ± in this subsection.
To change basis of H 1 (F ) to either {x + = v + , y = (−1, 0)} or {x − = v − , y = (−1, 0)}, let The Seifert matrices with respect to these bases {x ± , y} are respectively. Let m + = −l − 1 and m − = l. We can apply Theorem 4.4(b) to A ± with x = x ± and m = m ± , respectively, since a = 0. Observe that we can choose as shown in Figure 2 when l = 3. Here, the property T m,n = T −m,−n has been used. For a prime power q, a prime p dividing |(m + 1) q − m q | = (l + 1) q − l q , and any integer s relatively prime to p, Theorem 4.4(b) implies that x ± ⊗ s/p defines a character. Abusing notation, x ± ⊗ s/p will also denote its corresponding character. We have where s i are integers such that 0 < s i < p and s i ≡ (1 + (m ± ) * ) i s mod p. Since a connected sum K 1 #K 2 can be considered as a satellite of K 1 with orbit K 2 and winding number 1, by Theorem 4.5 we have Lemma 5.2. For any prime power q, any prime p dividing (l + 1) q − l q , and any integer s with p ∤ s, where s i are integers such that 0 < s i < p and s i ≡ (1 + (m ± ) * ) i s mod p.
Now we need to estimate σ r (T l,−l−1 ).
(b) For any constant C 0 , there is an integer l 0 ≥ 2 such that, for any l ≥ l 0 , any prime power q, any prime p dividing (l + 1) q − l q , and any m = m ± , there is an integer s such that where s i are integers such that 0 < s i < p and s i ≡ (1 + m * ) i s mod p.
To show (b), let l 0 be an integer such that l 0 ≥ 2 and 3 8 l 2 0 − l 0 − 2 ≥ 2C 0 . Let l ≥ l 0 , q a prime power, and p a prime dividing (l + 1) q − l q . It is easy to see that 1 + (m ± ) * ≡ 0 mod p. For simplicity, let a = 1 + (m ± ) * and e the multiplicative order of a mod p. Then e divides p − 1 and let f = (p − 1)/e. For any integer z not divisible by p, let E(z) = {a i z ∈ Z/p} i=0,...,e−1 , that is, a coset of the multiplicative group Z/p * = Z/p − {0} modulo a . Then there are integers z 1 , . . . , z f such that ∪ j E(z j ) = Z/p * and E(z j ) are all disjoint.
Note that if s i ∈ P , then 1 4 ≤ s i /p ≤ by the definition of l 0 .
Summing these, we have This completes the proof.

5.3.
Homology of prime power fold cyclic branched covers. We need some algebraic background. The resultant of two non-constant integral polynomials f (t) and g(t) is defined as follows: We may factor completely the polynomials f and g in some extension ring of Z as: Then the resultant of f and g, denoted R(f, g), is a m b n n i=1 m j=1 (α i − β j ). It is easy to see that R(f, g) = 0 if and only if f and g have a common root in a field over Z. We remark that this is also valid when working modulo a prime p.
It is known by Fox (see [18] for a proof) that the order of the homology of the nfold cyclic cover of S 3 branched over a knot J is the absolute value of the resultant, |R(t n − 1, ∆ J (t))|, of t n − 1 and the Alexander polynomial ∆ J (t) of J.
Proposition 5.4. For any knot K and any integer k > 0, there are infinitely many distinct primes each of which divides |H 1 (M q )| for some prime q, where M q is the q-fold cyclic cover of S 3 branched over D k (K).
First, we will show that R q1 and R q2 are relatively prime for distinct primes q 1 and q 2 . Suppose to the contrary there is a prime p dividing both R q1 and R q2 . Then, working modulo p, for j = 1, 2, ∆ k and t qj − 1 have a common root r j in an extension field of the finite field F p of p elements.
We claim that the three polynomial ∆ k , t q1 − 1, and t q2 − 1 have a common root, r, mod p. If r 1 = r 2 , this is obvious. Suppose r 1 = r 2 . Then ∆ k has two distinct roots r 1 and r 2 mod p and hence ∆ k must be quadratic over F p . In particular, k = 0 in F p . Thus, r 1 r 2 = (−k)/(−k) = 1 or r 2 = 1/r 1 . So, r q1 2 = 1/r q1 1 = 1 mod p and r 2 is a common root, r, mod p of the three polynomials.
Since q 1 and q 2 are distinct primes, there are a and b such that aq 1 + bq 2 = 1. So r = r aq1+bq2 = (r q1 ) a (r q2 ) b ≡ 1 mod p. This implies that 1 is a root of ∆ k mod p. However, this is a contradiction since ∆ k (1) = 1 ≡ 0 mod p. Thus there are no primes p dividing both R q1 and R q2 , implying they are relatively prime.
It now suffices to show that R q > 1 for any large primes q. We will show that R q → ∞ as prime q → ∞. If G is a matrix associated with the isometric structure of D k (K) and N q is the kernel of (G q − (G − I) q ) ⊗ id Q/Z as before, then R q = | det (G q − (G − I) q ) | since R q = |H 1 (M q )| = |N q |. A Seifert matrix for D k (K) corresponding to the Seifert surface in Figure 1 . Let u = (1 + √ 4k + 1)/2, w = (1 − √ 4k + 1)/2, and P = ( w u 1 1 ) . Then P −1 GP = ( u 0 0 w ) and P −1 (G − I)P = −w 0 0 −u . We now see that, for any odd integer q, P −1 (G q − (G − I) q )P = (u q + w q ) I and hence R q = (u q + w q ) 2 . Since |u| > 1, |w/u| < 1, and R q → ∞ as prime q → ∞. In particular, R q > 1 for any large primes q. This completes the proof. Suppose to the contrary that n > 0 and nD l(l+1) (K) is zero in Γ + . Then nD l(l+1) (K) has vanishing Gilmer slice obstruction. Use the same notation F , M q , N , etc. as before for D l(l+1) (K) and put a subscript n on the objects corresponding to nD l(l+1) (K). Let Z n be a metabolizer satisfying the conclusion of Theorem 2.1(a) for the surface F n . By Lemma 5.1, Z n contains an integral vector v that is either for some a > 0 and e ≥ n/2. By Proposition 5.4 we can find a prime p and an odd prime q such that p divides |H 1 (M q )|, p does not divide a, and τ (nD l(l+1) (K), (N q n ) p ∩ (Z n ⊗ Q/Z)) vanishes, where (N q n ) p denotes the p-primary component subgroup of N q n . Recall that N q ∼ = ker((G q −(G−I) q )⊗id Q/Z ). From the proof of Proposition 5.4, G q − (G − I) q is the identity matrix multiplied by an integer h = |u q + w q | = (l +1) q −l q when q is odd and k = l(l +1). Thus N q = {z ⊗1/h | z ∈ H 1 (F )}. Since N q n is the direct sum of n copies of N q , we have N q n = {z n ⊗ 1/h | z n ∈ H 1 (F n )}. Since the prime p divides |H 1 (M q )| = h 2 , p divides h and hence z n ⊗ 1/p ∈ (N q n ) p for any z n ∈ H 1 (F n ). In particular, for the vector v ∈ Z n chosen above, v ⊗ 1/p ∈ (N q n ) p . Moreover, since v ∈ Z n , v ⊗ 1/p ∈ (N q n ) p ∩ (Z n ⊗ Q/Z) and hence σ 1 τ nD l(l+1) (K), v ⊗ 1/p = 0.
On the other hand, let s be the constant from Lemma 5.3(b) determined by l, p, and q chosen above together with m = m + or m − depending on whether v = a( Since p was chosen not to divide a, by multiplying an integer to v, we may further assume that a ≡ s mod p. For i = 0, . . . , q − 1, j = e + 1, . . . , n, let s i , s ij be integers such that 0 < s i , s ij < p, s i ≡ (1 + m * ) i a mod p, and s ij ≡ (1 + m * ) i a j mod p.
By the additivity of σ 1 τ , it is equal to By Lemma 5.3, we now see that Thus σ 1 τ nD l(l+1) (K), v ⊗ 1/p = 0, which is a contradiction. This proves (a). Next, suppose that l ≥ 2 and σ r (K) ≥ 0 for all r. Under the same contradiction hypothesis and notation as above except: let s = [p/2], instead of choosing it from Lemma 5.3(b). Observe that 1/l(l + 1) < 1 3 ≤ s/p ≤ 1 2 and hence σ s/p (T l,−l−1 ) ≥ 2 by Lemma 5.3(a). We may assume a ≡ s mod p as before. Then we have This is a contradiction, completing the proof of the first part of (b).
In addition, suppose that σ r0 (K) > 0 for some r 0 . We only need to check the case l = 1. By the definition of the averaged signature σ r , there are r 1 and r 2 such that 0 < r 1 < r 2 ≤ 1 2 and σ r (K) > 0 for any r with r 1 < r < r 2 . We can take p arbitrary large as above so that r 1 < s/p < r 2 for some integer s. Apply a similar argument as above. The difference from the previous case is that the role of σ s/p (T l,−l−1 ) is switched with that of σ s/p (K).
Proof of Theorem 1.2. The Alexander polynomial of D k (K) is ∆ k = −kt 2 + (2k + 1)t − k. We will see that all ∆ k are coprime in Q[t ±1 ]. Let k = l and let g be the greatest common divisor of ∆ k and ∆ l in Q[t ±1 ]. Then g divides l∆ k −k∆ l = (l−k)t that is a unit in Q[t ±1 ]. So, ∆ k and ∆ l are relatively prime in Q[t ±1 ] for any distinct pair of integers k and l.
Let L be the set of integers l ≥ 0 for which D l(l+1) (K) has infinite order in Γ + . Theorem 4.2 implies L contains all but finitely many nonnegative integers. Note that under the hypotheses of (b) L contains all but 0, 1 or only 0. Suppose that there are distinct l i ∈ L such that e 1 D l1(l1+1) (K)#e 2 D l2(l2+1) (K)# · · · #e n D ln(ln+1) (K) is zero in Γ + for some integers e 1 , . . . , e n . Since ∆ eiD l i (l i +1) (K) (t) = ∆ li(li+1) ei , all ∆ eiD l i (l i +1) (K) (t) are pairwise coprime. Also, note that all e i D li(li+1) (K) have nonsingular Seifert forms. Applying Theorem 1.1 inductively, each e i D ki (K) must be zero in Γ + and hence each e i must be 0. This completes the proof.
Proof of Corollary 1.3. Part (a) is an immediate consequence of Theorem 1.2(b) since σ r (K) = 0 for any r if K is the unknot. For part (b), the torus knots T l,−l−1 for l > 1 are such examples with σ r ≥ 0 for all r and σ r > 0 for some r.

Non-ribbonness of linear combinations of twisted doubles
In this section we will estimate the Casson-Gordon invariants of all twisted doubles D k (K) for double branched covers and prove Theorem 4.3 and Theorem 1.4. 6.1. Estimation of the Casson-Gordon invariants. We estimate the Casson-Gordon invariants of D k (K) for k > 0 in this subsection. Recall from Theorem 4.1 that if k ≥ 0, then D k (K) has finite order in the algebraic concordance group G.