Abelian invariants of doubly slice links
AABELIAN INVARIANTS OF DOUBLY SLICE LINKS
ANTHONY CONWAY AND PATRICK ORSON
Abstract.
We provide obstructions to a link in S arising as the cross section of any number ofunlinked spheres in S . Our obstructions arise from the multivariable signature, the Blanchfieldform and generalised Seifert matrices. We also obtain obstructions in the case of surfaces ofhigher genera, leading to a lower bound on the doubly slice genus of links. Introduction
A knot is doubly slice if it arises as the intersection of a locally flat unknotted 2-sphere in S with the equatorial S . In this article, we study several natural generalisations of this notion tolinks. Given an n -component oriented link L ⊂ S , and some 1 ≤ µ ≤ n , the link is called µ -doubly slice if it arises as the equatorial cross-section of µ oriented unlinked 2-spheres in S inducing the given orientation on L . The cases µ = 1 and µ = n have previously been studiedin [Don15, McD20, MM21] and are called respectively weakly and strongly doubly slice.Here is a sample of our results. Theorem.
Let L ⊂ S be an oriented n -component link.i) If L is weakly doubly slice, then its single variable rational Blanchfield form Bl Q L is hyperbolicand its Levine-Tristram signature vanishes identically: σ LTL ( ω ) = 0 for all ω ∈ S .ii) If L is strongly doubly slice, then its multivariable Blanchfield form Bl L is hyperbolic, and itsmultivariable signature vanishes identically: σ L ( ω ) = 0 for all ω ∈ ( S ) n . From now on all links are assumed to be oriented. In this paper we will develop severalobstructions to a link being µ -doubly slice using the multivariable signature, the multivariableBlanchfield form and generalised Seifert matrices; in other words, all these obstructions comefrom multivariable abelian invariants. Multivariable abelian invariants are in general defined withrespect to a choice of µ -colouring of an oriented link L (cf. [CF08]); that is a choice of an orderedpartition of the components into µ sublinks L = L ∪ · · · ∪ L µ . These µ -colourings naturally arisein the study of doubly slice links, as if L is µ -doubly slice then each component of L can be givena colour according to which 2-sphere it belongs. Except when µ = 1 or n , it may be the case thatsome µ -colourings of a link arise from µ -double slicing where some others may not. Thus for afixed µ , our task will be to individually rule out all µ -colourings from arising this way.We will also consider an extension of these ideas to higher genus surfaces, all of which weassume to be compact and orientable. The doubly slice genus of a knot is the minimal genusamong unknotted surfaces in S for which the knot appears as a cross section, and has beenstudied recently in [Che20, McD19, OP20]. Given an oriented n -component link L and 1 ≤ µ ≤ n we consider the µ -doubly slice genus (1) g µds ( L ) = min (cid:8) g ( (cid:70) µi =1 Σ i ) | L i = Σ i ∩ S for each i and (cid:70) µi =1 Σ i ⊂ S is unlinked (cid:9) , where a collection of surfaces in S is unlinked if it bounds a disjoint collection of handlebodiesin S . When µ = 1, one could call this the weak doubly slice genus, and this quantity is alwaysdefined. To see this, consider that every oriented link bounds a connected oriented surface in S and when such a surface is pushed into D , then doubled along the boundary, the result is unlinkedin S . When µ (cid:54) = 1, the set in equation (1) may be empty, and in that case we define g µds ( L ) := ∞ .Besides the abelian invariants that we will describe further below, we note there also existmore straightforward tools for studying µ -double sliceness. These will be introduced, as needed,to analyse specific links later in the paper, but we mention some now. For example, a µ -doubly a r X i v : . [ m a t h . G T ] J a n ANTHONY CONWAY AND PATRICK ORSON slice link is moreover ( µ − µ -component unlink of 2-spheresbounds µ disjoint 3-balls in S which we may assume meet the equatorial S transversely. Thusa µ -doubly slice link bounds a collection of µ disjoint surfaces in S (with the i -coloured sublinkbounding the i th surface). For µ (cid:54) = 1, this is a fairly restrictive condition and indeed for µ = n isthe well-known notion of a boundary link .Before stating our results, we briefly recall previous results concerning doubly slice knots andthe doubly slice genus of knots, for context.1.1. Doubly slice knots and doubly slice genus of knots.
Among the early results on doublyslice knots, Sumners [Sum71] proved that if a knot K is doubly slice, then there exists a Seifertsurface and a basis of curves so that the Seifert matrix is hyperbolic : A = (cid:18) BB T (cid:19) . A knot with a hyperbolic Seifert matrix is called algebraically doubly slice . In fact it is now knownthat every Seifert matrix for an algebraically doubly slice knot is hyperbolic [Ors17, Theorem 4.14].For any knot K , the Levine-Tristram signature σ LTK ( ω ) at ω ∈ S is the signature of the complexHermitian matrix (1 − ω ) A + (1 − ω ) A T , where A is any Seifert matrix for the knot (the signatureof this matrix is independent of the choice of A ). If K is algebraically doubly slice then theLevine-Tristram signature vanishes identically on S . It is also known that K is algebraic doublyslice if and only if its Blanchfield form is hyperbolic. Recall that the Blanchfield form of a knot K is a non-singular, sesquilinear, Hermitian pairingBl K : H ( X K ; Z [ t ± ]) × H ( X K ; Z [ t ± ]) −→ Q ( t ) / Z [ t ± ] , where X K = S (cid:114) νK denotes the knot exterior. The form is called hyperbolic if there are twomodules P , P ⊂ H ( X K ; Z [ t ± ]) such that P ⊕ P = H ( X K ; Z [ t ± ]) and P i = P ⊥ i for i = 1 , H ( X K ; Q [ t ± ]) splitsas G ⊕ G for some Q [ t ± ]-module G ; see [Lev89, Kea75, Kea81]. Here, we write G for the Q [ t ± ]-module whose underlying abelian group is G but with module structure given by p ( t ) · g = p ( t − ) g for g ∈ G and p ( t ) ∈ Q [ t ± ].Beyond algebraic double sliceness, metabelian invariants are also known to obstruct doublesliceness [GL83, LM15], as are L -invariants [Kim06, CK19]. In the smooth category, obstructionshave been obtained using the correction terms from Heegaard-Floer homology [Mei15, KK18].Concerning the doubly slice genus, less is known, although the field seems to be advancingbriskly. In [OP20], the second named author and Mark Powell proved that for every ω ∈ S | σ K ( ω ) | ≤ g ds ( K ) . Other lower bounds were obtained earlier by W. Chen using Casson-Gordon invariants [Che20],while upper bounds were described by McDonald [McD19].1.2.
Multivariable signature obstructions.
Recall that our strategy to obstruct µ -doublesliceness of links will be to obstruct it for each given µ -colouring for the link. Definition 1.1.
Let L = L ∪ · · · ∪ L µ be an n -component µ -coloured link.i) The link L is doubly slice (with respect to the µ -colouring) if there exists a locally flat orderedoriented µ -component unlink S = S (cid:116) . . . (cid:116) S µ ⊂ S such that L i = S i ∩ S for i = 1 , . . . , µ ,where S ⊂ S is the standard equator.ii) The doubly slice genus g ds ( L ) of L (with respect to the µ -colouring) is the minimal genusamong ordered unlinked µ -component locally flat, embedded, closed, oriented surfaces in S for which L arises as a cross section: g ds ( L ) = min (cid:8) g ( (cid:70) µi =1 Σ i ) | L i = Σ i ∩ S for each i and (cid:70) µi =1 Σ i ⊂ S is unlinked (cid:9) . If L does not arise in this way we write g ds ( L ) = ∞ . BELIAN INVARIANTS OF DOUBLY SLICE LINKS 3
As mentioned above, the Levine-Tristram signature of a doubly slice knot vanishes and providesa lower bound on the doubly slice genus. To describe the generalisation of this result to links, weset T µ = ( S ) µ and recall from [CF08] that the multivariable signature of an oriented µ -colouredlink L is a function σ L : T µ −→ Z that generalises the Levine-Tristram signature σ LTL : S → Z . In fact for µ = 1, the multivariablesignature coincides with σ LTL . The multivariable signature is known to obstruct links from bound-ing collections of surfaces in D [CF08, CNT20, Vir09] (and in more general 4-manifolds [CN20])and from arising as the cross section of surfaces in S for certain proper subsets of T µ [CF08,Theorem 7.3].Our first result, proved in Theorem 3.4, is the following lower bound on the doubly slice genusof a µ -coloured link; note that this bound holds for all ω ∈ T µ . Theorem 1.2. If L is a µ -coloured link, then for all ω ∈ T µ one has | σ L ( ω ) | ≤ g ds ( L ) . In particular, the multivariable signature of a doubly slice coloured link vanishes.
Applying Theorem 1.2 with µ = 1, we obtain: Corollary 1.3.
If an oriented link L arises as the cross section of a single unknotted genus g surface Σ ⊂ S , then for all ω ∈ S , one has | σ LTL ( ω ) | ≤ g. In particular, the Levine-Tristramsignature of a weakly doubly slice link vanishes.
Corollary 1.3 immediately implies that following result.
Corollary 1.4.
If an n -component ordered link L arises as the cross section of an n -componentordered unlinked genus g surface Σ = Σ (cid:116) . . . (cid:116) Σ n ⊂ S , then for all ω ∈ S , one has | σ LTL ( ω ) | ≤ g. In particular, the Levine-Tristram signature of a strongly doubly slice link vanishes.
Applying Theorem 1.2 with µ = n , we obtain: Corollary 1.5.
If an n -component ordered link L arises as the cross section of an n -componentunlinked genus g surface Σ = Σ (cid:116) . . . (cid:116) Σ n ⊂ S , then for all ω ∈ T n , one has | σ L ( ω ) | ≤ g. Inparticular, the multivariable signature of a strongly doubly slice link vanishes.
Given a µ -coloured link L , Cimasoni and Florens also introduced a multivariable nullity function η L : T µ → Z [CF08, Section 2]. This function is non-zero at ω ∈ ( S (cid:114) { } ) µ =: T µ ∗ if and onlyif ∆ L ( ω ) = 0 and, among other properties, is known to obstruct L from arising as a cross sectionof surfaces in S for certain proper subsets of T µ ∗ [CF08, Theorem 7.3]. We show for all ω ∈ T µ ∗ that the nullity η L provides an obstruction to a coloured link being doubly slice. Proposition 1.6.
If a µ -coloured link L is doubly slice, then η L ( ω ) ≥ µ − for all ω ∈ T µ ∗ . The Blanchfield pairing of doubly slice links.
As we mentioned above, the Blanchfieldpairing of a doubly slice knot K is hyperbolic. The second goal of this paper is to generalise thisresult to links. We define:Λ := Z [ t ± , . . . , t ± µ ] , Λ S := Z [ t ± , . . . , t ± µ , (1 − t ) − , . . . , (1 − t µ ) − ] , Q := Q ( t , . . . , t µ ) , and denote by X L := S (cid:114) νL the link exterior, i.e. the complement of an open tubular neigh-bourhood of L . We also write H ∗ ( X L ; Λ) for the homology of the Z µ -cover of X L that arisesfrom the homomorphism π ( X L ) → Z µ ; γ (cid:55)→ ( (cid:96)k ( L , γ ) , . . . , (cid:96)k ( L µ , γ )). Deck transformationsendow H ∗ ( X L ; Λ) with the structure of a Λ-module and we set H ( X L ; Λ S ) = H ( X L ; Λ) ⊗ Λ Λ S ,referring to Section 2 for further discussion of twisted homology. Before focusing on the torsion sub-module of H ( X L ; Λ S ), we note the following fact about its rank (proved in Proposition 2.9), whichis sometimes referred to as the Alexander nullity of the coloured link: β ( L ) := rk Λ H ( X L ; Λ). Proposition 1.7.
If a µ -coloured link L is doubly slice, then β ( L ) ≥ µ − . In particular, for µ > ,the Alexander polynomial of a doubly slice µ -coloured link L vanishes identically. ANTHONY CONWAY AND PATRICK ORSON
In this article, the Blanchfield pairing of a µ -coloured link L will refer to a sesquilinear Hermitianpairing on the torsion submodule T H ( X L ; Λ S ) ⊂ H ( X L ; Λ S )Bl L : T H ( X L ; Λ S ) × T H ( X L ; Λ S ) −→ Q/ Λ S , whose precise definition can be found in Subsection 4.1. If µ = 1 and if we work over the PID Q [ t ± , (1 − t ) − ] instead of Z [ t ± , (1 − t ) − ], then we refer to the resulting pairing as the (single-variable) rational Blanchfield pairing of L . Remark 1.8.
It is also possible to define a Blanchfield pairing on
T H ( X L ; Λ) ⊂ H ( X L ; Λ),but there are several algebraic advantages to working over the ring Λ S , and we will use these inthe paper. Firstly H ( X L ; Λ S ) always admits a square presentation matrix [CF08, Corollary 3.6],while this is not true for H ( X L ; Λ) [CS69]. Secondly Bl L can be described using generalisedSeifert matrices [Con18, CFT18] while no such result is known over Λ. Thirdly, over Λ S there arecriteria under which Bl L is known to be nonsingular [Hil12]. In particular, if L is a boundary link,then Bl L is non-singular; see Remark 4.2 and Proposition 4.3.We say the Blanchfield form Bl L is hyperbolic if there are submodules P , P ⊂ T H ( X L ; Λ S )such that T H ( X L ; Λ S ) = P ⊕ P and P i = P ⊥ i for i = 1 ,
2. What follows is a particular case ofTheorem 4.7 which is our main result about the Blanchfield pairing of doubly slice coloured links.
Theorem 1.9.
Let L be an n -component oriented link.i) If L is strongly doubly slice, then its n -variable Blanchfield pairing is hyperbolic.ii) If L is weakly doubly slice, then its single-variable rational Blanchfield pairing is hyperbolic.iii) If L is weakly doubly slice and also a boundary link, then its single-variable Blanchfield pairingis hyperbolic. Since these conditions are difficult to apply in practice, we also formulate criteria that areeasier to verify. Let R be a Noetherian ring that is a unique factorisation domain. The order ofan R -module H is the greatest common divisor of the ideal in R generated by all m × m minorsof an m × n -presentation matrix for H . For instance, the (multivariable) Alexander polynomialof a coloured link L is the order of H ( X L ; Λ). Given f = f ( t , . . . , t µ ) ∈ Q ( t , . . . , t µ ), we write f for the rational function f ( t − , . . . , t − µ ). The following result, which is a corollary of the proof ofTheorem 1.9, provides a restriction on the torsion submodule T H ( X L ; Λ S ) of the Λ S -Alexandermodule of a doubly slice link. Proposition 1.10.
The following assertions hold:i) if L is strongly doubly slice, then there exist submodules G , G ⊂ T H ( X L ; Λ S ) with Ord( G ) =Ord( G ) such that T H ( X L ; Λ S ) = G ⊕ G ; ii) if L is weakly doubly slice, then there exists a submodule G ⊂ T H ( X L ; Q [ t ± , (1 − t ) − ]) such that T H ( X L ; Q [ t ± , (1 − t ) − ]) = G ⊕ G ; iii) if L is weakly doubly slice and also a boundary link, then there exist submodules G , G ⊂ T H ( X L ; Z [ t ± ]) with Ord( G ) = Ord( G ) such that T H ( X L ; Z [ t ± ]) = G ⊕ G . The obstruction given by Proposition 1.10 is fairly computable. Indeed, for boundary links [Con18,Theorem 4.6] shows that a presentation matrix for
T H ( X L ; Λ S ) can be calculated by using gen-eralised Seifert matrices, and we describe this in more detail in Subsections 1.4 and 5.4.We speculate that a completely general coloured version of Theorem 1.9 might hold if weconsider the Blanchfield form as a pairing on (cid:98) tH ( X L ; Λ S ), where given a Λ S -module M , wewrite (cid:98) tM = T M/zM for the quotient of
T M by its so-called maximal pseudo-null submodule zM ;see Remark 4.2. We will not pursue this question here as the resulting coloured obstruction wouldbe unwieldy.Use Σ ( L ) to denote the double branched cover of S along L . Arguments similar, but simpler,than those from the proof of Theorem 1.9 also give the following result. BELIAN INVARIANTS OF DOUBLY SLICE LINKS 5
Proposition 1.11.
The linking form λ Σ ( L ) : T H (Σ ( L )) × T H (Σ ( L )) → Q / Z of a weaklydoubly slice link L is hyperbolic, and T H (Σ ( L )) = G ⊕ G for some finite abelian group G . Generalised Seifert matrices of doubly slice links.
Since doubly slice knots admithyperbolic Seifert matrices, it is also natural to investigate constraints imposed by double slicenesson (generalisations of) the Seifert matrix. For simplicity, we only state our results for stronglydoubly slice links, referring to Section 5 for the general coloured case.Strongly doubly slice links are boundary links. Associated to a boundary link L = K ∪ . . . ∪ K n is a boundary Seifert matrix A = ( A ij ) ≤ i ≤ j ≤ n which consists of n square matrices where A ii isa Seifert matrix for K i and A Tij = A ij [Ko87]. Each of these matrices determines a bilinear pair-ing H i × H j → Z , and a boundary Seifert matrix is hyperbolic if there are submodules G ± , . . . , G ± n such that rk( G ± i ) = rk( H i ) , H i = G − i ⊕ G + i , and A vanishes on G ± i × G ± j . When n = 1, thisrecovers the usual notion of a hyperbolic matrix. A particular case of Theorem 5.8, which is ourfirst obstruction to double sliceness in terms of Seifert matrices, reads as follows: Theorem 1.12.
A strongly doubly slice link admits a hyperbolic boundary Seifert matrix.
We describe how Theorem 1.12 can be used to recover some of our previous obstructions.By [Cim04, Lemma 1], every µ -coloured link L bounds a C -complex, i.e. a collection of Seifertsurfaces F = F ∪ . . . ∪ F µ that are either disjoint or intersect pairwise along clasp intersections;see Definition 5.10 for details. Following [Coo82b, Coo82a, Cim04], given a sequence of signs ε =( ε , . . . , ε µ ), there is a generalised Seifert matrix A ε that is obtained by computing linking numbersof the form (cid:96)k ( i ε ( x ) , y ), where x, y ∈ H ( F ) and i ε ( x ) denotes the curve obtained from x by pushingit off F in the direction prescribed by ε . Summarising, a choice of a C -complex F and of a basisfor H ( F ) leads to a collection of generalised Seifert matrices { A ε } for L . One can then combinethese 2 µ matrices to obtain a C -complex matrix H ( t , . . . , t µ ) = (cid:88) ε µ (cid:89) i =1 (1 − t ε i i ) A ε . When µ = 1, a C -complex is a Seifert surface, A := A − is a Seifert matrix for the oriented link L (with A + = A T ) and a C -complex matrix is (1 − t − ) A + (1 − t ) A T . Furthermore, generalisedSeifert matrices (and the resulting C -complex matrices) provide a natural generalisation of theclassical Seifert matrix: they can be used to calculate the Alexander module [CF08, Theorem 3.2],the potential function [Cim04], the multivariable signature [CF08, Section 2] and the Blanchfieldform [CFT18, Con18].However, while the multivariable signature, the Alexander polynomial and the Blanchfieldform all provide obstructions to sliceness (and now double sliceness), no obstruction in termsof generalised Seifert matrices or C -complex matrices was known until now. Using Theorem 1.12,Theorem 5.13 establishes a generalisation of the following result: Theorem 1.13.
Any strongly doubly slice link admits a collection of generalised Seifert matri-ces { A ε } where A ε is hyperbolic for each ε , and also admits a hyperbolic C -complex matrix. As we mentioned above, Theorem 1.13 allows us to recover particular cases of Theorems 1.2and 4.7 (see Corollary 5.14), but the result might also be of independent interest as we now outline.
Remark 1.14.
While Levine’s algebraic concordance group [Lev69] has been generalised toboundary links [CS80, Ko89, She03, She06, RS06], in general, link concordance currently lacksa meaningful notion of algebraic concordance. In particular, there are no known restrictions in-duced by sliceness on generalised Seifert matrices, nor on C -complex matrices. For this reason,it is encouraging that in the doubly slice case such results can be established. Finally, note thatthis paper contains two proofs of Theorem 1.9, one more algebraic and one relying on generalisedSeifert surfaces. We hope that these two perspectives will offer future insights into the possibledefinitions of algebraic (double) concordance of links. ANTHONY CONWAY AND PATRICK ORSON
Examples.
Here are some of the examples where we apply our results. Example 6.5 exhibitsan oriented 4-component link that is weakly doubly slice and 2-doubly slice, but that is neither3-doubly slice nor strongly doubly slice, Example 6.6 shows the limitations of our methods bystudying a weakly doubly slice link with vanishing torsion submodule of its Λ S -Alexander moduleand vanishing multivariable signature. Proposition 6.7 shows that there are no strongly doublyslice links with 11 or fewer crossings. In Section 6.3 we determine the weakly doubly slice statusof all links with 9 or fewer crossings, with the exception of 3 links (with various orientations).This is mainly achieved using our abelian obstructions, but in some cases when our obstructionsare ineffective we use more ad hoc arguments based on linking numbers. Of special interest is thecase of the Borromean rings, which are not weakly doubly slice, and for which we use an entirelydistinct argument that exploits the triple linking. Finally, Example 6.15 exhibits an example of a2-component link that is weakly doubly slice with one quasi-orientation, but not with the other,thereby answering a question of McCoy and McDonald [MM21, Question 3]. Organisation.
In Section 2, we review some general results we will require about the homologyof abelian covers and twisted homology. Section 3 concerns multivariable signatures and we provethe lower bound on the doubly slice genus from Theorem 1.2. Section 4 is about the Blanchfieldpairing, and we prove it is hyperbolic for strongly doubly slice links. In Section 5, we turn to whatcan be said about various notions of Seifert matrix for a doubly slice link and use these methodsto provide alternative proofs for some of the results so far. In Section 6, we study examples.
Acknowledgments.
The authors would like to thank Chris W. Davis for helpful conversationsand recalling an argument of Peter Teichner for us, which we used in Proposition 6.13. We thankPeter Feller for suggesting Lemma 5.15 to us and for helpful conversations. We thank DuncanMcCoy and Clayton McDonald for drawing our attention to [MM21, Question 3].
Conventions.
Links are assumed to be oriented, unless otherwise stated. Surfaces are assumedto be compact and orientable. Given a ring R with an involution x (cid:55)→ x and a R -module H , wewrite H for the R -module whose underlying abelian group is H but with module structure givenby p · h = ph for h ∈ H and p ∈ R . From now on, all manifolds are assumed to be compact, basedand oriented. An element ω ∈ T µ will always have coordinates denoted by ω = ( ω , . . . , ω µ ). Wework in the topological category with locally flat embeddings unless otherwise stated, but notethis means our results will also hold in the smooth category.2. Twisted homology
In this section, we review some facts about twisted homology. Here, given a CW-pair (
X, Y ),a commutative domain R with involution and a ( R, Z [ π ( X )])-module M , twisted homology andcohomology refers to the R -modules H ∗ ( X, Y ; M ) = H ∗ (cid:16) M ⊗ Z [ π ( X )] C ∗ ( (cid:101) X, (cid:101) Y (cid:17) ,H ∗ ( X, Y ; M ) = H ∗ (cid:16) Hom Z [ π ( X )] ( C ∗ ( (cid:101) X, (cid:101) Y ) , M ) (cid:17) , where p : (cid:101) X → X denotes the universal cover and (cid:101) Y = p − ( Y ). Throughout this article, weassume some familiarity with twisted homology, but refer to [KL99] for a general reference, orto [Con17, Chapter 5] for a reference whose conventions match ours. Subsection 2.1 collects factsabout twisted homology with coefficients in M = R = Z [ t ± , . . . , t ± µ , (1 − t ) − , . . . , (1 − t µ ) − ],while Subsection 2.2 is concerned with a coefficient system over R = C .2.1. Homology of free abelian covers.
Use Λ := Z [ Z µ ] = Z [ t ± , . . . , t ± µ ] to denote the ringof Laurent polynomials in µ variables. Let Λ S be the ring obtained from Λ by localising themultiplicative set generated by the 1 − t i . Let ( X, Y ) be a CW pair with a map H ( X ) → Z µ .We can then consider the twisted homology modules H i ( X, Y ; Λ) and H i ( X, Y ; Λ S ). Remark 2.1.
For R = Λ , Λ S , the homology R -module H ∗ ( X, Y ; R ) can equivalently be describedas the homology of the chain complex R ⊗ Λ C ∗ ( (cid:98) X, (cid:98) Y ), where p : (cid:98) X → X is the Z µ -cover of X and (cid:98) Y = p − ( Y ). Since Λ S is flat over Λ, note also that H ∗ ( X, Y ; Λ S ) = Λ S ⊗ Λ H ∗ ( X, Y ; Λ).
BELIAN INVARIANTS OF DOUBLY SLICE LINKS 7
We recall the description of the 0-th homology module of a space X as well as the mainsimplification afforded by the use of Λ S coefficients. Lemma 2.2.
Let X be a connected CW complex, and let ϕ : π ( X ) → Z µ be a homomorphism.i) If ϕ is surjective, then H ( X ; Λ) = Z .ii) If z ∈ im( ϕ ) is non-trivial, then H ( X ; Z [ Z µ ][( z − − ]) = 0 .iii) If ψ : π ( X × S ) → Z µ is a homomorphism such that π ( S ) → π ( X × S ) ψ −→ Z µ sends agenerator to a non-trivial element z of Z µ , then H ∗ ( X × S ; Z [ Z µ ][( z − − ]) = 0 .Proof. The first two assertions follow from the usual computation of the 0-the twisted homologygroup [HS97, Chapter VI.3]. The third statement is in [CFT18, Lemma 2.2]. (cid:3)
Let X be a CW complex, and let ϕ : π ( X ) → Z µ be a homomorphism. In what follows, wewill frequently refer to H ( X ; Λ) as the Alexander module of X and to H ( X ; Λ S ) as the Λ S -Alexander module of X . The next lemma describes the Λ S -Alexander module of a space with freefundamental group; for instance the exterior of an unlinked surface in S . Lemma 2.3. If X is a CW complex with π ( X ) = F µ , then H ( X ; Λ S ) is free of rank µ − .Proof. The first twisted homology group of a space X only depends on π ( X ). Since π ( X ) = F µ ,we may therefore work with Y = ∨ µi =1 S instead of X . Use (cid:98) Y to denote the free abelian cover of Y ,so that H ( Y ; Λ S ) = Λ S ⊗ Λ H ( (cid:98) Y ); recall Remark 2.1. Since Y is a graph, we have H ( (cid:98) Y ) = ker( ∂ ),where ∂ is the boundary map of the chain complex0 −→ C ( (cid:98) Y ) ∂ −−→ C ( (cid:98) Y ) −→ . Endow Y with the cell structure with a single 0-cell z and µ x , . . . , x µ . Fix a lift (cid:101) z of z to (cid:98) Y and lifts (cid:101) x i of the x i with (cid:101) z as a start point. It follows that C ( (cid:98) Y ) = ⊕ µi =1 Λ (cid:101) x i , that C ( (cid:98) Y ) = Λ (cid:101) z ,and that the boundary map is given by ∂ (cid:101) x i = ( t i − (cid:101) z . Some linear algebra over Λ S shows thata basis for Λ S ⊗ Λ H ( (cid:98) Y ) is given by (1 − t ) (cid:101) x − (1 − t ) (cid:101) x , . . . , (1 − t µ − ) (cid:101) x µ − (1 − t µ ) (cid:101) x µ − . Thisestablishes that H ( X ; Λ S ) = H ( Y ; Λ S ) is free of rank µ − (cid:3) We now focus on link exteriors. Given a µ -coloured link L with exterior X L := S (cid:114) νL , weconsider the map π ( X L ) → Z µ , γ (cid:55)→ ( (cid:96)k ( L , γ ) , . . . , (cid:96)k ( L µ , γ )). The Alexander module of L isthen the Λ-module H ( X L ; Λ). Remark 2.4.
By applying [COT03, Proposition 2.11], we deduce that for any n -component µ -coloured link L , we have rk Λ H ( X L ; Λ) ≤ n − Lemma 2.5.
Suppose L is a link that bounds an ordered collection F = F ∪ · · · ∪ F µ of disjointoriented surfaces. Then with respect to the associated µ -colouring, Λ µ − S is a direct summandof H ( X L ; Λ S ) . When µ = n , it is moreover true that H ( X L ; Λ S ) = T H ( X L ; Λ S ) ⊕ Λ n − S .Proof. By tubing together components of the same colour, we can and will assume that F i isconnected for all i = 1 , . . . , µ . Thicken F to F × [ − , Z µ -cover p : (cid:98) X L → X L associated tothe µ -colouring may be described as follows.We abuse notation and write the intersection F ∩ X L from now on as F . The space (cid:98) X L (cid:114) p − ( F )is a disjoint union (cid:70) a ∈ Z µ Y a , where p : Y a → X L (cid:114) F × [ − ,
1] is a homeomorphism for all a ∈ Z µ .Note that ∂ (cl( Y a )) ∼ = ( F × {− , } ) ∪ ( ∂X L (cid:114) F × [ − , . Write νY a := Y a ∪ F × ( {− } × ( − ε, ∪ ( F × { } × [0 , ε )), for an open collar in X L alongthe F × {− , } part of the boundary. A Mayer-Vietoris decomposition for the Z µ -cover (cid:98) X L isthen given by (cid:71) a ∈ Z µ µ (cid:71) i =1 F ai × ( − ε, ε ) −−→ (cid:71) a ∈ Z µ νY a −→ (cid:98) X L . ANTHONY CONWAY AND PATRICK ORSON
Writing Y := X L (cid:114) F × [ − , . . . −−→ H ( (cid:98) X L ) ∂ −−→ H ( F ) ⊗ Z Λ ϕ −−→ H ( Y ) ⊗ Z Λ −→ . . . With respect to the obvious bases, ϕ is given by the matrix (1 − t − t · · · − t µ ). Passingto the ring Λ S , this matrix is seen to have kernel the free module Λ µ − S , similarly to the proof ofLemma 2.3. As this kernel is the image of ∂ , the claimed result follows because free modules areprojective and so ∂ splits.In the case that µ = n , write the decomposition so far achieved as H ( X L ; Λ S ) ∼ = T ⊕ Λ n − S .As the rank of H ( X L ; Λ S ) is at most n − n -component link (by Remark 2.4), the rankof T must be 0 and so T is torsion. It follows that T ∼ = T H ( X L ; Λ S ). (cid:3) The next corollary shows that the splitting of Lemma 2.5 into free and torsion parts holds forall colourings of a boundary link.
Corollary 2.6.
If an n -component µ -coloured link L is also a boundary link, then H ( X L ; Λ S ) = T H ( X L ; Λ S ) ⊕ Λ n − S Proof.
In this proof, we temporarily set Λ n,S = Z [ t ± , . . . , t ± n , (1 − t ) − , . . . , (1 − t n ) − ]. ByProposition 1.10, we have the decomposition H ( X L ; Λ n,S ) = T H ( X L ; Λ n,S ) ⊕ Λ n − n,S . Tensoringthe result with Λ S we obtain(2) Λ S ⊗ Λ n,S H ( X L ; Λ n,S ) = (cid:0) Λ S ⊗ Λ n,S T H ( X L ; Λ n,S ) (cid:1) ⊕ Λ n − S . We apply the universal coefficient spectral sequence with E p,q = Tor Λ n,S p ( H q ( X L ; Λ n,S ) , Λ S ) andwhich converges to H ∗ ( X L ; Λ S ). Since H ( X L ; Λ n,S ) = 0 by Lemma 2.2, a computation usingthis spectral sequence shows that H ( X L ; Λ S ) = Λ S ⊗ Λ n,S H ( X L ; Λ n,S ). Using Remark 2.4, itfollows that both sides of (2) have rank n −
1. In turn, this implies that Λ S ⊗ Λ n,S T H ( X L ; Λ n,S )coincides with the torsion submodule of H ( X L ; Λ S ). (cid:3) Homology with C ω coefficients. Given ω ∈ T µ ∗ := ( S (cid:114) { } ) µ , the map Z µ → C , t i (cid:55)→ ω i endows C with the structure of a Λ-module, which we write C ω for emphasis. Note that C ω is amodule over Λ S ; indeed none of the coordinates of ω is equal to one. Let ( X, Y ) be a CW-pair witha homomorphism H ( X ) → Z µ . Composing the induced map Z [ π ( X )] → Λ with the map Λ → C which evaluates t i at ω i produces a morphism of rings with involutions. We write H i ( X, Y ; C ω )for the resulting twisted homology C -vector spaces. As in Remark 2.1, these vector spaces canequivalently be described via Z µ -covers as the homology of C ω ⊗ Λ C ∗ ( (cid:98) X, (cid:98) Y ). Lemma 2.7.
Let ( X, Y ) be a CW pair, let ϕ : H ( X ) → Z µ = (cid:104) t , . . . , t µ (cid:105) be a homomorphism,and let ω ∈ T µ ∗ . The following assertions holds:i) if X is connected and at least one generator t i is in the image of ϕ , then H ( X ; C ω ) = 0 ,H ( X ; C ω ) = C ω ⊗ Λ S H ( X ; Λ S ); ii) evaluation produces an isomorphism H k ( X, Y ; C ω ) ∼ = Hom C ( H k ( X, Y ; C ω ) , C ) .Proof. The proofs can be found in [CNT20, Lemmas 2.3 and 2.6]. (cid:3)
The next lemma focuses on spaces with free fundamental group.
Lemma 2.8.
Let X be a CW complex with π ( X ) = F µ . For ω ∈ T µ ∗ , we have H ( X ; C ω ) = C µ − . Proof.
We saw in Lemma 2.7 that H ( X ; C ω ) = C ω ⊗ Λ S H ( X ; Λ S ) , while Lemma 2.3 establishedthat H ( X ; Λ S ) = Λ µ − S . The result follows by combining these two facts. (cid:3) We conclude this section by studying the C ω -twisted homology of link exteriors in S andsurface exteriors in S . As in Subsection, 2.1, given a µ -coloured link L ⊂ S , we consider themap π ( X L ) → Z µ , γ (cid:55)→ ( (cid:96)k ( L , γ ) , . . . , (cid:96)k ( L µ , γ )). After fixing ω ∈ T µ ∗ , we therefore obtainthe C -vector space H ( X L ; C ω ). For later use, we record the following fact about boundary links. BELIAN INVARIANTS OF DOUBLY SLICE LINKS 9
Proposition 2.9.
Fix ≤ µ ≤ n and ω ∈ T µ ∗ . If L is an n -component link that bounds a orderedcollection F = F ∪ · · · ∪ F µ of disjoint oriented surfaces, theni) b ω ( X L ) ≥ µ − ;ii) if L is a boundary link, b ω ( X L ) ≥ n − with equality if ∆ torL ( ω ) (cid:54) = 0 .Proof. We prove the first item. Lemma 2.5 implies that H ( X L ; Λ S ) = M ⊕ Λ µ − S for someΛ S -module M . The result now follows from an application of Lemma 2.7: H ( X L ; C ω ) = C ω ⊗ Λ S H ( X L ; Λ S ) = ( C ω ⊗ Λ S M ) ⊕ C µ − . We now assume that L is a boundary link and prove the second assertion. We temporarily setΛ n := Z [ Z n ] and Λ n,S for the corresponding localised ring. View L as an n -component orderedlink. As L is a boundary link, Lemma 2.5 implies that H ( X L ; Λ n,S ) = T H ( X L ; Λ n,S ) ⊕ Λ n − n,S .For ω ∈ T µ ∗ , the Λ S -module C ω is also a module over Λ n,S . An application of Lemma 2.7 nowgives the isomorphisms H ( X L ; C ω ) = C ω ⊗ Λ n,S H ( X L ; Λ n,S ) = (cid:0) C ω ⊗ Λ n,S T H ( X L ; Λ n,S ) (cid:1) ⊕ C n − . This shows that b ω ( X L ) ≥ n −
1. To prove the last statement, we must show that if ∆ torL =Ord(
T H ( X L ; Λ n,S )) does not vanish at ω , then C ω ⊗ Λ n,S T H ( X L ; Λ n,S ) = 0. Since L is aboundary link, T H ( X L ; Λ n,S ) admits a square presentation matrix A ; see e.g. [Con18, proof ofTheorem 4.6]. In particular, det( A ) = ∆ torL . Evaluating the coordinates of this matrix at ω , weobtain a square presentation matrix A ( ω ) for C ω ⊗ Λ n,S T H ( X L ; Λ n,S ). This vector space vanishesif and only if ∆ torL ( ω ) = det( A ( ω )) (cid:54) = 0 and this concludes the proof of the proposition. (cid:3) Given a µ -coloured link L and ω ∈ T µ ∗ , the integer η L ( ω ) := b ω ( X L ) is also known as the multivariable nullity of L . As we recall in Proposition 5.12, this quantity can also be defined using C -complexes. Corollary 2.10.
If a µ -coloured link L is doubly slice, then η L ( ω ) ≥ µ − for all ω ∈ T µ ∗ and,in particular, η L ( ω ) ≥ µ − for all ω ∈ S ∗ .Proof. As L is doubly slice, it bounds an ordered collection F = F ∪ · · · ∪ F µ of disjoint orientedsurfaces in S . The first assertion now follows from Proposition 2.9, while the second is immediatesince η L ( ω ) = η L ( ω, . . . , ω ) for all ω ∈ S ∗ [CF08, Proposition 2.5]. (cid:3) Finally, we record a fact about unlinked surfaces in the 4-sphere.
Proposition 2.11. If Σ ⊂ S is a µ -component unlinked surface of genus g , then its Eulercharacteristic is χ ( X Σ ) = 2(1 − µ ) + 2 g , and b ω ( X Σ ) = µ − for ω ∈ T µ ∗ .Proof. An Euler characteristic argument applied to the decomposition S = X Σ ∪ ∂ ν Σ showsthat χ ( X Σ ) = 2 − χ (Σ) = 2 − µ + 2 g = 2(1 − µ ) + 2 g . This establishes the first assertion. For thesecond assertion, note that since Σ is unlinked one has π ( X Σ ) = F µ . By Lemma 2.8, it followsthat b ω ( X Σ ) = µ −
1, concluding the proof of the proposition. (cid:3) The multivariable signature and the doubly slice genus
The goal of this section is to prove Theorem 1.2 from the introduction which states that the mul-tivariable signature of a coloured link provides a lower bound on its doubly slice genus. To achievethis, Subsection 3.1 first recalls a definition of the multivariable signature, while Subsection 3.2 isconcerned with the proof of Theorem 1.2.3.1.
The multivariable signature.
Set T µ ∗ = ( S (cid:114) { } ) µ . We recall a four dimensional defini-tion of the multivariable signature σ L : T µ ∗ → Z of a µ -coloured link L ; further background on thisdefinition of σ L can be found in [CNT20, Section 3]. Then, we establish an upper bound on σ L ( ω )that will be useful to establish the lower bound on g ds ( L ). Definition 3.1. A coloured bounding surface for a µ -coloured link L is a union F = F ∪ . . . ∪ F µ of locally flat, embedded, oriented surfaces F i ⊂ D with ∂F i = L i and that intersect transversallyin at worst double points. The exterior of a coloured bounding surface F ⊂ D will always be written as W F . The homol-ogy group H ( W F ) is freely generated by the meridians of the components F i [CNT20, Lemma 3.1].Summing up the meridians of the same colour gives rise to an epimorphism H ( W F ) (cid:16) Z µ whichextends the epimorphism H ( X L ) (cid:16) Z µ described in Section 2. We therefore obtain twistedhomology modules H ∗ ( W F ; Λ) and H ∗ ( W F ; Λ S ). Additionally, given ω ∈ T µ ∗ , we obtain a C -valued twisted-coefficient intersection form λ W F , C ω on H ( W F ; C ω ). The signature σ ω ( W F ) :=sign( λ W F , C ω ) depends only on the coloured link L [DFL18, Theorem 4.6], justifying the followingdefinition. Definition 3.2.
The multivariable signature of a µ -coloured link L at ω ∈ T µ ∗ is defined as σ L ( ω ) := σ ω ( W F ) , where F ⊂ D is any coloured bounding surface for L .In its 3-dimensional definition, which we recall in Section 5, σ L ( ω ) is obtained as the signatureof a matrix H ( ω ) that vanishes when one of the coordinates of ω is equal to 1. This explains bothwhy we restrict ourselves to T µ ∗ = ( S (cid:114) { } ) µ and why the results in the introduction were statedon the whole of T µ = ( S ) µ . We establish some bounds for the absolute value of this signature. Proposition 3.3.
Let ω ∈ T µ ∗ , and let F ⊂ D be a coloured bounding surface for a µ -colouredlink L . The following assertions hold:i) for every k ≥ , the inclusion induces isomorphisms H ∗ ( X L ; Λ S ) ∼ = H ∗ ( ∂W F ; Λ S ) ,H ∗ ( X L ; C ω ) ∼ = H ∗ ( ∂W F ; C ω ); ii) the following inequality holds: | σ L ( ω ) | ≤ b ω ( W F ) − b ω ( X L ) + b ω ( W F ) − b ω ( W F ) . Proof.
We prove the first assertion. The boundary of W F decomposes as ∂W F = X L ∪ ∂ M F ,where M F is a certain plumbed 3-manifold. It can then be shown that H ∗ ( M F ; Λ S ) = 0 (see [CFT18,Lemma 5.2]) and the first assertion then follows from a Mayer-Vietoris sequence (use the thirditem of Lemma 2.2 to deduce that H ∗ ( ∂X L ; Λ S ) = 0). The argument over C ω is analogous, as ismentioned in [CNT20, Lemma 3.10].We prove the second assertion. Denote by Im ω the image of H ( ∂W F ; C ω ) → H ( W F ; C ω ) andnote that Im ω lies in the radical of the intersection form λ W F , C ω . As F is a coloured boundingsurface for L , we have σ L ( ω ) = σ ω ( W F ) by definition. We deduce | σ L ( ω ) | ≤ dim C H ( W F ; C ω )dim(Im ω ) . (3)We compute the dimension of Im ω . Consider the following portion of the long exact sequence ofthe pair ( W F , ∂W F ) with C ω coefficients:0 −→ Im ω −→ H ( W F ; C ω ) −→ H ( W F ; ∂W F ; C ω ) −→ H ( ∂W F ; C ω ) −→ H ( W F ; C ω ) −→ H ( W F , ∂W F ; C ω ) −→ . By item i), H ( ∂W F ; C ω ) ∼ = H ( X L ; C ω ). By Poincar´e duality and the universal coefficient theo-rem, we have H ( W F , ∂W F ; C ω ) ∼ = H ( W F ; C ω ). We also have H ( W F , ∂W F ; C ω ) ∼ = H ( W F ; C ω ).We deduce from these facts that the dimension of Im ω can be expressed as(4) dim C Im ω = b ω ( X L ) − b ω ( W F ) + b ω ( W F ) . The second assertion now follows by combining (3) with (4). (cid:3)
BELIAN INVARIANTS OF DOUBLY SLICE LINKS 11
A lower bound on the doubly slice genus.
Recall the doubly slice genus g ds ( L ) fromDefinition 1.1. We now restate Theorem 1.2 from the introduction, and prove it. Theorem 3.4.
Given a µ -coloured link L , for all ω ∈ T µ ∗ one has | σ L ( ω ) | ≤ g ds ( L ) . Proof.
Assume that the µ -coloured link L ⊂ S is a cross section of a genus g µ -componentunlinked surface Σ ⊂ S . Decompose S as the union D ∪ S D along its equatorial sphere, anddefine coloured bounding surfaces for L by setting F := D ∩ Σ and G := D ∩ Σ. It follows thatthe surface exterior X Σ := S (cid:114) ν Σ decomposes as X Σ = W F ∪ X L W G , where W F = D (cid:114) νF and W G = D (cid:114) νG are the exteriors of the coloured bounding surfaces F and G . The C ω coefficient system on X Σ restricts to W F , W G and X L . An Euler characteristic argument usingProposition 2.11 shows that(5) 2(1 − µ ) + 2 g = χ ( X Σ ) = − b ω ( W F ) − b ω ( W G ) + (cid:88) i = F,G ( b ω ( W i ) − b ω ( W i )) . By Proposition 2.11, H ( X Σ ; C ω ) = C µ − , and so the Mayer-Vietoris exact sequence for X Σ yields H ( X Σ ; C ω ) −→ H ( X L ; C ω ) ι −−→ H ( W F ; C ω ) ⊕ H ( W G ; C ω ) −→ C µ − −→ . This implies that dim C (im( ι )) ≤ b ω ( X L ) and we deduce the inequality(6) b ω ( W F ) + b ω ( W G ) ≤ b ω ( X L ) + µ − . Applying the bound on σ L ( ω ) from Proposition 3.3 twice (once for F and once for G ) and addingthe results together, we obtain the following inequality:2 | σ L ( ω ) | ≤ − b ω ( X L ) + ( b ω ( W F ) + b ω ( W G )) + (cid:88) i = F,G ( b ω ( W i ) − b ω ( W i )) . Combining this inequality with (5) and (6) gives2 | σ L ( ω ) | ≤ − b ω ( X L ) + ( b ω ( W F ) + b ω ( W G )) + (cid:88) i = F,G ( b ω ( W i ) − b ω ( W i )) ≤ − b ω ( X L ) + 2( b ω ( X L ) + µ −
1) + 2(1 − µ ) + 2 g = 2 g. Dividing both sides of this inequality by two yields the required result. (cid:3) The Blanchfield pairing of doubly slice links
In this section we will prove that the Blanchfield pairing of a strongly doubly slice link ishyperbolic, as stated in Theorem 1.9 from the introduction.4.1.
The Blanchfield pairing of a coloured link.
We first recall the definition of the Blanch-field pairing of a coloured link; references include [Hil12, BFP16, Con18]. Recall the rings Λ S and Q defined in Subsection 1.3. Let L = L ∪ · · · ∪ L µ be a coloured link. Consider the composition (cid:98) Bl L : T H ( X L ; Λ S ) ( i ) −−→ T H ( X L , ∂X L ; Λ S ) ( ii ) −−→ ker( H ( X L ; Λ S ) −→ H ( X L ; Q )) ( iii ) −−→ H ( X L ; Q/ Λ S )ker( H ( X L ; Q/ Λ S ) BS → H ( X L ; Λ S )) ( iv ) −−→ Hom Λ S ( T H ( X L ; Λ S ) , Q/ Λ S )where the maps are as follows:( i ) The map H ( X L ; Λ S ) → H ( X L , ∂X L ; Λ S ) induced by the inclusion is an isomorphism (bythe third item of Lemma 2.2). Passing to the torsion submodules gives the first map, whichis an isomorphism. ( ii ) Since Q is flat over Λ S , we deduce that ker( H ( X L ; Λ S ) → H ( X L ; Q )) = T H ( X L ; Λ S ) andthus Poincar´e duality induces the second map, which is an isomorphism.( iii ) The third map is the isomorphism obtained by considering the Bockstein exact seqeunce . . . −→ H ( X L ; Q ) −→ H ( X L ; Q/ Λ S ) BS −−→ H ( X L ; Λ S ) −→ H ( X L ; Q ) −→ Indeed,
T H ( X L ; Λ S ) = ker( H ( X L ; Λ S ) → H ( X L ; Q )) is equal to im( BS ) ∼ = H ( X L ; Q/ Λ S )ker(BS) .( iv ) The fourth map is given by evaluation, and to justify it is well-defined we must showthat elements of ker( BS ) evaluate to zero on elements of T H ( X L ; Λ S ). Since ker( BS ) =im( H ( X L ; Q ) → H ( X L ; Q/ Λ S )), elements of ker( BS ) are represented by cocycles whichfactor through Q -valued homomorphisms. Since Q is a field, these latter cocycles vanish ontorsion elements, and thus so do the elements of ker( BS ). Definition 4.1.
The
Blanchfield pairing of a coloured link L is the pairingBl L : T H ( X L ; Λ S ) × T H ( X L ; Λ S ) → Q/ Λ S defined by Bl L ( a, b ) = (cid:98) Bl L ( b )( a ) . It follows from the definition of (cid:98) Bl L that the Blanchfield pairing is sesquilinear over Λ S , inthe sense that Bl L ( pa, qb ) = p Bl L ( a, b ) q for any a, b in T H ( X L ; Λ S ) and any p, q in Λ S . TheBlanchfield pairing is also known to be Hermitian; see e.g. [Con18, Corollary 1.2].
Remark 4.2.
The Blanchfield pairing of an ordered link L need not be nonsingular. However,the Blanchfield pairing is known to be non-singular on (cid:98) tH ( X L ; Λ S ). Here, given a Λ S -module M ,we write (cid:98) tM = T M/zM for the quotient of
T M by its so-called maximal pseudo-null submod-ule zM [Hil12, p. 30].In fact, when L is a boundary link, one has (cid:98) tH ( X L ; Λ S ) = T H ( X L ; Λ S ) [Hil12, Lemma 2.2].In particular, the Blanchfield pairing of a boundary link is non-singular. We prefer to give a moredirect proof of this fact as follows. Proposition 4.3.
If the Λ S -Alexander module of a µ -coloured link L admits a direct sum decom-position of the form H ( X L ; Λ S ) = T H ( X L ; Λ S ) ⊕ Λ bS for some b , then the Blanchfield pairingof L is non-singular.Proof. We have already argued that the maps ( i ) , ( ii ) and ( iii ) in the definition of the Blanchfieldpairing are isomorphisms. We therefore only need to prove that the evaluation map ( iv ) is anisomorphism. Given a Λ S -module H , set H ∗ := Hom Λ S ( H, Λ S ) , H := Hom Λ S ( H, Q ) as wellas H ∨ := Hom Λ S ( H, Q/ Λ S ). Set X := X L , consider the following commutative diagram:0 (cid:47) (cid:47) H ( X ; Λ S ) (cid:47) (cid:47) ev ∼ = (cid:15) (cid:15) H ( X ; Q ) (cid:47) (cid:47) ev ∼ = (cid:15) (cid:15) H ( X ; Q/ Λ S ) (cid:47) (cid:47) ev ∼ = (cid:15) (cid:15) H ( X ; Q/ Λ S ) / ker( BS ) (cid:47) (cid:47) ev (cid:15) (cid:15) (cid:47) (cid:47) H ( X ; Λ S ) ∗ (cid:47) (cid:47) H ( X ; Λ S ) ι Q/ Λ SQ (cid:47) (cid:47) H ( X ; Λ S ) ∨ incl ∨ (cid:47) (cid:47) T H ( X ; Λ S ) ∨ (cid:47) (cid:47) . Since H ( X L ; Λ S ) = 0, the universal coefficient spectral sequence gives the three vertical isomor-phisms. The top row is exact, arising from the Bockstein long exact sequence. We will prove thatthe bottom row is also exact, for then a simple diagram chase shows the right-most vertical arrowis an isomorphism. This vertical arrow is the map labelled ( iv ) in the definition of (cid:98) Bl L so this willcomplete the proof of the proposition.Set H := H ( X L ; Λ S ). By assumption, we have H/T H ∼ = Λ bS , so Ext S ( H/T H, Q/ Λ S ) = 0,and hence incl ∨ is surjective. Since all the evaluation maps but one are isomorphisms, it onlyremains to prove exactness at H ( X ; Λ S ) ∨ , i.e. to establish(7) im( i Q/ Λ S Q ) = ker(incl ∨ ) . By our assumption on the coloured link L , we have H = T H ⊕ Λ bS and so H ∗ = Λ bS and H = Q b so that im( ι Q/ Λ S Q ) = Hom Λ S (Λ bS , Q/ Λ S ). We now describe ker(incl ∨ ). Apply Hom Λ S ( − , Q/ Λ S )to the short exact sequence 0 → T H incl −−→ H → H/T H →
0. As
H/T H = Λ bS , the result isalso ker(incl ∨ ) = Hom Λ S (Λ bS , Q/ Λ S ), verifying (7) as required. (cid:3) BELIAN INVARIANTS OF DOUBLY SLICE LINKS 13
Combining Corollary 2.6 and Proposition 4.3, we obtain the following.
Corollary 4.4. If L is an n -component boundary link that is endowed with a µ -colouring, thenthe Blanchfield pairing Bl L is non-singular. We conclude with a remark specific to the one-variable setting. Given an oriented link, theexact same construction as in Definition 5.4 produces a rational Blanchfield pairing Bl Q L : T H ( X L ; Q [ t ± , (1 − t ) − ]) × T H ( X L ; Q [ t ± , (1 − t ) − ]) −→ Q ( t ) / Q [ t ± ] . Since the ring Q [ t ± , (1 − t ) − ] is obtained by localising the PID Q [ t ± ], it is itself a PID. Itfollows that H ( X L ; Q [ t ± , (1 − t ) − ]) splits into a free and torsion summand, and therefore theexact same proof as the one of Proposition 4.3 yields the following result. Corollary 4.5. If L is an oriented link, then the rational Blanchfield pairing Bl Q L is non-singular. We conclude with one final remark about boundary links in the case µ = 1. Remark 4.6. If L is an oriented boundary link and µ = 1, then multiplication by ( t −
1) inducesan isomorphism on
T H ( X L ; Z [ t ± ]) [FPP20, Lemma 3.3]. In particular, in this case T H ( X L ; Z [ t ± ]) = T H ( X L ; Z [ t ± , (1 − t ) − ]) . The Blanchfield pairing of a strongly doubly slice link is hyperbolic.
We are nowable to prove Theorem 1.9 and Proposition 1.10 from the introduction.
Theorem 4.7.
Let L be an n -component boundary link that is endowed with a µ -colouring. If L is doubly slice as a µ -coloured link, theni) the Blanchfield pairing of L is hyperbolic;ii) for some Λ S -submodules G , G ⊂ T H ( X L ; Λ S ) with Ord( G ) = Ord( G ) , the torsion sub-module of the Λ S -Alexander module of L splits as T H ( X L ; Λ S ) = G ⊕ G . By Remark 4.6, for µ = 1 , these results hold not only over Z [ t ± , (1 − t ) − ] , but also over Z [ t ± ] .Furthermore, for any weakly doubly slice oriented link L , the rational Blanchfield form Bl Q L ishyperbolic and T H ( X L ; Q [ t ± , (1 − t ) − ]) = G ⊕ G for some Q [ t ± , (1 − t ) − ] -module G . Note in particular, the hypotheses of Theorem 4.7 are satisfied when L is a strongly doublyslice link and when L is a weakly doubly slice boundary link. Proof.
Assume that the coloured link L ⊂ S is a cross section of an unlink Σ ⊂ S . Decompose S as the union D ∪ S D along its equatorial sphere, and define coloured bounding surfaces for L by setting F := D ∩ Σ and G := D ∩ Σ. It follows that the unlink exterior X Σ := S (cid:114) ν Σdecomposes as X Σ = W F ∪ X L W G , where W F = D (cid:114) νF and W G = D (cid:114) νG are the exteriors ofthe coloured bounding surfaces F and G . We therefore obtain the following Mayer-Vietoris exactsequence with Λ S coefficients:(8) 0 −→ H ( X L ; Λ S ) ( ι F ι G ) −−−→ H ( W F ; Λ S ) ⊕ H ( W G ; Λ S ) −→ H ( X Σ ; Λ S ) −→ . Since L is a boundary link, we know from Corollary 2.6 that H ( X L ; Λ S ) = T H ( X L ; Λ S ) ⊕ Λ n − S .The same conclusion holds over the PID Q [ t ± , (1 − t ) − ] for any oriented link L . We also knowfrom Lemma 2.3 that H ( X Σ ; Λ S ) = Λ µ − S . Consequently the short exact sequence in (10) splitsand, in particular we deduce that T H ( X L ; Λ S ) ∼ = T H ( W F ; Λ S ) ⊕ T H ( W G ; Λ S ) , (9)Here note that since H ( X L ; Λ S ) splits into free and torsion parts, so does H ( W F ; Λ S ) ⊕ H ( W G ; Λ S )and therefore so do H ( W F ; Λ S ) and H ( W G ; Λ S ). From now on, we will write T ι F and T ι G forthe restriction of ι F and ι G to the torsion submodule of the Λ S -Alexander module of L : T ι F : T H ( X L ; Λ S ) −→ T H ( W F ; Λ S ) ,T ι G : T H ( X L ; Λ S ) −→ T H ( W G ; Λ S ) . We now have
T H ( X L ; Λ S ) = ker( T ι F ) ⊕ ker( T ι G ), and our objective is to show that these twokernels are metabolisers for the Blanchfield pairing Bl L : this will show that Bl L is hyperbolic. We prove that ker(
T ι F ) is a metaboliser of Bl L ; the proof for ker( T ι G ) is identical. We followthe strategy from [COT03, Theorem 4.4 and Lemma 4.5], applying it in a similar way to [Kim06,Proposition 2.10]. Claim.
The following sequence is exact:
T H ( W F , ∂W F ; Λ S ) ∂ −−→ T H ( ∂W F ; Λ S ) T ι F −−→ T H ( W F ; Λ S ) −→ . Proof of claim.
Observe that (11) shows
T ι F is surjective. We must show exactness at T H ( ∂W F ; Λ S ).We first assert that H ( W F , ∂W F ; Q ) → H ( ∂W F ; Q ) is injective. Since H ( X Σ ; Q ) = 0 (as X Σ isa connected sum of S D .) using the Mayer-Vietoris sequence for X Σ = W F ∪ X L W G , we knowthat the inclusion induced map H ( X L ; Q ) (cid:16) H ( W F ; Q ) ⊕ H ( W G ; Q ) is surjective. Since wehave H ( ∂W F ; Q ) = H i ( X L ; Q ) by Proposition 3.3, we deduce that H ( ∂W F ; Q ) (cid:16) H ( W F ; Q ) issurjective. By exactness, this implies that H ( W F ; Q ) → H ( W F , ∂W F ; Q ) is the zero map. Byexactness, this implies that H ( W F ; ∂W F ; Q ) → H ( ∂W F ; Q ) is injective, as assertedAs in [BFP16, proof of Theorem 1.1] we consider the following commutative diagram with exactcolumns: 0 (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) T H ( W F , ∂W F ; Λ S ) (cid:15) (cid:15) (cid:47) (cid:47) T H ( ∂W F ; Λ S ) (cid:15) (cid:15) (cid:47) (cid:47) T H ( W F ; Λ S ) (cid:15) (cid:15) H ( W F , ∂W F ; Λ S ) (cid:15) (cid:15) (cid:47) (cid:47) H ( ∂W F ; Λ S ) (cid:15) (cid:15) (cid:47) (cid:47) H ( W F ; Λ S ) (cid:15) (cid:15) (cid:47) (cid:47) H ( W F , ∂W F ; Q ) (cid:47) (cid:47) H ( ∂W F ; Q ) (cid:47) (cid:47) H ( W F ; Q ) . The middle row is exact and the bottom row is exact thanks to the assertion. By the “sharp 3 × (cid:3) We wish to define a pairing Bl F : T H ( W F ; Λ S ) × T H ( W F , ∂W F ; Λ S ) → Q/ Λ S . For this,consider the following commutative diagram. The pairing we desire is the adjoint of the leftmostcolumn. T H ( W F , ∂W F ; Λ S ) ∂ (cid:47) (cid:47) ∼ = PD (cid:15) (cid:15) T H ( ∂W F ; Λ S ) T ι F (cid:47) (cid:47) ∼ = PD (cid:15) (cid:15) T H ( W F ; Λ S ) (cid:47) (cid:47) T H ( W F ; Λ S ) BS − ∼ = (cid:15) (cid:15) ι ∗ F (cid:47) (cid:47) T H ( ∂W F ; Λ S ) BS − ∼ = (cid:15) (cid:15) T H ( W F ; Q/ Λ S )ker BSev ∼ = (cid:15) (cid:15) ι ∗ F (cid:47) (cid:47) T H ( ∂W F ; Q/ Λ S )ker BSev ∼ = (cid:15) (cid:15) Hom Λ S ( T H ( W F ; Λ S ) , Q/ Λ S ) T ι ∨ F (cid:47) (cid:47) Hom Λ S ( T H ( ∂W F ; Λ S ) , Q/ Λ S ) . The top row is the exact sequence from Claim 4.2. We briefly justify this various isomorphisms inthe diagram. The vertical maps BS − are defined as the map ( iii ) in the definition of (cid:98) Bl L and arejustified to be isomorphisms similarly. The bottom vertical map in the central column is justifiedto be an isomorphism similarly to the proof of Proposition 4.3. Similarly, the proof that the bottomleft evaluation map is an isomorphism uses the splitting H ( W F ; Λ S ) = T H ( W F ; Λ S ) ⊕ Λ bS forsome b ; just as in the proof of Proposition 4.3. We omit the repetition of the details.Now set P := ker( T ι F ) = im( ∂ ). We claim that P = P ⊥ . The inclusion P ⊂ P ⊥ follows fromthe commutativity of the diagram above: given ∂ ( z ) ∈ im( ∂ ) = P and x ∈ ker( T ι F ) = P , onehas Bl L ( x, ∂ ( z )) = Bl F ( T ι F ( x ) , z ) = 0. It remains to prove the reverse inclusion, namely P ⊥ ⊂ P .Let x ∈ P ⊥ so that Bl L ( x )( p ) = 0 for all p ∈ P . In other words, Bl( x ) defines an elementin Hom( T H ( ∂W F ; Λ S ) /P, Q/ Λ S ). By Claim 4.2, we know that T ι F is surjective and therefore, BELIAN INVARIANTS OF DOUBLY SLICE LINKS 15 since P = ker( T ι F ), it induces an isomorphism T H ( ∂W F ; Λ S ) /P ∼ = T H ( W F ; Λ S ). It followsthat we obtain an isomorphism T ι ∨ F : Hom( T H ( W F ; Λ S ) , Q/ Λ S ) ∼ = −−→ Hom(
T H ( ∂W F ; Λ S ) /P, Q/ Λ S ) . Thus Bl( x ) lives in the image of T ι ∨ F . By the commutativity of diagram above (and especially thefact that all of the left vertical maps are isomorphisms), it follows that x ∈ im( ∂ ) = P .Summarising, we have proved that ker( T ι F ) = ker( T ι F ) ⊥ . The proof that ker( T ι G ) = ker( T ι G ) ⊥ is identical and we have therefore established that ker( T ι F ) and ker( T ι G ) are metabolisers of Bl L with ker( T ι F ) ⊕ ker( T ι G ) = T H ( X L ; Λ S ). This concludes the proof that Bl L is hyperbolic.It remains to prove that Ord(ker( T ι F )) = Ord(ker( T ι G )). First, note that the isomorphismdisplayed in (11) implies that ker( T ι G ) = T H ( W F ; Λ S ). The left vertical column in the previouscommutative diagram then shows thatker( T ι F ) = T H ( W F , ∂W F ; Λ S ) ∼ = Hom Λ S ( T H ( W F ; Λ S ) , Q/ Λ S ) ∼ = Hom Λ S (ker( T ι G ) , Q/ Λ S ) . As ker(
T ι G ) is torsion, we deduce that Hom Λ S (ker( T ι G ) , Q/ Λ S ) is isomorphic to Ext S (ker( T ι G ) , Λ S ).By [FNOP19, Lemma 15.7 items (3) and (4)], one deduces that the order of this latter moduleequals Ord(ker( T ι G )); here note that [FNOP19, Lemma 15.7] is stated for Λ, but the proofs gothrough for any localisation of Λ.The exact same proof works for Q [ t ± , (1 − t ) − ] in place of Λ S , but now since Q [ t ± , (1 − t ) − ]is a PID, Ord(ker( T ι F )) = Ord(ker( T ι G )) implies that ker( T ι F ) = ker( T ι G ). This concludes theproof of the theorem. (cid:3) The same methods as those used in the proof of Theorem 4.7 can also be applied to the linkingform on the double branched cover of L . The linking form of a compact, connected, oriented3-manifold M is a symmetric bilinear pairing λ M : T H ( M ) × T H ( M ) → Q / Z . When M isclosed, λ M is nonsingular, for instance by using the same arguments as in Proposition 4.3 with Z and Q in place of Λ S and Q . In particular, the 2-fold branched cover along a link L comes with anon-singular linking form λ Σ ( L ) . Proposition 4.8. If L is weakly doubly slice, then λ Σ ( L ) hyperbolic. Furthermore, for some finitegroup G we have T H (Σ ( L )) = G ⊕ G. Proof.
Assume that the ordered link L ⊂ S is a cross section of a 2-sphere S ⊂ S . Decompose S as the union D ∪ S D along its equatorial sphere, and define coloured bounding surfaces for L bysetting F := D ∩ S and G := D ∩ S . It follows that Σ ( S ) = Σ ( F ) ∪ Σ ( L ) Σ ( G ), where Σ ( F )and Σ ( G ) respectively denote the 2-fold covers of D branched along F and G . Since S consistsof a single 2-sphere, we deduce that the 2-fold branched cover Σ ( S ) is diffeomorphic to S . AMayer-Vietoris argument therefore shows that the inclusion induced maps induce the followingisomorphism:(10) (cid:18) ι F ι G (cid:19) : H (Σ ( L )) ∼ = −−→ H (Σ ( F )) ⊕ H (Σ ( G )) . Recall that every abelian group splits as a free part and a torsion part. Consequently the previousisomorphism from (10) restricts to an isomorphism on torsion summands: (cid:18)
T ι F T ι G (cid:19) : T H (Σ ( L )) ∼ = −−→ T H (Σ ( F )) ⊕ T H (Σ ( G )) . (11)Here, we wrote T ι F and T ι G for the restriction of ι F and ι G to the torsion subgroups. Thesame argument as in the proof of Theorem 4.7, now shows that T H (Σ ( F )) = ker( T ι G ) and T H (Σ ( G )) = ker( T ι F ) are metabolisers for the linking form λ Σ ( L ) , and this shows that λ Σ ( L ) is hyperbolic. It remains to show that
T H (Σ ( F )) and T H (Σ ( G )) are isomorphic abelian groups. Thiswill follow from the following sequence of isomorphisms T H (Σ ( F )) ∂, ∼ = ←−− T H (Σ ( S ) , Σ ( F )) ∼ = T H (Σ ( G ) , Σ ( L )) ∼ = T H (Σ ( G )) ∼ = Ext Z ( T H (Σ ( G )) , Z ) ∼ = T H (Σ ( G )) . The first isomorphism comes from the long exact sequence of the pair (using that Σ ( S ) ∼ = S ),the second is excision, the third is Poincar´e-Lefschetz duality, the fourth is the universal coefficienttheorem and the last follows from properties of the Ext functor. (cid:3) The Seifert surface approach
The goal of this section is to prove the general coloured version of Theorem 1.12 from theintroduction about Seifert matrices of doubly slice links. In Subsections 5.1 and 5.2 we describe acoloured version of the boundary Seifert matrices of [Ko87] and how such a collection of matricescan be associated to a boundary coloured link. In Subsection 5.3, we prove that if a coloured link isdoubly slice then it admits a doubly isotropic coloured boundary Seifert matrix. In Subsection 5.4,we show how this result can be used to recover results about multivariable invariants.5.1.
Boundary Seifert matrices.
A matrix A over Z is a called a Seifert matrix for a knot if A − A T is invertible. A matrix A over Z is a Seifert matrix for a link if A − A T is congruent tothe block sum of a nonsingular matrix and a 0 matrix. Definition 5.1.
A matrix A = ( A ij ) ≤ i ≤ j ≤ µ with entries given by a collection of µ matrices A ij over Z is called a coloured boundary Seifert matrix if for some a i ∈ N we havei) for i = 1 , . . . , µ , the matrix A ii is a size a i Seifert matrix for a link;ii) for i (cid:54) = j , the matrix A ij is an ( a i × a j )-matrix with A ij = A Tji .If the A ii are Seifert matrices for knots, then we call A a boundary Seifert matrix. A colouredboundary Seifert matrix A = ( A ij ) has an associated collection of µ bilinear pairings betweenbased free Z -modules H i of rank a i , namely as A ij : H i × H j → Z . In the case where the A ii are Seifert matrices for knots, Definition 5.1 recovers the notion of aboundary Seifert matrix due to Ko [Ko87, p. 668]. Definition 5.2.
Let A = ( A ij ) ≤ i ≤ j ≤ µ be a coloured boundary Seifert matrix with associatedpairings A ij : H i × H j → Z .i) An isotropic family for A is a collection G = { G i } ≤ i ≤ j ≤ µ of non-zero direct summands G i ⊂ H i such that for all i, j the restriction of A ij to G i × G j is the trivial pairing. If A admitsan isotropic family, it is called isotropic .ii) If A admits two isotropic families G ± such that G − i ⊕ G + i = H i for each i , then we call A doubly isotropic . Remark 5.3.
We make some comments justifying the terminology.i) An isotropic submodule for a pairing A : Z g × Z g → Z is a direct summand G ⊂ H suchthat the restriction of A to G is the trivial pairing; G is a metaboliser if, additionally, G is half rank. If G = { G i } ≤ i ≤ j ≤ µ is an isotropic family for a coloured boundary Seifertmatrix A = ( A ij ), then G ⊕ . . . ⊕ G µ is an isotropic submodule for A .ii) A pairing A : Z g × Z g → Z is called hyperbolic if it admits two metabolisers G ± suchthat H = G − ⊕ G + ; if the G ± are merely isotropic, then we call A doubly isotropic . Notethat the diagonal blocks of a doubly isotropic matrix might in general not be of the same size;it is for this reason we chose to introduce the new terminology ‘doubly isotropic’. If G ± = { G ± i } ≤ i ≤ j ≤ µ are isotropic families for a coloured boundary Seifert matrix A = ( A ij ), as inthe second item of Definition 5.2, then A is doubly isotropic. BELIAN INVARIANTS OF DOUBLY SLICE LINKS 17 iii) We observe that if a boundary Seifert matrix is doubly isotropic then for each i , the matrix A ii is in fact hyperbolic. To see this, consider that if a pairing A on H is doubly isotropic withrespect to a decomposition G − ⊕ G + ∼ = H , then so is A − A T . But by definition A − A T is nonsingular when A comes from a Seifert matrix for a knot. An isotropic summand of anonsingular pairing can be at most half rank. As the ranks of G + and G − add up to therank of H , they each have exactly half rank and so A is hyperbolic.iv) If a boundary Seifert matrix A is doubly isotropic, via G ± , . . . , G ± n then, as an integral matrix,it is hyperbolic, with orthogonal metabolisers G ± := G ± ⊕ . . . ⊕ G ± n . To see, this combinethe first and third items above.5.2. Boundary Seifert surfaces.
In this subsection, we introduce the natural coloured general-isation of the notion of a boundary link. We then show how to obtain a coloured boundary Seifertmatrix for this generalisation.
Definition 5.4. A µ -coloured link L = L ∪ . . . ∪ L µ is boundary if there are disjoint (possiblydisconnected) Seifert surfaces F , . . . , F µ such that ∂F i = L i . The collection F = F (cid:116) . . . (cid:116) F µ ofSeifert surfaces is called a boundary Seifert surface for L .For µ = n , Definition 5.4 recovers the usual notion of a boundary link, while for µ = 1 infact any oriented link satisfies the definition. In our context, the key observation is that when acoloured link is doubly slice, then it is boundary in the sense above. Remark 5.5.
As the reader familiar with boundary links might expect, Definition 5.4 has a morealgebraic characterisation as follows. Let X L be a µ -coloured link together with a homomorphism φ : π ( X L ) → F (cid:104) s , . . . , s µ (cid:105) , the free group on µ letters, that for each i sends every i -colouredmeridian to s i . This homomorphism determines a map X L → ∨ µi =1 ( S ) i , up to homotopy. Foreach i choose a regular point x i ∈ ( S ) i , away from the wedge point, and take a transversepreimage. This will determine a boundary Seifert surface F φ for L .Conversely, given a boundary Seifert surface F write m F : X L → ∨ i ( S ) i for the continuousmap given by int( F i ) × [ − , pr −−→ [ − , e −−→ ( S ) i , where e ( x ) := exp( πi ( x + 1)), and sending X L (cid:114) (int( F i ) × [ − , φ of the type described above, and suchthat F = F φ when we choose the points x i to be m F ( F i × { } ).Next, we describe how to associate a coloured boundary Seifert matrix (as in Definition 5.1) toa boundary µ -coloured link. Let F = F (cid:116) . . . (cid:116) F µ be a boundary Seifert surface for a boundary µ -coloured link L = L ∪ . . . ∪ L µ . Pushing curves off F in the negative normal direction thenproduces a homomorphism i − : H ( F ) → H ( S (cid:114) F ). The assignment θ ( x, y ) := (cid:96)k ( i − ( x ) , y )gives rise to a pairing on H ( F ) and to a coloured boundary Seifert matrix for L (the details areidentical to [Ko87, p.670] where the µ = n case is treated). Since H ( F ) decomposes as the directsum of the H ( F i ), by choosing bases for the surfaces F i , the restriction of θ to H ( F i ) × H ( F j )produces matrices A ij . For i (cid:54) = j , these matrices satisfy A ij = A Tji , while A ii is a Seifert matrixfor the sublink L i . It follows that A is a coloured boundary Seifert matrix for L in the sense ofDefinition 5.1.5.3. Doubly slice links have doubly isotropic Seifert matrices.
In this subsection, we provethe general coloured version of Theorem 1.12 from the introduction: we show that doubly slicecoloured links have doubly isotropic coloured boundary Seifert matrices.
Definition 5.6. A µ -coloured link L = L ∪ · · · ∪ L µ is S -boundary slice if there exist disjointlocally flatly embedded connected oriented ordered 3-manifolds M , . . . , M µ ⊂ S such thati) ∂M i = S ,ii) M i ∩ S =: F i ∪ G i , where F i is a Seifert surface for L i and G i is a closed surface. Note that if a µ -coloured link is S -boundary slice, it is automatically boundary (as in Defini-tion 5.4). One notices furthermore that if a µ -coloured link is doubly slice then it is in particu-lar S -boundary slice. Remark 5.7.
Definition 5.6 should not be confused with Ko’s notion of boundary sliceness.In [Ko87, Remark 2.12], Ko defines an n -component boundary link L to be boundary slice if thereare disjoint discs D , . . . , D n ⊂ D and a boundary Seifert surface F = F (cid:116) . . . (cid:116) F n ⊂ S suchthat the F i ∪ ∂ D i bound pairwise disjoint 3-manifolds in D . By doubling D along its equator,one sees that if an ordered boundary link is boundary slice, then it is S -boundary slice in oursense. It is however unclear whether the converse holds. Theorem 5.8.
Let L be a µ -coloured link.i) If L is S -boundary slice, then it admits an isotropic coloured boundary Seifert matrix.ii) If L is doubly slice, then it admits a doubly isotropic coloured boundary Seifert matrix.Proof. Suppose L is S -boundary slice and that M , . . . , M µ are as in Definition 5.6, determininga Seifert surface F = F ∪ · · · ∪ F µ . The equatorial S ⊂ S divides S = D ∪ S D − , inducinga division M i = M + i ∪ F i M − i so that ∂M ± i ⊂ D ± . Exactly as in [Ko87, proof of Lemma 3.3], weargue that B + i := ker( H ( F i ) → H ( M + i )) is isotropic (the argument for the “ − ” version is entirelysimilar). For each i , choose any splitting H i ∼ = B + i ⊕ C + i , where H i := H ( F i ). Then if α i ∈ B + i and β j ∈ B + j the value A ij ( α i , β j ) may be computed as follows. Let P i ⊂ M + i and Q j ⊂ M + j besurfaces bounded by α i and β j . The value of A ij ( α i , β j ) is then the algebraic intersection countin S between P i and a push-off of Q j in the positive normal direction of M + j . When i (cid:54) = j thefact that M + i is disjoint from M + j implies this intersection is empty. When i = j , consider thata positive push-off of M + i does not meet M + i as the normal bundle is trivial. Hence P i does notintersect the push-off of Q j .When L is moreover doubly slice, we may assume M i is a 3-ball for all i . For each i , thereis a Mayer-Vietoris sequence H ( B ) → H ( F i ) → H ( M + i ) ⊕ H ( M − i ) → H ( B ), so the inclu-sions induce isomorphisms H ( F i ) ∼ = H ( M + i ) ⊕ H ( M − i ). Under this isomorphism, we see thatnow B + i ∼ = H ( M − i ) and B − i ∼ = H ( M + i ). (cid:3) Combining Proposition 5.8, item ii) with Remark 5.3, items iii) and iv), we obtain:
Corollary 5.9.
A strongly doubly slice link admits a boundary Seifert matrix A such that eachmatrix A ii is hyperbolic, and A is hyperbolic as an integral matrix. Doubly slice links have doubly isotropic C -complex matrices. In this subsection,we show how Theorem 5.8 gives information about abelian invariants of doubly slice colouredlinks, thus recovering a subset of the results from Theorems 3.4 and 4.7. To achieve this, we firstrecall the notion of a C -complex [Coo82b, Coo82a] and the 3-dimensional interpretation of themultivariable signature [CF08, Section 2]. Definition 5.10. A C -complex for a µ -coloured link L = L ∪ · · · ∪ L µ is a union F = F ∪ · · · ∪ F µ of surfaces in S such that:i) F i is a Seifert surface for the sublink L i (possibly disconnected but with no closed compo-nents);ii) F i ∩ F j is either empty or a union of clasps for all i (cid:54) = j ;iii) F i ∩ F j ∩ F k is empty for all i, j, k pairwise distinct.Every coloured link admits a C -complex [Cim04, Lemma 1]. Note that for µ = 1, a C -complexfor a 1-coloured link L is a Seifert surface for L . Furthermore, a boundary Seifert surface for aboundary coloured link L (in the sense of Definition 5.4) is a C -complex for L .Given a C -complex F for a coloured link L and a sequence ε = ( ε , . . . , ε µ ) of ± i ε : H ( F ) → H ( S (cid:114) F ) as follows. Any homology class in H ( F ) can be represented by anoriented cycle x which behaves as illustrated in Figure 1 whenever crossing a clasp. Define i ε ([ x ]) BELIAN INVARIANTS OF DOUBLY SLICE LINKS 19
Figure 1.
An arc, forming part of a 1-cycle, running through a clasp intersection.as the class of the 1-cycle obtained by pushing x in the ε i -normal direction off F i for i = 1 , . . . , µ .The generalised Seifert form associated to F and ε is then defined as α ε : H ( F ) × H ( F ) −→ Z ( x, y ) (cid:55)→ (cid:96)k ( i ε ( x ) , y ) . Fix a basis of H ( F ) and denote by A ε the matrix of α ε . The resulting 2 µ matrices are called generalised Seifert matrices for the coloured link L . Note that since A − ε = ( A ε ) T for all ε , thereare only 2 µ − matrices to calculate in practice. Definition 5.11.
Given a µ -coloured link L , a choice of C -complex F , and a choice of basisfor H ( F ), the C -complex matrix is H ( t , . . . , t µ ) = (cid:88) ε µ (cid:89) i =1 (1 − t ε i i ) A ε where the A ε are the generalised Seifert matrices described above.When µ = 1, the definition of a C -complex is that of a Seifert surface, and a C -complex matrixis one of the form (1 − t − ) A + (1 − t ) A T , where A is a Seifert matrix for the oriented link L .The original definition of the multivariable signature σ L : T µ → Z was in terms of C -complexmatrices [CF08, Subsection 2.2], not the way it is stated in Definition 3.2. The connection be-tween the definition we gave and the original definition is as follows, and was proved in [CNT20,Proposition 1.1]. Proposition 5.12.
Let H := H F ( t , . . . , t µ ) be an ( n × n ) C -complex matrix coming from thechoice of a C -complex F for a coloured link L . Write β ( F ) for the number of connected com-ponents of F . For any ( ω , . . . , ω µ ) ∈ T µ ∗ , the multivariable signature and nullity of L can bedescribed as σ L ( ω , . . . , ω µ ) = sign( H F ( ω , . . . , ω µ )) ,η L ( ω , . . . , ω µ ) = null( H F ( ω , . . . , ω µ )) + β ( F ) − . While it is always possible to use a C -complex matrix to compute the Blanchfield form of acoloured link (see [Con18, Theorem 1.1]), it is shown in [Con18, Theorem 4.6] that in the case ofboundary links a particularly clean presentation of the Blanchfield form is possible. We now useideas from the proof of that theorem to show that doubly slice links admit doubly isotropic C -complex matrices; this will prove Theorem 1.13 from the introduction. Theorem 5.13.
Any doubly slice coloured link admits a collection of generalised Seifert matri-ces { A ε } where A ε is doubly isotropic for each ε , and also admits a doubly isotropic C -complexmatrix. Furthermore, a strongly doubly slice link admits a collection of generalised Seifert matri-ces { A ε } where A ε is hyperbolic for each ε , and also admits a hyperbolic C -complex matrix.Proof. Let F be a boundary Seifert surface for a µ -coloured link L , and let A = ( A ij ) be a colouredboundary Seifert matrix associated to F . Viewing F as a C -complex for L , we recall how A canbe used to describe a C -complex matrix for L , generalising the argument for the µ = n casefrom [Con18, Theorem 4.6].If i (cid:54) = j , since F is a coloured boundary Seifert surface, A εij is independent of ε and is equal tothe block A ij of the coloured boundary Seifert matrix A . Similarly, for each ε with ε i = −
1, the restriction of A ε to H ( F i ) × H ( F i ) is equal to the block A ii (for ε i , it equals A Tii ). CombiningTheorem 5.8 and Remark 5.3, it follows that A ε is doubly isotropic for each ε . The additionalstatement for µ = n follows as in Corollary 5.9.We now prove the assertions about the C -complex matrices. Let H i = (1 − t i ) A Tii + (1 − t − i ) A ii denote the corresponding C -complex matrix for the sublink L i and let u denote (cid:81) µj =1 (1 − t j ).Using Definition 5.11 and arguing as in [Con18, Proof of Theorem 4.6], we see that the C -complexmatrix H associated to F is uu (1 − t ) − (1 − t − ) − H uuA . . . uuA µ uuA uu (1 − t ) − (1 − t − ) − H . . . uuA µ ... . . . . . . ... uuA µ uuA µ . . . uu (1 − t µ ) − (1 − t − µ ) − H µ . Since L is doubly slice, each A ij is doubly isotropic by Theorem 5.8. Viewing the A ij aspairings A ij : H i × H j → Z , this means that there are submodules G ± i for i = 1 , . . . , µ suchthat H i = G − i ⊕ G + i and A ij vanishes on G i × G j . Arguing as in the second item of Remark 5.3,this implies that H is itself doubly isotropic: the two isotropic submodules are given by ⊕ µi =1 G − i and ⊕ µi =1 G + i . In the case that µ = n , this implies that H is hyperbolic; recall the fourth item ofRemark 5.3. (cid:3) As a corollary, we can give alternative proofs of our multivariable obstructions to a link beingdoubly slice, first stated in Theorems 3.4 and 4.7.
Corollary 5.14.
Let L be a µ -coloured link.i) If L is doubly slice, then σ L ( ω ) = 0 for all ω ∈ T µ ∗ .ii) If L is strongly doubly slice, then Bl L is hyperbolic.Proof. We recalled in the first item of Proposition 5.12 that σ L ( ω , . . . , ω µ ) = sign H ( ω , . . . , ω µ )for any C -complex matrix H . Since L is doubly slice, it admits a doubly isotropic C -complexmatrix (by Theorem 5.13), and the first statement therefore reduces to the known fact that thesignature of a doubly isotropic complex Hermitian matrix vanishes. We verify this algebraic factbelow in Lemma 5.15.We prove the second statement. Assume that L is strongly doubly slice, so that L is a boundarylink. Looking at [Con18, proof of Theorem 4.6], we obtain the following: for µ = n , the C -complexmatrix H described during the proof of Theorem 5.13 (whose size we denote m ) is invertible over Q ,presents T H ( X L ; Λ S ) and Bl L is isometric to the pairing λ H : Λ mS /H T Λ mS × Λ mS /H T Λ mS → Q/ Λ S ([ x ] , [ y ]) (cid:55)→ y T H − x. The statement therefore reduces to proving the following algebraic statement: if H is a hyperbolicmatrix over Λ S with non-zero determinant, then λ H is a hyperbolic linking form. The proof of thisis standard in the theory of linking forms; for instance the metabolisers are obtained as in [Fri03,Proposition C.1] and they will clearly be complementary direct summands. This concludes theproof of the corollary. (cid:3) It remains to prove the following algebraic lemma, which was used in the proof of Corollary 5.14.
Lemma 5.15.
Every doubly isotropic complex Hermitian matrix has vanishing signature.Proof.
Let A be such a matrix and let G − and G + be the complementary isotropic submodules.Choosing bases for G − and G + , there is a block decomposition A ∼ (cid:18) CC T (cid:19) where C is a complex matrix and C denotes the complex conjugate. We assume the bases areordered so that C determines a linear map G − → G + . Choose a new basis for G − extending a BELIAN INVARIANTS OF DOUBLY SLICE LINKS 21 basis for ker( C ) ⊂ G − and choose a new basis for G + extending a basis for im( C ). Order the newbases so that with respect to them this linear map is (cid:18) D
00 0 (cid:19) : G − −→ G + where D is invertible. We thus have congruences A ∼ D
00 0 0 0 D T ∼ (cid:18) DD T (cid:19) ⊕ (cid:18) (cid:19) , and the latter clearly has vanishing signature. (cid:3) Examples
In this section we will present some applications of the obstructions developed in this article.We introduce some useful terminology.
Definition 6.1. A quasi-orientation for an unoriented link L is an equivalence class of orientationsfor L , where two orientations are equivalent if they differ by reversing the orientation on eachcomponent of the link.If a link L is weakly doubly slice with a given orientation then the link with the reversedorientation is also weakly doubly slice (this is easily seen by reversing the orientation for theunknotted 2-sphere of which L is a cross-section). So given an unoriented link, the task of checkingwhich orientations correspond to weak double slicings reduces to checking one orientation perquasi-orientation class. Remark 6.2.
There is a coloured version of this concept that we won’t utilise, but we mention forcompleteness. Given an ordered partition of a link L = L ∪ · · · ∪ L µ , one could define a colouredquasi-orientation with respect to the ordered partition to be an equivalence class of orientationsfor L where equivalence is the transitive closure of the relation determined by an overall orientationreversal on L j , for some 1 ≤ j ≤ µ . If a µ -coloured link L is doubly slice then it is doubly slice withrespect to the colouring determined by any orientation in the coloured quasi-orientation class.For the convenience of the reader, we summarise our obstructions: a strongly doubly slice linkmust have σ L ≡ σ LTL ≡ β ( L ) = β ( L ) − L ≡ β ( L ) > T H ( X L ; Λ S ) = G ⊕ G with Ord( G ) = Ord( G ); it must also be a slice boundarylink with doubly slice components. In particular, a strongly doubly slice link must have vanishinglinking matrix. A weakly doubly slice link must have T H (Σ ( L )) = G ⊕ G for some abeliangroup G (thus the link determinant must be a square number), vanishing Levine-Tristram signatureand T H ( X L ; Q [ t ± , (1 − t ) − ]) = G ⊕ G for some Q [ t ± , (1 − t ) − ]-module G .On the constructive side, we recall a geometric operation originally described in [Iss19, Lemma4.9.2]. An ( a, b ) -tangle T is a collection of properly embedded oriented arcs in D × [0 ,
1] withboundary consisting of a points in D × { } and b points in D × { } . Suppose D × [0 , ⊂ S intersects a link L in an ( a, b )-tangle T and consider the tangle replacement operation depicted inFigure 2. Observe that if L is a µ -coloured link then the effect of the tangle replacement operationinherits a natural orientation and µ -colouring; cf. Figure 3. The following statement is moreovertrue. Lemma 6.3.
Suppose L is a µ -coloured doubly slice link and D × [0 , ⊂ S intersects L inan ( a, b ) -tangle T . Then the effect of the tangle replacement operation depicted in Figure 2 is alsoa µ -coloured doubly slice link.Proof. This is an immediate corollary of [MM21, Lemma 2.1]. (cid:3) ... ...... ... T a b ... ... ... ... T (cid:101) T T a b a b
Figure 2.
Issa’s folding construction. Left: A dashed rectangle represent-ing D × [0 ,
1] and containing the ( a, b )-tangle T . Right: A dashed rectanglerepresenting D × [0 ,
1] and containing the replacement ( a, b )-tangle. Here, (cid:101) T isthe effect of reflecting T in a vertical plane, then changing all the crossings andreversing the orientation.We will refer to the tangle replacement operation depicted in Figure 2 as Issa’s folding con-struction for reasons that become clear if one reads the proof of [MM21, Lemma 2.1] (which wedo not reproduce here). We present a basic application of the folding construction, for later use.
Example 6.4.
Consider the pretzel link P ( a , a , . . . , a k ). For any 1 ≤ j ≤ k , by choosing T tobe the (2 , a j half-twists region and performing Issa’s foldingconstruction we obtain P ( a , . . . , a j − , a j , − a j , a j , a j +1 , . . . a k ). This immediately provides for µ -double sliceness of various pretzel links via Lemma 6.3. For example, P ( a ) is the unknot for all a ∈ Z and hence, by iterating the observation above, we have that P ( a, − a, a, − a, . . . , a ) is weaklydoubly slice for all a , with respect to the unique quasi-orientation class inheritable from P ( a ) underthe folding construction. It is not necessarily true that other quasi-orientations of these pretzellinks are weakly doubly slice; see e.g. the link L6n1 in Section 6.3 below. Similarly, P ( a, − a ) isthe two-component unlink for all a ∈ Z and hence P ( a, − a, a, − a, . . . , a, − a ) is 2-doubly slice withrespect to any orientation and 2-colouring inheritable from the 2-coloured unlink (hence this linkis also weakly doubly slice with respect to these orientations). Both of these facts were partiallyderived by Donald using a different method [Don15, Corollary 2.7].6.1. Links that are m -doubly slice but not ( m + 1) -doubly slice.Example 6.5. Consider the 4-component link L in Figure 3 (right), obtained by applying Issa’sfolding construction to the tangle in the link in Figure 3 (left). As the 2-component 2-colouredunlink is doubly slice, so L is µ -doubly slice for µ = 1 ,
2. Since L has vanishing pairwise linking Figure 3.
The 2-coloured link L (right) for Example 6.5 built using Issa’s foldingconstruction on the boxed tangle in the 2-coloured unlink (left).numbers, its strong sliceness cannot immediately be ruled out using this easy check. However acomputation shows that its Alexander nullity equals 1, ruling out µ -double sliceness for µ = 3 , L is neither strongly slice nor aboundary link. Example 6.6.
Consider the link L = L10n36. It is a 2-component link with one unknottedcomponent and component whose knot type is 3 − , both of which are doubly slice and do notlink each other. One can show that L is isotopic to the pretzel link P (2 , − , , −
3) which was proved
BELIAN INVARIANTS OF DOUBLY SLICE LINKS 23 to be weakly doubly slice by Donald [Don15, Proposition 2.10]; in particular Corollary 1.3 ensuresthat σ LTL ≡
0. A C -complex computation shows that H ( X L ; Λ S ) = Λ S so that ∆ L = 0 , ∆ torL = 1.Since σ LTL ≡
0, it follows from [CF08, Theorem 4.1] that σ L ≡ L is not strongly doubly slice since it isnot a boundary link [Cro71, Section 6].6.2. Strongly doubly slice links with up to 11 crossings.
In order to detect potentialstrongly doubly slice links, we first use LinkInfo [ ? ] to list low crossing links with vanishing Mura-sugi signature σ LTL ( − η L ( − b ( L ) − ω (cid:54) = − Proposition 6.7.
Among all prime oriented links with or fewer crossings, there are no stronglydoubly slice links.Proof. Among all oriented prime links with 11 or fewer crossings there are only three with vanishingMurasugi signature, linking matrix and Alexander polynomial and whose Murasugi nullity is atleast b ( L ) −
1: the 2-component links L10n32, L10n36 and L11n247; each time for all orientations.However, L10n32 has a non doubly slice knot as a component (a Stevedore’s knot 6 ) so itcannot be strongly doubly slice, L = L11n247 has T H (Σ ( L )) = Z so it is not even weaklydoubly slice (by Proposition 1.11), and we already mentioned in Example 6.6 that the link L10n36is not strongly doubly slice. All three of these arguments hold regardless of the orientations. (cid:3) Weakly doubly slice links with up to 9 crossings.
We now consider the weakly doublyslice status of links of 9 or fewer crossings. We determine the status for all such links except three9-crossing links that stubbornly resist our efforts:
Question 6.8.
Using the notation of LinkInfo, are the following links weakly doubly slice?L9a53 (all orientations) , L9n21 { , } , L9n21 { , } , L9n21 { , } , L9n25 (all orientations) . The vast majority of links with 9 or fewer crossings are seen to be not weakly doubly slice usingour most basic abelian invariants. For the handful of cases for which this is not true we either usead hoc arguments to show they are weakly doubly slice or apply linking numbers to show they arenot. The case of the Borromean rings is particularly interesting and for this we use an entirelydifferent argument to show it is not weakly doubly slice.First, we list the weakly doubly slice links we found.
Proposition 6.9.
Using the notation of LinkInfo, the following quasi-oriented links are weaklydoubly slice:L6n1 { , } , L6n1 { , } , L6n1 { , } , L8n8 { , , } , L8n8 { , , } , L8n8 { , , } , L8n8 { , , } , L8n8 { , , } , L8n8 { , , } . Proof.
Up to isotopy and reordering of the components the listed quasi-orientation for L6n1 coin-cide. In fact L6n1 { , } is the pretzel link P ( − , , −
2) with the orientation inherited from Issa’sfolding construction applied to the unknot as in Example 6.4, so this link is weakly doubly slice.Similarly, up to isotopy and reordering of the components, the listed quasi-orientations L8n8coincide. In fact L8n8 { , , } is the pretzel link P ( − , , − ,
2) with an orientation inheritablefrom Issa’s folding construction applied to the 2-component unlink as in Example 6.4, so this linkis weakly doubly slice. (cid:3)
We will now obstruct all remaining links from being weakly doubly slice. We record the followingobservation about linking numbers; cf. [MM21, Lemma 3.1].
Lemma 6.10.
Let L be a link that is an equatorial cross section of some -knot S (cid:44) → S .i) If L has two components, then (cid:96)k ( L , L ) = 0 .ii) If L has 3 components, then for some ordering (cid:96)k ( L , L ) = − (cid:96)k ( L , L ) = (cid:96)k ( L , L ) . Proof.
Denote by F and G the two surfaces into which the knotted S is divided by the equato-rial S ⊂ S .For the first item, consider that one of the surfaces F or G must be connected, and thereforean annulus. Hence the other surface is two disjoint discs, which implies (cid:96)k ( L , L ) = 0.For the second item, we must consider two possible configurations for { F, G } depicted in Fig-ure 4. In the left-hand image we suppose one surface, F say, is connected. This implies G must S FG S FG Figure 4.
The two configurations for a 2-sphere in S meeting an equatorial S in a 3-component link.consist of 3 disjoint discs, and again this implies all pairwise linking must vanish, so the claimedstatement follows. Now suppose neither of F or G is connected. Then each of F and G consistsof the disjoint union of an annulus and a disc, as depicted in Figure 4 (right). With the orderingdepicted, we now see that0 = (cid:96)k ( L , L ∪ L ) = (cid:96)k ( L , L ) + (cid:96)k ( L , L ) , (cid:96)k ( L ∪ L , L ) = (cid:96)k ( L , L ) + (cid:96)k ( L , L ) . Rearranging, the result follows. (cid:3)
Proposition 6.11.
A link with 9 or fewer crossings is one of the links listed in Proposition 6.9,the Borromean rings (with some orientation), one of the links listed in Question 6.8, or is notweakly doubly slice.Proof.
A weakly doubly slice link has vanishing Murasugi signature and the determinant must bea square number. Aside from the links listed in Proposition 6.9, Borromean rings and the linkslisted in Question 6.8, the following are the only links with 9 or fewer crossings, square determinantand vanishing Murasugi signature, according to LinkInfo:L8a19 { , } , L8a19 { , } , L8n3 { , } , L8n3 { , } , L9a45 { , } , L9a45 { , } , L9a45 { , } , L9a46 { , } , L9a46 { , } , L9a48 { , } , L9a48 { , } . The following table of pairwise linking numbers (for some ordering of the link components) shows,using Lemma 6.10, that the links in the table are not weakly doubly slice.Link (cid:96)k ( L , L ) (cid:96)k ( L , L ) (cid:96)k ( L , L )L8a19 { } -1 1 0L8a19 { } { } { } -1 1 2L9a46 { } { } { } { } L denoteL9a45 with any of the orientations listed. We compute that H (Σ ( L )) ∼ = Z / ⊕ Z /
18 and henceby Proposition 4.8, these links are not weakly doubly slice. (cid:3)
We now treat the case of the Borromean rings. This link resists all our abelian invariants asthe next example shows.
BELIAN INVARIANTS OF DOUBLY SLICE LINKS 25
Example 6.12 (Borromean rings) . For the Borromean rings L = L a
4, and for any orientation,some calculations show that σ LTL ≡ H ( X L ; Q [ t ± , (1 − t ) − ]) = 0 and H (Σ ( L )) = Z ⊕ Z ;additionally the linking matrix of L is identically zero.However, the Borromean rings are not weakly doubly slice. This will follow from the muchstronger result below, which is known to experts but does not appear in print as far as we cantell. We are grateful to Peter Teichner for the argument in the proof of the following proposition. Proposition 6.13.
The Borromean rings (with any orientation) do not bound a properly embeddedconnected genus 0 surface in D . The argument requires us to recall some more tools from 4-manifold topology and in particularwill use the τ invariant of an immersed 2-sphere in a 4-manifold. We refer the reader to [ST01]for a detailed account of the τ invariant, which we only briefly discuss now.A generically immersed sphere S in a topological 4-manifold W is said to be s -characteristic if S · S (cid:48) ≡ S (cid:48) · S (cid:48) (mod 2) for every generically immersed sphere S (cid:48) in W , where · denotes the algebraicintersection number. Now suppose W is simply connected and that S is a generically immersed s -characteristic 2-sphere in M such that the signed count of the self-intersections of S is 0. In thissituation the following definition of the τ invariant may be used [ST01, Remark 5]. Choose aset { W i } of framed, generically immersed Whitney discs in M , pairing the self-intersection pointsof S , and such that each W i intersects S transversely in double points in the interior of { ˚ W i } .Given such a set, define a signed intersection count τ ( S, { W i } ) := (cid:88) i S · ˚ W i (mod 2) . The value of τ ( S, { W i } ) ∈ Z / τ ( S ) := τ ( S, { W i } ) for any such choice.Note that if S is embedded then clearly τ ( S ) = 0 as we choose the Whitney disc collection tobe empty. We recall as well that τ ( S ) is an invariant of the homotopy class [ S ] ∈ π ( M ) of agenerically immersed 2-sphere (computable from homotopy representatives with vanishing self-intersection); see e.g. [FMN + § Proof of Proposition 6.13. As π ( S ) = 0, and this unique homotopy class is represented by an embedded S ), this implies that every generically immersed2-sphere in S with vanishing self-intersection has vanishing τ invariant.Now suppose F ⊂ D is a properly embedded connected genus 0 surface bounding the Bor-romean rings L ⊂ S . Consider [CST14, Figure 1]. This depicts the Borromean rings L ⊂ S together with a specific choice of three embedded discs D , D , D in D , bounded by L , thatintersect each other exactly twice in D ∩ D = { q, q (cid:48) } , and with opposite orientations. Define agenerically immersed 2-sphere S := F ∪ D ∪ D ∪ D ⊂ S , and note it is s -characteristic simplybecause S has vanishing intersection form. In [CST14, Figure 1], there is furthermore depicted asingle framed, embedded Whitney disc W pairing the intersection points q and q (cid:48) , and meeting D exactly once in some point p . Thus we calculate that τ ( S ) = 1. This contradicts our previousreasoning that τ ( S ) = 0, and so implies F does not exist. (cid:3) Corollary 6.14.
The Borromean rings (with any orientation) are not weakly doubly slice.Proof.
Write L for the Borromean rings with some choice of orientation and suppose (for a contra-diction) L is weakly doubly slice. Then L is the cross section of a 2-sphere meeting an equatorial S in one of the configurations depicted in Figure 4. The figure on the left cannot occur: if it did,then L would be slice, which is impossible since µ ( L ) = 1 and Milnor’s invariants are invariantunder link concordance [Cas75]. For the figure on the right, tube the annulus of F to the disc of F to see a connected genus 0 surface, contradicting Proposition 6.13. (cid:3) Orientations and double sliceness.
In Section 6.3 we saw that the unoriented links L6n1and L8n8 are not weakly doubly slice with one quasi-orientation but are weakly doubly slice with another. McCoy and McDonald have asked about the existence of such examples among 2-component links [MM21, Question 3]. Our multivariable invariants are fairly sensitive to orienta-tion changes and we use this to provide such an example.
Example 6.15.
Set K := 8 , and consider the unoriented link L = K ∪ K obtained as the (2 , K . We argue that L is weakly doubly slice with respect to one of its quasi-orientations butnot with respect to the other. If we orient K and K in opposite directions, then the fact that K is slice implies L is weakly doubly slice [McD19, Proposition 6]. Next, we orient L so that K and K (which are copies of K ) have “parallel” orientations and we use Corollary 1.3 to showthat L is not weakly doubly slice: Set ω := e πi/ . The Levine-Tristram signature of K vanishesidentically, except at ω and ω , where σ K ( ω ) = σ K ( ω ) = 1. Since with these orientations, L isobtained as the satellite on a winding number 2 pattern, we deduce from the cabling formula for theLevine-Tristram signature [Lit79, item (2) of the Remark on page 76] that σ LTL ( ω ) = σ K ( ω ) = 1 . Corollary 1.3 now implies that L is not weakly doubly slice. Finally, note that Theorem 1.2shows that the ordered link L is not strongly doubly slice since its multivariable signature is notidentically zero; this also follows since the components of L are not doubly slice. References [BFP16] Maciej Borodzik, Stefan Friedl, and Mark Powell. Blanchfield forms and Gordian distance.
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Massachusetts Institute of Technology, Cambridge MA 02139
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