Non-slice linear combinations of iterated torus knots

In 1976, Rudolph asked whether algebraic knots are linearly independent in the knot concordance group. This paper uses twisted Blanchfield pairings to answer this question in the affirmative for new large families of algebraic knots.


Introduction
A knot is algebraic if it arises as a link of an isolated singularity of a complex curve.Algebraic knots are special cases of iterated torus knots.In 1976, Rudolph [26] asked for a description of the subgroup of the knot concordance group generated by algebraic knots.For ease of reference, we refer to this question as a conjecture.
Conjecture 1 (Rudolph's Conjecture [26]).The set of algebraic knots is linearly independent in the smooth knot concordance group C.This question has been of particular interest due to its relevance to the slice-ribbon conjecture: a result of Miyazaki shows that non-trivial linear combinations of iterated torus knots are not ribbon [24,Corollary 8.4].In particular, if the slice-ribbon conjecture holds, then Rudolph's conjecture holds.Baker [2] and Abe-Tagami [1] recently noticed that the slice-ribbon conjecture implies a statement stronger than Rudolph's conjecture: Conjecture 2 (Abe-Tagami [1] and Baker [2]).The set of prime fibered strongly quasi-positive knots is linearly independent in the smooth knot concordance group C.This paper exhibits new large families of knots for which Conjectures 1 and 2 hold.
1.1.Statement of the results.Evidence of Rudolph's conjecture was first provided in 1979 by Litherland, who proved that positive torus knots are linearly independent in C [19].In 2010, Hedden, Kirk and Livingston showed that for an appropriate choice of positive integers {q n } ∞ n=1 , the set {T (2, q n ), T (2, 3; 2, q n )} ∞ n=1 is linearly independent in C, where T (p, q) and T (p, q; r, s) denote the (p, q)-torus knot and the (r, s)-cable of T (p, q), respectively, and p is coprime to qrs.It is known that an iterated torus knot T (p 1 , q 1 ; . . .; p k , q k ) is algebraic if and only if p i , q i > 0 and q i+1 > q i p i+1 p i for each i.Our main result, which relies on metabelian twisted Blanchfield pairings [3,23], reads as follows.
Theorem 1.1.Fix a prime power p.Let S p be the set of iterated torus knots T (p, q 1 ; p, q 2 ; . . .; p, q ), where the sequences (q 1 , q 2 , . . ., q ) of positive integers that are coprime to p satisfy (1) q is a prime; (2) for i = 1, . . ., − 1, the integer q i is coprime to q when > 1; The set S p is linearly independent in the topological knot concordance group C top .
As an immediate corollary of Theorem 1.1, we obtain the following.
Corollary 1.2.For every prime power p, the subset S alg p ⊂ S p of algebraic knots in S p is linearly independent in C top and therefore satisfies Conjecture 1.
Let λ p (J) denote the Q/Z-valued linking form on H 1 (Σ p (J)).Miller and Powell show that if for every Z p -invariant metaboliser G of λ p (J), there exists a prime power order character χ that vanishes on G and such that Bl α(p,χ) (J) is not metabolic, then J is not slice [23,Theorem 6.10].In order to make this obstruction more concrete, we now recall some terminology on linking forms and their metabolizers.
The Witt group of linking forms.We focus on linking forms over C[t ±1 ], referring to Section 4 for a discussion over more general rings.A linking form over C[t ±1 ] is a sesquilinear Hermitian pairing V ×V → C(t)/C[t ±1 ], where V is a torsion C[t ±1 ]-module.A linking form (V, λ) is metabolic if there is a submodule L ⊂ V such that L = L ⊥ ; such an L is called a metaboliser.The Witt group of linking forms, denoted W (C(t), C[t ±1 ]), consists of the monoid of non-singular linking forms modulo the submonoid of metabolic linking forms.We write λ 1 ∼ λ 2 if two linking forms agree in W (C(t), C[t ±1 ]).The Miller-Powell obstruction to sliceness, therefore, consists of deciding whether a certain twisted Blanchfield pairing Bl α(p,χ) (J) is zero in the group W (C(t), C[t ±1 ]).As we will now describe, one of our main ideas is to transfer a problem of linear independence in C top (namely Rudolph's conjecture) into a problem of linear independence in W (C(t), C[t ±1 ]).
In particular, any character on H 1 (Σ p (J)) can be written as χ = χ 1 ⊕ χ 2 ⊕ χ 3 ⊕ χ 4 .For each given Z p -invariant metaboliser M of λ p (J), the "sliceness-obstructing character" that we will produce will be of the form χ = χ 1 ⊕ χ 2 ⊕ θ ⊕ θ where θ denotes the trivial character.Using the definition of J, together with the direct sum decomposition of [3,Corollary 8.21], the Witt class of the metabelian Blanchfield pairing of J is given by Bl α(p,χ) (J) ∼ Bl α(p,χ1) (T (p, q; p, q 1 )) ⊕ − Bl α(p,χ2) (T (p, q 1 )) (2) ⊕ − Bl α(p,θ) (T (p, q; p, q 2 )) ⊕ Bl α(p,θ) (T (p, q 2 )).This expression can be further decomposed by applying the satellite formula for the metabelian Blanchfield forms given in [3,Theorem 8.19].Regardless of the final expression, the problem has been converted into a question of linear independence in W (C(t), C[t ±1 ]).In Proposition 4.3, we describe a criterion for linear independence in terms of roots of the orders of the underlying modules (recall that the order of a module is a Laurent polynomial in C[t ±1 ]; it is defined up to multiplication by units of C[t ±1 ]).Here is a simplified version of this statement.
Computation of twisted Alexander polynomials.In order to apply Proposition 1.5, we must, therefore, understand the roots of the metabelian twisted Alexander polynomials of T (p, q) associated to characters on H 1 (Σ p (T (p, q))).This is carried out in Section 3 and relies on our explicit understanding of the p-fold cover E p (T (p, q)) → E(T (p, q)) from Section 2. Since this computation of twisted polynomials might be of independent interest, we summarize it as follows.
Proposition 1.6 (Lemma 3.1 and Corollary 3.4).Let p, q > 0 be two coprime integers, and set ξ p = e 2πi/p .The abelian group of characters on We write χ a for the character associated to a.The metabelian twisted Alexander polynomial of the 0-framed surgery M T (p,q) associated to the character χ a : H 1 (Σ p (T (p, q))) → Z q is given by .
Main steps of the proof.We now return to the knot J = J(p, q, q 1 , q 2 ) from (1).Obstructing J from being slice has three main steps.In fact, the proof of Theorem 1.1 in Section 5 follows more complicated versions of these same steps.
(1) Firstly, we use the previously described ingredients to study the implications of Bl α(p,χ) (J) being metabolic on the characters χ 1 and χ 2 ; here χ = χ 1 ⊕ χ 2 ⊕ θ ⊕ θ with θ the trivial character.This is the content of Subsection 5.2.2.(2) Secondly, we show that for every metaboliser L of λ p (T p,q1 ) ⊕ −λ p (T p,q1 ), it is possible to build characters χ 1 and χ 2 that violate these conditions and such that χ 1 ⊕ χ 2 vanishes on L. This is the content of Subsection 5.2.3.(3) Finally, we combine these two steps to obstruct the sliceness of J: for every metaboliser G of λ p (J), we are able to build a character χ = χ 1 ⊕ χ 2 ⊕ θ ⊕ θ that vanishes on G and such that Bl α(p,χ) (K) is not metabolic.This is the content of Subsection 5.2.4.
Remark 1.7.When p = 2, Hedden, Kirk and Livingston also use an obstruction based on the Casson-Gordon set-up to show that for an appropriate choice of positive integers n=1 is linearly independent in C top [12].Our work differs from theirs in two main points: • While [12] uses a blend of discriminants and signatures to prove its linear independence result, we use metabelian Blanchfield pairings.In a nutshell, the Blanchfield pairing encapsulates both the discriminant and (most of) the signature invariants allowing us to both streamline and generalize several of the argument from [12].• The result of [12] is proved without having to study invariant metabolizers; see also [3,Section 9].This is a feature of iterated torus knots T (p, Q) with p = 2 and fails when p > 2.
Passing from our outline to obstruct the sliceness of J(p, q, q 1 , q 2 ) to the proof of Theorem 1.1 requires additional steps.As often in Casson-Gordon theory, the main technical difficulty to overcome concerns the metabolizers of the linking form of the knot in question.Regarding these metabolizers, our strategy can be summarized as follows: (1) Given a metabolizer, we isolate certain technical conditions which guarantee that a character violates the sliceness obstruction.This is the content of Lemma 5.9.(2) We distinguish a certain family of metabolizers called graph metabolizers, see Section 4.2.
(3) The construction of the required character, for any fixed non-graph metabolizer is not overly challenging, see Cases 1 and 2 in the proof of Lemma 5.9.(4) Dealing with graph metabolizers requires more work.In Case 3, we show that either there exists a character satisfying the conditions from Lemma 5.9, or the knot in question contains a slice summand K# − K, for some knot K. Consequently, once we cancel all the summands of the form K# − K, we are able to construct the desired obstructing character for any graph metabolizer, and finish the proof.
1.4.Assumptions and outlook.We conclude this introduction by commenting on the various technical assumptions that appear in Theorem 1.1. ( The assumption that the integers q i are coprime to q is used in Proposition 5.8 to ensure that certain Witt classes are linearly independent in W (C(t), C[t ±1 ]).This hypothesis has its roots in the notion of p-independence introduced in [12, Definition 6.2].(2) We assume that p is a prime power in order to use Casson-Gordon theory [4,5].
(3) We required that the q i be positive mostly because of our interest in Rudolph's conjecture: algebraic knots are iterated torus knots with positive cabling parameters.(4) We use that the q i, i are prime in order to obtain the decomposition in (21) and to ensure that F q i, i is a field.
Summarising, our assumptions are made for technical reasons: we have so far not encountered linear combinations of (algebraically slice) iterated torus knots whose sliceness is not obstructed by some Casson-Gordon invariants.Furthermore, this paper does not fully use the techniques developed in [3] to compute the Casson-Gordon Witt class.Therefore, it would be interesting to study how far these methods can be pushed to investigate Rudolph's conjecture.1.5.Organisation.This paper is organized as follows.In Section 2, we collect several results on the algebraic topology of the exterior of the torus knot T (p, q).In Section 3, we use these results to compute Alexander polynomials of T (p, q) twisted by metabelian representations.In Section 4, we review some facts about linking forms.Finally in Section 5, we prove Theorem 1.1.
Acknowledgments.AC thanks the MPIM for its financial support and hospitality.MHK was partly supported by the POSCO TJ Park Science Fellowship and by NRF grant 2019R1A3B2067839.WP was supported by the National Science Center grant 2016/22/E/ST1/00040.We wish to thank the CIRM in Luminy for providing excellent conditions where the bulk of this work was carried out.
Conventions.Manifolds are assumed to be compact and oriented.Throughout the paper, the pfold branched cover of a knot is denoted Σ p (K), and λ p (K) denotes the linking form on H 1 (Σ p (K)).

Branched covers of torus knots
The aim of this section is to describe the Z[Z p ]-module structure of H 1 (Σ p (T (p, q))) induced from the Z p -covering action on Σ p (T (p, q)) when q is a prime.Let E(T (p, q)) be the complement of the torus knot T (p, q), and let E p (T (p, q)) be its p-fold cyclic cover.In Subsection 2.1, to set up some notation, we recall the decomposition of E(T (p, q)) coming from the standard genus 1 Heegaard splitting of S 3 , as described in [8,Example 1.24].In Subsection 2.2, this decomposition of E(T (p, q)) is used to decompose E p (T (p, q)): after that, H 1 (Σ p (T (p, q))) can be computed via a Mayer-Vietoris sequence argument since Σ p (T (p, q)) is a union of E p (T (p, q)) with a solid torus glued along the torus boundary.
2.1.The homotopy type of E(T (p, q)).The goal of this subsection is to describe the homotopy type of E(T (p, q)), as well as describe explicit generators for π 1 (E(T (p, q))).To achieve this, we follow closely [8,Example 1.24].
Consider the standard decomposition S 3 = S 1 × D 2 ∪ D 2 × S 1 and denote S 1 × D 2 and D 2 × S 1 by H 1 and H 2 respectively; H 1 ⊂ R 3 being the bounded solid torus.We parametrize the (p, q)torus knot T (p, q) on the torus H 1 ∩ H 2 as follows: (3) Using this description of T (p, q), for each x ∈ S 1 , we see that T (p, q) intersects {x} × D 2 ⊂ H 1 in p equi-distributed points of {x} × ∂D 2 ; see Figure 1 for p = 3.
On the left hand side: the intersection T (p, q) ∩ ({x} × D 2 ).On the right hand side: the complement H 1 T (p, q) deformation retracts onto a 2-complex X p .
As depicted in the right hand side of Figure 1, the complement H 1 T (p, q) deformation retracts onto a 2-complex X p ⊂ H 1 which is the mapping cylinder of the degree p map f p : S 1 → c 1 , where c 1 is the core circle of H 1 .The same argument shows that H 2 T (p, q) deformation retracts onto the mapping cylinder X q of the degree q map f q : S 1 → c 2 , where c 2 is the core circle of H 2 .By perturbing X p near H 1 ∩ H 2 , we can arrange that X p and X q match up.Next, let X p,q be the union of X p and X q .Note that X p,q is homeomorphic to the double mapping cylinder of the maps f p and f q , defined by X p,q := S 1 × [0, 1] ∪ c 1 ∪ c 2 /∼ where (z, 0) ∼ f p (z) and (z, 1) ∼ f q (z) for all z ∈ S 1 (see Figure 2).By van Kampen's theorem, . Summarizing, we have the following proposition which is implicit in [8, Example 1.24]: Proposition 2.1 ([8, Example 1.24]).There is a deformation retraction E(T (p, q)) → X p,q sending H 1 T (p, q) and H 2 T (p, q) to X p and X q .In particular, π 1 (E(T (p, q))) where c i is the core circle of H i for i = 1, 2.
We first use Subsection 2.1 to describe a deformation retract of E p (T (p, q)).Using (3), we see that the torus knot T (p, q) links respectively q and p times the core circles c 1 and c 2 .Consequently, c 1 and c 2 are homologous to qµ and pµ in H 1 (E(T (p, q))), where µ = c k 1 c l 2 is a meridian of T (p, q) and pk + ql = 1.Use (X p,q ) p to denote the pre-image π −1 (X p,q ), and observe that by Proposition 2.1, E p (T (p, q)) deformation retracts onto (X p,q ) p .
The double mapping cylinder X p,q obtained by gluing S 1 × [0, 1] with the circles c 1 and c 2 by the degree p and q maps f p and f q .
We describe π 1 (E p (T (p, q))) by studying the homotopy type of (X p,q ) p .The (restricted) covering map π : (X p,q ) p → X p,q corresponds to the homomorphism π 1 (X p,q ) → Z p sending c 1 to q ∈ Z p and c 2 to 0 ∈ Z p .We use π * : π 1 ((X p,q ) p ) → π 1 (X p,q ) to denote the induced map.Let a be the pre-image π −1 (c 1 ) and let b 0 , . . ., b p−1 be the components of the pre-image π −1 (c 2 ); we choose the indices of the b i 's so that Since π is a covering map, the induced map π * : π 1 ((X p,q ) p ) → π 1 (X p,q ) is injective.For this reason, we shall often identify b i with µ i c 2 µ −i .Since X p,q is a double mapping cylinder, so is (X p,q ) p .More precisely as illustrated in Figure 3, we have where each S 1 i × {0} is identified with the circle a by the identity map, and S 1 i × {1} is identified with the circle b i by the degree q map.By van Kampen's theorem, we deduce that Since E p (T (p, q)) deformation retracts onto (X p,q ) p , we obtain the following proposition.Proposition 2.2.Let π : E p (T (p, q)) → E(T (p, q)) be the p-fold cyclic covering and let b 0 , b 1 , . . ., b p−1 be the homotopy classes of the components of π p copies Figure 3.The p-fold cyclic cover (X p,q ) p of X p,q is also a double mapping cylinder, where f 1 and f q denote the degree 1 and the degree q maps, respectively.
Next, we use this description of π 1 (E p (T (p, q))) to obtain generators of the finite abelian group First, note that Proposition 2.2 shows that H 1 (E p (T (p, q))) ∼ = Z⊕Z p−1 q has generators b 0 , b 1 , . . .b p−1 and relations qb i = qb j for each i, j.In the remainder of this section, we describe a set of generators that will be more convenient for the twisted Alexander polynomial computations of Section 3.
Remark 2.3.While the meridian µ of T (p, q) does not lift to E p (T (p, q)), a loop representing µ p does.Since the projection induced map π * : π 1 (E p (T (p, q))) → π 1 (E(T (p, q))) is injective, we slightly abuse notations and also write µ p for the homotopy class of this lift in π 1 (E p (T (p, q))).
In what follows, we make no notational distinction between elements in π 1 (E p (T (p, q))) and elements in H 1 (E p (T (p, q))), despite switching from multiplicative to additive notations.In some rare instances, we will also use the multiplicative notation in homology.Keeping this in mind, The next proposition describes the homology group H 1 (Σ p (T (p, q))) as a Z[Z p ]-module.
Proposition 2.4.The abelian group H 1 (Σ p (T (p, q))) ∼ = Z p−1 q is generated by the x i = b i − µ p , and these elements satisfy the following relations: (1) where t denotes the covering transformation of Σ p (T (p, q)).
In particular, there exists an isomorphism of Z[t ±1 ]-modules Proof.The proof has four steps.Firstly, we establish a criterion for an element in H 1 (Σ p (T (p, q))) to be torsion; secondly, we prove that the x i are torsion; thirdly, we show that that x i generate T H 1 Σ p (T (p, q)) as an abelian group; fourthly and finally we prove that the x i satisfy the two identities stated in the lemma.
We assert that an element )) to zero and maps the infinite cyclic summand isomorphically onto pZ ∼ = Z c 2 . 1 In particular, a class x ∈ H 1 (E p (T (p, q))) is torsion if and only if π * (x) = 0. On the other hand, using Proposition 2.2, we deduce that π induces the following map on homology, concluding the proof of the assertion: We move on to the second step: we prove that the homology classes x 0 , . . ., x p−1 are torsion.Using the criterion, we must show that π * (x i ) = 0 for each i.Since π * (b i ) = 1, this reduces to showing that π * (µ p ) = 1.We start by computing the abelianization of µ p .Since µ = c k 1 c l 2 , we notice that in π 1 (E p (T (p, q))), the following equality holds: (5) In order to compute the abelianisation of this expression, we claim that for any 0 ≤ s ≤ p − 1, and any k, the equation Using consecutively (5), the equation µ s c 2 µ −s = c ks 1 c 2 c −ks 1 that we just established, and the identification b i = µ i c 2 µ −i from (4) (as well as the presentation in Proposition 2.1 and qk+pl = 1), we obtain the following sequence of equalities in H 1 (E p (T (p, q))): and therefore each of the x i is torsion.This concludes the second step of the proof.
Thirdly, we show that every element of T H 1 (E p (T (p, q))) can be written as a linear combination of the x i for i = 0, 1, . . ., p−1: given x = p−1 i=0 a i b i , adding and substracting µ p , using p−1 i=0 a i = 0 (which holds thanks to the first step) and the definition of x i , we obtain Fourthly and finally, we establish the relations x 0 + x 1 + • • • + x p−1 = 0 and x i = t i x 0 .The latter relation is clear (since b i = t i b 0 and tµ p = µ p ) and so we focus on the former.Using consecutively (6), the relation qb i = qb j , and the fact that pl + qk = 1, we notice that the following equation holds in H 1 (E p (T (p, q))): 1 For any knot K and prime power n, one has the decomposition H 1 (En(K)) = T H 1 (En(K)) ⊕ Z, where the Z summand is generated by a lift of the n-fold power of the meridian.
The conclusion now promptly follows from the definition of the x i , establishing the proposition.
Assume that q is a prime.In this case H 1 (Σ p (T (p, q))) becomes an F q -vector space.The covering action t is then an F q -linear endomorphism of V p,q .

Twisted polynomials of torus knots
In this this section, we compute the Alexander polynomial of the 0-framed surgery M T (p,q) twisted by a metabelian representation α T (p,q) (p, χ) : π 1 (M T (p,q) ) → GL p (C[t ±1 ]) that frequently appears in Casson-Gordon theory [13].In Subsection 3.1, we recall the definition of α K (p, χ) for a general knot K, in Subsection 3.2, we restrict to torus knots, and in Subsection 3.3, we compute the relevant twisted Alexander polynomials.
3.1.The metabelian representation α K (p, χ).In this subsection, given a knot K and a positive integer p, we recall the definition of the representation α K (p, χ) : [13].In what follows, E K denotes the exterior of K and M K denotes its 0-framed surgery.Finally, we use ξ m := e 2πi/m to denote the m-th primary root of unity.
We use (2) to denote the Alexander module of K.In what follows, we shall frequently identify , as for instance in [6,Corollary 2.4].Consider the following composition of canonical projections: (1) .Combining these notations, we consider the following representation: Note that α(p, χ) can equally well be defined on π 1 (M K ) instead of π 1 (E K ): the definition can be adapted verbatim, and we use the same notation: A closely related observation is that α(p, χ) is a metabelian representation and therefore vanishes on the longitude of K; this also explains why α K (p, χ) descends to π 1 (M K ) .

3.2.
An explicit description of α T (p,q) (p, χ).We use the presentation of π 1 (E T (p,q) ) from Proposition 2.1 to describe the representation α T (p,q) (p, χ).In this subsection, we will often set K := T (p, q) in order to avoid cumbersome notations such as q T (p,q) .We recall the definition of the generators x 0 , . . ., x p−1 of H 1 (Σ p (K)) ∼ = Z p−1 q described in Proposition 2.4, referring to Section 2 for further details.Using the notations of that section, we set x i = b i − µ p , where µ is a meridian of K. Thinking of x i as the abelianisation of µ −p b i , and using Proposition 2.2 to identify b i with µ i c 2 µ −i , we have (9) Recall furthermore that Proposition 2.4 also established the relations x 0 + • • • + x p−1 = 0 as well as t K x i = x i+1 .The next result follows immediately from these considerations.
Recall that Proposition 2.1 described a two-generator one-relation presentation for the knot group π 1 (E T (p,q) ): the generators were denoted by c 1 and c 2 , and the unique relator was c p 1 c −q 2 .The next proposition describes the image of these generators under α(p, χ) := α T (p,q) (p, χ).This will be useful in Proposition 3.3 when we compute the twisted Alexander polynomial of E T (p,q) .Proposition 3.2.Let p, q > 0 be two coprime integers.For a character χ = χ a on H 1 (Σ p (T (p, q))), the representation α(p, χ) is conjugated to a representation α (p, χ) such that Proof.We first compute α(p, χ)(c 2 ).We know that φ K (c 2 ) = p and A p (t) p = t • id.In order to compute the diagonal matrix which appears in the definition of α(p, χ)(c 2 ) (recall ( 8)), we use (9) and Lemma 3.
First, for any x ∈ H 1 (Σ p (K)) a computation shows that the following equation holds: α(x)A p (t) q α(x) −1 = A p (t) q α((t −q K − 1)x).Define y := q K (µ −q c 1 ) so that α(p, χ)(c 1 ) = A p (t) q α(y).Consequently, if we set X := α(z) (for any z ∈ H 1 (Σ p (K))), use the definition of y, the fact that α(y) and X commute (both are diagonal), and the aforementioned identity, then we obtain Therefore, if we choose z := −(t −q K − 1) −1 y, then (10) holds.For this to make sense however, we must argue that (t −q K − 1) is an automorphism of H 1 (Σ p (K)).This is indeed the case: as t K − 1 is an automorphism of H 1 (Σ p (K)), the inverse is given by (t ), where qk ≡ 1 mod p.Such a k exists because p and q are coprime.We have therefore found X such that (10) and (11) hold, and this concludes the proof of the proposition.

3.3.
The computation of the twisted polynomial.In this subsection, we compute the twisted Alexander polynomial of the 0-framed surgery M T (p,q) with respect to α(p, χ).
Recall that given a space X and a representation ρ : π 1 (X) → GL p (C[t ±1 ]), the twisted Alexander polynomial ∆ ρ 1 (X) is defined as the order of the twisted Alexander module H 1 (X; C[t ±1 ] p ρ ).More generally, we write ∆ ρ i (X) for the order of the C[t ±1 ]-module H i (X; C[t ±1 ] p ρ ).Recall that the ∆ ρ i (X) are defined up to multiplication by units of , where E T (p,q) denotes the exterior of T (p, q).Proposition 3.3.Let p, q > 0 be coprime integers.For χ = χ a : H 1 (Σ p (T (p, q))) → GL p (C[t ±1 ]), the metabelian twisted Alexander polynomial of E T (p,q) is given by .
Proof.We use τ α(p,χ) (E K ) to denote the Reidemeister torsion of a knot exterior E K twisted by α(p, χ) := α K (p, χ).We refer to [7] for further references on the subject, but simply note that τ α(p,χ) (E K ) is defined since the chain complex we are reduced to computing τ α(p,χ) (E T (p,q) ).By [17, Theorem A], this torsion invariant can be expressed via Fox calculus.In our case, using the presentation of π 1 (E T (p,q) ) resulting from Proposition 2.1, we obtain (12) ∆ .
We will now compute the numerator of ( 12) and show that it equals (1−t q ) p−1 .Recall from (8) that for g ∈ π 1 (E K ), the metabelian representation α K (p, χ) is given by α where D g is the diagonal matrix with ξ as its i-th diagonal component.An inductive argument involving the properties of the Fox derivative shows that We will now apply α(p, χ) to g.We recall from Proposition 3.2 that α (p, χ)(c 1 ) = A p (t) q , and we now work over C[t ±1/p ].Indeed, as observed in [13, page 935], in this ring, the matrix A p (t) is conjugated to the diagonal matrix B p (t) := diag(t 1/p , ξ p t 1/p , ξ 2 p t 1/p , . . ., ξ p−1 p t 1/p ).
Since (12) only depends on the conjugacy class of the representation α(p, χ), we can work with B p (t) instead of A p (t).We use ∼ to denote the conjugacy relation.Since B p (t) is diagonal, its powers are easy to compute, and as a consequence, we obtain Taking the determinant of this expression, we deduce that Plugging ( 13) and ( 14) into (12) concludes the proof of the proposition.Using Proposition 3.3, we can compute the twisted polynomial of the 0-framed surgery M T (p,q) .Corollary 3.4.Let p, q > 0 be two coprime integers.For χ = χ a : H 1 (Σ p (T (p, q))) → GL p (C[t ±1 ]), the metabelian twisted Alexander polynomial of M T (p,q) is given by .

Linking forms and their metabolisers
This section collects some facts about linking forms and their metabolizers.This will be useful in Section 5 since both the metabelian Blanchfield pairing and λ p (T (p, q)) are linking forms.In Subsection 4.1, we recall some basics on linking forms and their Witt groups.In Subsection 4.2, we prove a result on metabolisers of linking forms of the type ( 4.1.The Witt group of linking forms.Let R be a PID with involution, and let Q denote its field of fractions.This subsection is concerned with linking forms.Firstly, we recall the definition of the Witt group W (Q, R) of linking forms.Secondly, we collect some facts about W (C(t), C[t ±1 ]) that are used in Section 5 below.
A linking form over R is a pair (V, λ), where V is a torsion R-module, and λ : V ×V → Q/R is a sesquilinear and Hermitian pairing.A linking form (V, λ) is non-singular if its adjoint λ • : V → V * , x → λ(x, −) is an isomorphism.In the sequel, our linking forms will be either over Z or C[t ±1 ].From now on, we also assume that all linking forms are non-singular.Given a linking form (V, λ) over R, a submodule L ⊂ V is isotropic if L ⊂ L ⊥ and is a metaboliser if L = L ⊥ .A linking form is metabolic if it admits a metabolizer.Definition 4.1.The Witt group of linking forms, denoted W (Q, R), consists of the monoid of linking forms modulo the submonoid of metabolic linking forms.Two linking forms (V, λ) and (V , λ ) are called Witt equivalent if they represent the same element in W (Q, R).
The Witt group of linking forms is known to be an abelian group under direct sum, where the inverse of the class [(V, λ)] is represented by (V, −λ).Next, we collect some facts on W (C(t), C[t ±1 ]) that will be used in Section 5 below.
Remark 4.2.The Witt group W (C(t), C[t ±1 ]) is known to be free abelian and is detected by the signature jumps δσ (V,λ) [3,Sections 4 and 5].In particular, a linking form (V, λ) over C[t ±1 ] is metabolic if and only if all its signature jumps vanish [3,Theorem 5.3].Reformulating, [V, λ] = 0 in W (C(t), C[t ±1 ]) if and only if δσ (V,λ) (ω) = 0 for all ω ∈ S 1 .We refer to [3, Sections 4 and 5] for further details regarding signatures of linking forms but note that a linking form (V, λ) will have a trivial jump at ω ∈ S 1 if the order Ord(T ) of the C[t ±1 ]-module T does not have a root at ω.
In particular, Remark 4.2 implies the following result about linear independence in and Ord(V 2 ) have distinct roots, then the following assertions hold: Proof.We only prove the first assertion as the second assertion follows immediately.Assume = 0 for some integers n 1 and n 2 .Remark 4.2 implies that all the signature jumps of n 1 λ 1 ⊕ n 2 λ 2 must vanish.Since λ 1 is not metabolic, Remark 4.2 also implies that λ 1 admits a non-trivial signature jump at some ω 1 ∈ S 1 .As a consequence of these two assertions, we infer that n 1 λ 1 and n 2 λ 2 must have a non-trivial signature jump at ω 1 .Since Ord(V 1 ) and Ord(V 2 ) have distinct roots, we deduce that n 1 = 0.The same reasoning shows that n 2 = 0, thus establishing the linear independence of [V 1 , λ 1 ] and [V 2 , λ 2 ] and establishing the proposition.4.2.Graph metabolisers.Given linking forms (V 1 , λ 1 ), (V 2 , λ), we prove a result on metabolisers of linking forms of the type (V 1 ⊕V 2 , λ 1 ⊕−λ 2 ).More precisely, Proposition 4.4 provides a criterion for when such a metabolizer must be a graph.This result will be used in Section 5 when we study metabolisers of λ p (T (p, q)) N ⊕ −λ p (T (p, q)) N .
Given linking forms (V 1 , λ 1 ) and (V 2 , λ 2 ), a morphism of linking forms is an R-linear homomorphism f : V 1 → V 2 such that λ 2 (f (x), f (y)) = λ 1 (x, y) for all x, y ∈ V 1 .Observe that if the forms are non-singular, then a morphism is necessarily injective.An isometry of linking forms is a bijective morphism of linking forms.The graph The next proposition provides an assumption under which the converse also holds.
The following assertions hold: (2) if we additionally work over R = Z, suppose that V 1 and V 2 are equipped with an isometric Z p −action, and L is a Z p -invariant metaboliser, then the isometry f is Z p -equivariant.
Proof.We prove the first assertion.The isometry f will be defined by using the canonical projections pr Set W i := pr i (L), for i = 1, 2, and define f as the composition Since f is an isomorphism of R-modules, it remains to check that it is a morphism of linking forms.First however, we use the definition of f to observe that The fact that f is a morphism now follows from the fact that L is isotropic: for any v, w ∈ W 1 , the pairs (v, f (v)), (w, f (w)) belong to L, and therefore we have Looking at (15), it only remains to show that V 1 = W 1 and V 2 = W 2 .Since f is an isomorphism, we have ord(W 1 ) = ord(W 2 ) and therefore (15) implies that ord(L) 2 = ord(W 1 ) ord(W 2 ).Since L is a metaboliser, we deduce that ( 16) By way of contradiction, assume that ord(W 1 ) divides ord(V 1 ), but that ord(W 1 ) = ord(V 1 ); we write ord(W 1 ) ord(V 1 ).A glance at (16) shows that ord(V 2 ) ord(W 2 ), contradicting the inclusion W 2 ⊂ V 2 .We conclude that ord(W i ) = ord(V i ) and consequently W i = V i , for i = 1, 2. This concludes the proof of the first assertion.
We prove the second assertion.Use t to denote a generator of Z p .As the metaboliser and thus f is Z p -equivariant, as desired.This concludes the proof of the proposition.

Non-slice linear combinations of iterated torus knots
This section aims to prove Theorem 1.1 from the introduction, whose statement we now recall.For an integer p ≥ 2 and a sequence Q = (q 1 , q 2 , . . ., q ) of integers that are relatively prime to p, we use the following notation for iterated torus knots: T (p, Q) := T (p, q 1 ; p, q 2 ; . . .; p, q ).Our main result reads as follows.
Theorem 5.1.Fix a prime power p.Let S p be the set of iterated torus knots T (p, q 1 ; p, q 2 ; . . .; p, q ), where the sequences (q 1 , q 2 , . . ., q ) of positive integers that are coprime to p satisfy (1) q is a prime; (2) for i = 1, . . ., − 1, the integer q i is coprime to q when > 1; The set S p is linearly independent in the topological knot concordance group C top .
To prove Theorem 5.1, we must obstruct the sliceness of linear combinations of knots belonging to S p .The first step, which is carried out in Subsection 5.1, is to determine which of these linear combinations are algebraically slice.In Subsection 5.2, we use metabelian twisted Blanchfield pairings to obstruct the sliceness of such algebraically slice linear combinations.

5.1.
Algebraically slice linear combinations of algebraic knots.Fix an integer p ≥ 2. For i = 1, . . ., k, fix sequences Q i = (q i,1 , q i,2 , . . ., q i, i ) of i positive integers each of which is coprime to p, and let n 1 , . . ., n k ∈ Z.The goal of this subsection is to determine when the following knot is algebraically slice: In order to provide a convenient criterion, we define the s-level of K to be the following knot: Here, it is understood that T (p, As an example of this notation, we see that if Q = (q 1 , . . ., q ), then K s (T (p, Q)) = T (p, q −s ) for 0 ≤ s ≤ − 1 and K s (T (p, Q)) = U , for s ≥ .In particular, the cabling formula for the classical Blanchfield form implies that ( 18) Indeed, for a knot L, the cabling formula reads as Bl(L p,q )(t) = Bl(T (p, q))(t) ⊕ Bl(L)(t p ) [22].
Next, we move on to a slightly more involved example.
In particular, observe that the formula displayed in (18) also holds for J.As we shall use in Proposition 5.4 below, it holds for the linear combination of (32).
For later use, we note that the 0-level of K is the most important to us: the first homology of its p-fold branched cover equals that of K.
The next proposition uses s-levels to exhibit a criterion for the algebraic sliceness of K.
Proposition 5.4.Fix an integer p ≥ 2 and choose sequences of positive integers Q i = (q i,1 , . . ., q i, i ) that are relatively prime to p, for i = 1, 2, . . ., k.The following statements are equivalent: (1) the knot Proof.We first assert that the polynomials ∆ Ks(K) (t p s ) and ∆ Ku(K) (t p u ) have distinct roots if s = u.For a positive integer m, we set ξ m := e 2πi/m .The roots of ∆ T (p,q) (t) occur at those ξ a pq where the integer 1 ≤ a ≤ pq is such that neither p nor q divides a, i.e. ξ a pq p = 1 and ξ a pq q = 1.Consequently, the roots of ∆ T (p,q) (t p s ) occur at ξ a p s+1 q such that 1 ≤ a ≤ p s+1 q and neither p nor q divides a.
We argue that if s = u, then ∆ T (p,q1) (t p s ) and ∆ T (p,q2) (t p u ) have distinct roots.Assume to the contrary that they have a common root.This root must be of the form ξ a p s+1 q1 = ξ b p u+1 q2 where q 1 , p (resp.q 2 , p) do not divide a (resp.b).Without loss of generality, assume that s < u so that 1 = (ξ a p s+1 q1 ) p s+1 q1 = (ξ b p u+1 q2 ) p s+1 q1 = ξ bq1 p u−s q2 .This implies that p u−s q 2 divides bq 1 .However, by assumption, p divides neither q 1 nor b, yielding the desired contradiction.
Next, recall from the definition of the s-level that Thus, if s = u, then ∆ Ks(K) (t p s ) and ∆ Ku(K) (t p u ) have distinct roots.This proves the assertion.Assume that K is algebraically slice.By the cabling formula for the Blanchfield pairing (see Example 5.2), (19) Bl is metabolic.By the assertion and Proposition 4.3, we deduce that each Bl(K s (K))(t p s ) is metabolic.It follows that the jump function of each Bl(K s (K))(t p s ) is trivial which is simply a reparametrization of the jump function of Bl(K s (K))(t) where the parameter t ∈ S 1 is changed to t p r .Hence K s (K) is a connected sum of torus knots such that the jump function of σ ω (K s ) is trivial.Since Litherland showed in [19, Lemma 1] that the jump functions of σ ω (T (p, q)) are linearly independent, K s (K) is slice as desired.
Assume that each K s (K) is slice.As a linking form over Z[t ±1 ], Bl(K s (K)) is metabolic.Combining this with the decomposition displayed in (19), we deduce that Bl(K) is metabolic, as a linking form over Z[t ±1 ].This is equivalent to K being algebraically slice [15], completing the proof of Proposition 5.4.
When K is algebraically slice, we obtain a convenient description of the 0-level of K.
Corollary 5.5.Suppose that K, p, i and Q i , for i = 1, . . ., k, are as in Proposition 5.4.If K is algebraically slice, then k is even and, after renumbering if necessary, the 0-level of K is Proof.By Proposition 5.4, K 0 (K) is a slice linear combination of torus knots.Since torus knots are linearly independent in the knot concordance group, the conclusion follows.5.2.Linear independent families of iterated torus knots.Fix a prime power p.The goal of this section is to prove Theorem 5.1 whose statement we briefly recall.Let S p be the set of iterated torus knots T (p, Q), where the sequences Q = (q 1 , q 2 , . . ., q ) of i positive integers are coprime to p and satisfy (1) q is a prime; (2) for i = 1, . . ., − 1, the integer q i is coprime to q when > 1; Theorem 5.1 states that S p is linearly independent in the topological knot concordance group C top .
For i = 1, . . ., k, we therefore choose sequences Q i = (q i,1 , q i,2 , . . ., q i, i ) of positive integers where q i, i is prime for all i, and the integer q i,j is coprime to p and to q i, i for all j.We also let n 1 , . . ., n k ∈ Z be integers.We will use metabelian Blanchfield pairings [3,23] to obstruct the sliceness of the knot The sliceness obstruction that we will use, and which is due to Miller-Powell [23,Theorem 6.10], reads as follows.If for every Z p -invariant metaboliser G of λ p (K), there exists a prime power order character χ that vanishes on G and such that Bl α(p,χ) (K) is not metabolic, then K is not slice.Here, we use α(p, χ) := α K (p, χ) to denote the metabelian representation that was described in Subsection 3.1.
Remark 5.6.The metabelian Blanchfield pairing is a linking form Bl α(p,χ) (K) : where ) denotes the homology of the 0-framed surgery of K twisted by α(p, χ).The precise definition of Bl α(p,χ) (K) is not needed in this paper (the interested reader can nonetheless find it in [23] and [3]).All we need is the behavior of Bl α(p,χ) (K) under satellite operations, and this will be recalled as the argument proceeds.
The strategy behind the proof of Theorem 5.1 is as follows.
(2) Secondly, we study the consequences of Bl α(p,χ) (K) being metabolic.This will impose substantial restrictions on χ. (3) Thirdly, we build characters that violate these restrictions.(4) Finally, we combine these first three steps to conclude the proof.
The reader that wishes to see how these steps combine might consider starting with a glance at the end of the argument, after the conclusion of the proof of Lemma 5.9; see Subsection 5.2.4.

5.2.1.
Characters on H 1 (Σ p (K)).Assume that K is slice.The first step is to study the possible characters on the p-fold branched cover of K. Since K is algebraically slice, Corollary 5.5 implies that k is even and, after renumbering if necessary, for some prime r (which is one of the q j, j ) and some integers m 1 , . . ., m k/2 , we can write # j=2 m j T (p, q j, j )# − T (p, q j, j ) , where q i, i = r if and only if 1 ≤ i ≤ 2m 1 .It follows that if we set M j = m 1 + m 2 + • • • + m j−1 , for j = 2, . . ., k/2, then after further possible renumbering, the knot K can be rewritten as ), the description of K 0 (K), the primary decomposition, and the fact that the q i, i are prime shows that The linking form λ p (K) on Σ p (K) decomposes analogously.
From now on, θ denotes the trivial character.Also, since , we write characters H 1 (Σ p (T (p, r))) → Z r as χ a where a ∈ Z p r .Since r is distinct from q i, i for i > 2m 1 , the decomposition of (21) implies that any character χ : H 1 (Σ p (K)) → Z r must be of the form (22) χ where {a j } m1 j=1 and {b j } m1 j=1 are sequences of p elements in Z r .
Remark 5.7.Recall that the Miller-Powell obstruction requires that for every Z p -invariant metaboliser G of λ p (K), we construct a prime power order character χ that vanishes on G and such that Bl α(p,χ) (K) is not metabolic.The primary decomposition implies that every such metabolizer decomposes as a direct sum of metabolisers of the summands in (21).Consequently, thanks to the form of the character in (22), it is sufficient to prove the following result: for every Z p -invariant metaboliser L of λ p (T (p, r)) m1 ⊕ −λ p (T (p, r)) m1 , there is a prime power order character m1 i=1 (χ a i ⊕ χ b i ) that vanishes on L and such that Bl α(p,χ) (K) is not metabolic, with χ as in (22).5.2.2.The metabelian Blanchfield pairing of K. We now study the metabelian Blanchfield pairing of K. We first use satellite formulas to decompose it, and we then study the implications of it being metabolic.We use α(p, χ) := α K (p, χ) to denote the metabelian representation that was described in Subsection 3.1.The behavior of metabelian Blanchfield pairings under connected sums [3,Corollary 8.21] implies that Bl α(p,χ) (K) is Witt equivalent to the following linking form: For a sequence S = (q 1 , . . ., q k ), we use T (p, S) to denote the iterated torus knot T (p, q 1 ; . . .; p, q k−1 ).Next, we apply the satellite formula for the metabelian Blanchfield pairing [3,Theorem 8.19] to Summarising the content of these claims, we have shown that if the metabelian Blanchfield pairing Bl α(p,χ) (K) is metabolic, then the linking forms B χ 3 (q, s) ⊕ B 4 (q, s) are metabolic for all q, s.This concludes the second part of the proof.5.2.3.Building the characters that vanish on metabolisers.The third part consists in showing that for every Z p -invariant metaboliser L of λ p (T (p, r)) m1 ⊕ −λ p (T (p, r)) m1 there are characters χ a = m1 i=1 χ a i and χ b = m1 i=1 χ b i such that χ a ⊕ χ b vanishes on L, but for which the linking forms B χ 3 (q, s) ⊕ B 4 (q, s) are not all metabolic, where χ = χ a ⊕ χ b ⊕ θ is as in (22).The next proposition describes characters for which B χ 3 (q, s) ⊕ B 4 (q, s) is not metabolic.Proposition 5.8.Let q, s > 0 be positive integers with q coprime to p.If a character m1 i=1 χ a i ⊕ χ b i satisfies one of the following conditions: (1) χ b k = θ for every k ∈ I 2 (q, s) and χ a k 0 = θ for some k 0 ∈ I 1 (q, s), or, (2) χ a k = θ for every k ∈ I 1 (q, s) and χ b k 0 = θ for some k 0 ∈ I 2 (q, s), then the linking form B χ 3 (q, s) ⊕ B 4 (q, s) is not metabolic.
Proof.We will only consider case (1).In order to give the proof in case (2) just exchange the roles of χ a and χ b .Assume that χ b k = θ for every k ∈ I 2 (q, s) and χ a k 0 = θ for some k 0 ∈ I 1 (q, s).
We assert that the orders of the modules underlying the summands of the right hand side of (31) have distinct roots.First, note that r is coprime to q: as k ∈ I 1 (q, s), we know that q ∈ Q i for some i < 2m 1 , and since Q i = (q i,1 , q i,2 , . . ., q i, i−1 , r) for i < 2m 1 , this follows from the assumption of Theorem 5.1.It is known that ∆ T (p,q) (ξ a1 r t) and ∆ T (p,q) (ξ a2 r t) have distinct roots whenever a 1 = a 2 and r and q are coprime [12,Theorem 7.1].This establishes the assertion.
Thanks to the assertion, we may apply Proposition 5.8.Indeed, the fact that χ a k 0 = θ and Proposition 4.3 now guarantees that the linking form on the right-hand side of (31) is not metabolic.This concludes the proof of Proposition 5.8.
Before constructing the required characters, we introduce some terminology.We say that the knot K is simplified, if there are no indices k 1 ∈ I 1 (q, s) and k 2 ∈ I 2 (q, s) such that Lemma 5.9.Let p be a prime power.If the knot K is simplified, then for any Z p -invariant metabolizer L ⊂ H 1 (Σ p (T (p, r))) m1 ⊕H 1 (Σ p (T (p, r))) m1 there exist q, s and a character χ a ⊕χ b = m1 i=1 χ a i ⊕ χ b i vanishing on L such that one of the following conditions is satisfied: (1) χ b k = θ for every k ∈ I 2 (q, s) and χ a k 0 = θ for some k 0 ∈ I 1 (q, s), or, (2) χ a k = θ for every k ∈ I 1 (q, s) and χ b k 0 = θ for some k 0 ∈ I 2 (q, s).
onto the i-th factor.The proof is divided into three separate cases.Case 1: pr 1 (L) is a proper subspace of H 1 (Σ p (T (p, r))) m1 .In this case, we can define the characters χ a and χ b as follows: χ b = θ and It is not difficult to see that χ a and χ b satisfy (1) and are such that χ a ⊕ χ b vanishes on L.
Case 2: pr 2 (L) is a proper subspace of H 1 (Σ p (T (p, r))) m1 .In this case, we exchange the roles of χ a and χ b and repeat the argument from the first case.This way, we obtain characters χ a and χ b that satisfy (2) and are such that χ a ⊕ χ b vanishes on L.
Next, we use these sets and the anti-isometry g to describe a sufficient criterion to obtain the characters χ a , χ b required by the statement of Lemma 5.9.Claim 4. If there exist q, s such that g(S I1(q,s) ) = S I2(q,s) , then there are characters χ a , χ b satisfying either (1) or (2) and such that χ a ⊕ χ b vanishes on L.
Such choices of χ a and χ b satisfy condition (1).We verify that χ a ⊕χ b vanishes on L; where we recall that L is the graph of g.
If on the other hand, we assume that S I2(q,s) g(S I1(q,s) ) = ∅, and the argument is nearly identical.Choose v ∈ S I2(q,s) g(S I1(q,s) ) and write once more H 1 (Σ p (T (p, r))) m1 = v ⊕ W and define the required characters as These choices of χ a and χ b satisfy condition (2) and χ a ⊕ χ b vanishes on L. This concludes the proof of the claim.By Claim 4, to prove Lemma 5.9, it is enough to show that there always exist q, s such that g(S I1(q,s) ) = S I2(q,s) .Assume by way of contradiction that we have g(S I1(q,s) ) = S I2(q,s) for all q, s.We will show in Claim 5 below that this assumption implies that K is not simplified.This is a contradiction since we assumed that K is simplified.This proves Lemma 5.9 modulo Claim 5.
Claim 5.If g(S I1(q,s) ) = S I2(q,s) for all q, s, then K is not simplified.
We will need the following properties of these sets: The advantage of writing X(k 0 ) and Y (k 0 ) as intersections of the I j (q k0, k 0 −s , s) is that the action of g on S X(k0) can be described as This concludes the third part of the proof.We can now conclude.

Conclusion of the proof.
We can now prove Theorem 5.1.
Proof of Theorem 5.1.Let K be a (non-trivial) linear combination of iterated torus knots of the form T (p, Q i ) for i = 1, . . ., k.Here, the Q i = (q i,1 , q i,2 , . . ., q i, i ) are sequences of i positive integers where q i, i is prime for all i, and the integer q i,j is coprime to p and to q i, i for all j.
Assume that K is slice to obtain a contradiction.In particular K is algebraically slice and, as we saw in (20), we can therefore assume without loss generality that it is of the form ) .
-modules is acyclic [4, Corollary after Lemma 4].Since E K has torus boundary, by [7, Proposition 2, item 5], the twisted Reidemeister torsion and twisted Alexander polynomial are related by