Filtering Topologically Slice Knots

# Filtering Smooth Concordance classes of Topologically Slice Knots

Tim D. Cochran Department of Mathematics MS-136, P.O. Box 1892, Rice University, Houston, TX 77251-1892 Shelly Harvey Department of Mathematics MS-136, P.O. Box 1892, Rice University, Houston, TX 77251-1892  and  Peter Horn Department of Mathematics, Syracuse University, 215 Carnegie Building, Syracuse, NY, 13244-1150
July 17, 2019
###### Abstract.

We propose and analyze a structure with which to organize the difference between a knot in bounding a topologically embedded -disk in and it bounding a smoothly embedded disk. The -solvable filtration of the topological knot concordance group, due to Cochran-Orr-Teichner, may be complete in the sense that any knot in the intersection of its terms may well be topologically slice. However, the natural extension of this filtration to what is called the -solvable filtration of the smooth knot concordance group, is unsatisfactory because any topologically slice knot lies in every term of the filtration. To ameliorate this we investigate a new filtration, , that is simultaneously a refinement of the -solvable filtration and a generalization of notions of positivity studied by Gompf and Cochran. We show that each has infinite rank. But our primary interest is in the induced filtration, , on the subgroup, , of knots that are topologically slice. We prove that is large, detected by gauge-theoretic invariants and the , , -invariants; while the non-triviliality of can be detected by certain -invariants. All of these concordance obstructions vanish for knots in . Nonetheless, going beyond this, our main result is that has positive rank. Moreover under a “weak homotopy-ribbon” condition, we show that each has positive rank. These results suggest that, even among topologically slice knots, the fundamental group is responsible for a wide range of complexity.

###### 2000 Mathematics Subject Classification:
Primary Secondary
Partially supported by the National Science Foundation DMS-1006908
Partially supported by NSF CAREER DMS-0748458
Partially supported by NSF Postdoctoral Fellowship DMS-0902786

## 1. Introduction

One of the most surprising mathematical developments of the last years was the discovery that , in stark contrast to all other dimensions, has an infinite number of inequivalent differentiable structures. This was a consequence of the work of Fields medallists Michael Freedman and Simon Donaldson  [15, 20]. In the intervening years many topological -manifolds have been shown to admit an infinite number of smooth structures, distinct up to diffeomorphism. Indeed, as of this writing there is not a single topological -manifold that is known to admit a finite (non-zero) number of smooth structures. This striking difference between the topological and smooth categories can, in a sense, be traced to the failure of the Whitney Trick in dimension  [26, Thm. 9.27]. This may be thought of as the inability to replace a topologically embedded -dimensional disk by a smoothly embedded disk. Locally, given a -disk, , topologically embedded in the -ball so that is a knot in , we cannot necessarily find a smoothly embedded disk with as boundary (as first investigated by Fox, Milnor and Kervaire in the 1950’s). Thus this local failure may be viewed as a paradigm for the chasm between the categories on the global scale of -manifolds.

Despite this proliferation of smooth structures on -manifolds the authors know of no attempt to organize the set of all such structures. Here we propose a scheme for organizing this difference in categories for the local problem. A knotted circle in is said to be topologically slice if it is the boundary of a topologically embedded -disk (with a product regular neighborhood) in . A knot is said to be a slice knot is the boundary of a smooth embedding of a -disk in . We propose a method for organizing the difference between these notions (generalizing  [27]). We also give examples exhibiting new behavior among knots that are topologically slice but not smoothly slice. Our proposed organizational scheme uses a known group structure on certain equivalence classes of knots, which we now review.

A knot is the image of a tame embedding of an oriented circle into . Two knots, and , are (smoothly) concordant if there exists a proper smooth embedding of an annulus into that restricts to the knots on . Let denote the set of concordance classes of knots. It is known that the connected sum operation endows with the structure of an abelian group, called the smooth knot concordance group. The identity element is the class of the trivial knot. It is elementary to see that this equivalence class is precisely the set of slice knots. The inverse of is the class of the mirror-image of with the circle orientation reversed, denoted .

In  [12] a filtration by subgroups of the topological knot concordance group was defined. This filtration has provided a convenient framework for many recent advances in the study of knot concordance. For example, the classical invariants of Milnor, Levine, Tristram, and Casson-Gordon are encapsulated in the low-order terms. The filtration is also significant because of its strong natural connection to the techniques of A. Casson and M. Freedman on the topological classification problem for -manifolds. This filtration my be complete, in that its intersection may be precisely the set of topologically slice knots. More recent papers (e.g.  [8, p.1423][9]) and the present paper are concerned only with a filtration of the smooth knot concordance group suggested by that in  [12]),

 ⋯⊂Fn+1⊂Fn.5⊂Fn⊂⋯⊂F1⊂F0.5⊂F0⊂C,

called the ()-solvable filtration of . The -solvable filtration (just like the filtration of the topological concordance group) is highly non-trivial; each of the associated graded abelian groups contains  [8, 9].

However the -solvable filtration of the smooth knot concordance group is not useful in distinguishing among knots that are slice in the topological category, but not slice in the smooth category, which is the main focus of this paper. Indeed, if denotes the subgroup of smooth concordance classes of knots that are topologically slice, then it was observed in  [8, p.1423] (using  [20, Section 8.6]) that

 T⊂∩∞n=1Fn.

Yet itself is known to be highly non-trivial. It was first shown in  [17] using gauge-theoretic techniques of Furuta and Fintushel-Stern that has infinite rank. Recent work of Hedden-Livingston-Ruberman, Hom and Hedden-Kirk shows that much finer structure exists in  [34, 35, 33]. Yet no proposal has been made to organize this structure.

It is the purpose of the present work to propose and investigate new filtrations of that, like the -solvable filtration, are highly non-trivial and yet are superior to that filtration in that they induce non-trivial filtrations of . Our filtration can also been seen as a generalization of Gompf’s notion of kinkiness  [27]. Our filtration thus retains the strong connection to the tower techniques of A. Casson and M. Freedman.

In Section 2, we define nested submonoids, and , of , which we call the -positive and negative knots, respectively. A knot is -positive (respectively -negative) if it bounds a smoothly embedded -disk, , in a smooth, compact, oriented, simply-connected -manifold, , with , where the intersection form on is positive definite (respectively negative definite), and where in . It follows that is homeomorphic to a punctured . To motivate the definition of for , note that if were diffeomorphic to a punctured and if the corresponding ’s were embedded in the complement of , they could be (smoothly) blown-down, resulting in an actual slice disk in . Therefore we say a knot is positive if admits a basis of disjointly embedded surfaces in the exterior of , where, loosely-speaking, these surfaces are more and more like -spheres as increases (the precise definitions are in Section 2). The sets of -positive and -negative knots induce monoid filtrations of , called the -positive filtration, , and the -negative filtration, respectively. Then we observe that the intersection , which we call n-bipolar knots, yields a filtration of by subgroups:

 {0}⊂⋯⊂Bn+1⊂Bn⊂⋯⊂B0⊂C.

We show that membership in (and ) is obstructed by (the sign of) many well-known knot concordance invariants. For example,

###### Proposition 1.1.

Suppose is a knot.

1. If then the Levine-Tristram signature function of vanishes, so has finite order in the algebraic concordance group (Corollary 4.2) ;

2. If then is algebraically slice (Corollary 5.7) ;

3. If then its Casson-Gordon slicing obstructions vanish, as do all metabelian signature obstructions (Theorems 10.1and  5.8) .

In fact, in Section 5 we show that these new filtrations are (essentially) refinements of and so the -order signature obstructions that were used to study also obstruct membership in . Thus, even though the terms of are much smaller than those of , we are still able to show that the new filtration is highly non-trivial using the same techniques as were used to show that was non-trivial.

###### Theorem 7.1.

For each , there exists

 Z∞⊂BnBn+1.

while for each there exists

 Z∞2⊂BnBn+1.

But the examples of Theorem 7.1 are not topologically slice. So we turn our attention to the intersection which yields a filtration of by subgroups:

 {0}⊂⋯⊂Tn+1⊂Tn⊂⋯⊂T0⊂T.

The advantage of over is that is an interesting non-trivial filtration of whereas is a trivial filtration of (each term is itself). Evidence that is natural is provided by showing that known invariants fit well into this structure. We are able to analyze most of the known invariants that obstruct a knots being smoothly slice and prove (more generally) that they obstruct membership in certain terms of . Evidence for non-triviality is provided by showing non-triviality for certain successive quotients .

Specifically, we show that, even among topologically slice knots, membership in is obstructed by (the sign of) many well-known knot concordance invariants.

###### Proposition 1.2.

If then

• the Levine-Tristram signature function of is non-positive (Proposition 4.1) ;

• (Ozsváth-Szabó see Proposition 4.8);

• (Kronheimer-Mrowka see Proposition 4.11 ) ;

• If, additionally, the -signatures of vanish and the -fold cover of branched over is a homology sphere, then (Corollary 6.11) ;

• if is -surgery on then (Corollary 6.7) ;

• If, additionally, the -signatures of vanish, then the corresponding Hedden-Kirk slice obstructions (extending the Fintushel-Stern invariants) obstruct membership in (see Theorem 4.7);

• If , then (see Proposition 4.10).

Here is the concordance invariant of Ozsváth-Szabó and Rasmussen defined from Heegard Floer homology  [45], is the concordance invariant of Hom  [35], is Rasmussen’s concordance invariant defined from Khovanov homology  [49], and refers to the invariants of Manolescu-Owens  [41] and Jabuka  [37] (Ozsváth-Szabó d-invariants associated to prime-power branched covers and certain specific -structures  [44]). The term Fintushel-Stern obstructions (as generalized in  [33, Theorem 1]) refers to the invariant of those authors that obstructs a rational homology -sphere from being the boundary of a -manifold with positive definite intersection form  [18], as applied to a -fold cyclic cover of branched over .

If then a similar result holds, so that if then the invariants in Proposition 1.2 and are zero.

In fact we show that:

###### Theorem 4.7.

The family of topologically slice pretzel knots considered by Endo generates a

 Z∞⊂T/T0.

The proof uses Endo’s original argument. Similarly, using the extension of the Fintushel-Stern-Furuta strategy (and the calculations) due to Hedden-Kirk  [33, Theorem 1], the set of Whitehead doubles of the torus knots () has the same property.

Hence is quite rich and the invariants of Proposition 1.2 are very useful for proving that a knot is not in , but none is directly useful beyond that.

In Section 6 we prove that the (signs of) Ozsváth-Szabo -invariants associated to prime-power branched covers can do slightly better: they obstruct membership in hence obstruct membership in .

###### Theorem 6.5.

If and is the -fold cyclic cover of branched over , then there is a metabolizer for the -linking form on ; and there is a structure on such that for all , where is the Poincare dual of . Furthermore we may take to be a structure corresponding to a spin structure on .

A similar but sharper result (Corollary 6.6) holds for the -invariants. Taken together (varying ) these invariants yield a homomorphism

 T0T1→×∞i=1Z,

but as of now too few calculations have been done for topologically slice knots to prove that the image is infinitely generated.

None of the invariants above is capable of detecting non-triviality in . Specifically, the Casson-Gordon slicing obstructions and the -invariant slicing obstructions vanish for knots in . Given this, it is surprising that we can show:

###### Theorem 8.1.

The group

 T1T2

has positive rank.

This is shown using a combination of -invariants and Casson-Gordon invariants.

We also sketch, in Theorem 8.3, the proof of a result only slightly weaker than the desired end result that is non-zero for every . Namely we exhibit topologically slice knots in that do not lie in a “weakly homotopy ribbon” fashion (there exists no -positon as in Definition 2.2 wherein the inclusion induces a surjection on Alexander modules). This is shown using a combination of -invariants and von Neumann signature invariants.

There is still a lot of room for improvement. Neither of and (iterated Whitehead doubles of the right-handed trefoil) lie in (as detected, say, by their -invariants), yet we offer no new invariants with which to distinguish them. Moreover, the positive and bipolar filtrations are still not as discriminating as we could hope for certain knots with Alexander polynomial . For example, in Corollary 3.7 we show that the (untwisted) Whitehead double of any knot in in fact lies in for every . Specifically, the Whitehead double of the figure eight knot lies in the intersection of all . To detect such knots, a different filtration and truly new invariants are needed.

Finally we remark that, although the bipolar filtration is far superior to the solvable filtration when studying , when considering the entire concordance group , the solvable filtration is still useful (perhaps more useful). Indeed is a bifiltration that is finer that either individual filtration.

## 2. Definitions of the Filtrations

In this section we define various new relations on and use these relations to define the new filtrations of (and ) that will be our objects of study. To accomplish this one should consider relaxing the condition that a knot bounds an embedded -disk in . There are two obvious paths (although they are not unrelated). One possibility is to relax the condition on the -disk and ask only that the knot bound a singular disk or a surface or a grope, for example. Alternatively, one can relax the condition on , and consider when a knot bounds an embedded disk in some other -manifold. Here we take the latter approach.

We say that two knots and are concordant in if is a smooth, compact, oriented, -manifold with and there exists an annulus, , smoothly and properly embedded in whose boundary gives the knots and and where the annulus is trivial in . We say that is slice in if is smooth, compact and oriented with and there is a -disk smoothly embedded in whose boundary is , and where the slice disk is trivial in . This last condition is important because any knot bounds a smoothly embedded disk in a punctured connected sum of ’s  [11, Lemma 3.4].

In  [7, Def. 2.1] Cochran and Gompf defined a relation, , on knots that generalized the relation that can be transformed to by changing only positive crossings. We generalize and filter this notion by adding an integer parameter.

###### Definition 2.1.

We say if is concordant to in a smooth -manifold such that

• ;

• the intersection form on is positive definite;

• has a basis represented by a collection of surfaces disjointly and smoothly embedded in the exterior of the annulus such that, for each , .

Here denotes the term of the derived series of the group , where and . Note that any group series satisfying a certain functorialty could be used here and would give, a priori, a different relation.

It is elementary to verify that these relations descend to relations on . In particular if is any slice knot, and is the unknot then and for every . Moreover it is easy to check that each relation is compatible with the monoidal structure on , that is, if then for any . These verifications are left to the reader. It follows from and that the matrix of the intersection form on with respect to this basis is an identity matrix. Condition ensures that is homologically trivial. Each of the relations is clearly reflexive and transitive, but fails to be symmetric and fails to be antisymmetric.

###### Definition 2.2.

Let , the set of n-positive knots, be the set of (concordance classes of) knots such that where is the trivial knot. Let , the set of n-negative knots, be the set of (concordance classes of) knots such that . Equivalently is n-positive (respectively, n-negative), if is slice in a smooth -manifold such that

• ;

• the intersection form on is positive definite (respectively, negative definite);

• has a basis represented by a collection of surfaces disjointly embedded in the exterior of the slice disk such that for each .

Such a is called an -positon for (respectively an -negaton for ).

Since the surfaces are disjoint, the intersection matrix with respect to this basis is diagonal. Since the intersection form is positive definite and unimodular, this matrix is the identity matrix. Thus these conditions imply (by work of Freedman on the classification of closed, smooth, simply-connected -manifolds up to homeomorphism) that any -positon is a smooth manifold that is homeomorphic to a (punctured) connected sum of ’s. Hence is a punctured connected sum of ’s, but with a possibly exotic differentiable structure.

The parameter in this definition can be motivated by the following observation: if the surfaces were actually spheres then, since (respectively ), their regular neighborhoods would be diffeomorphic to that of so they could be (differentiably) excised (blown-down), proving that is smoothly slice in endowed with a possibly exotic smooth structure. Condition . is intended to progessively approximate, as increases, this scenario.

###### Remark 2.3.

This definition extends to links and string links for which the components have zero pairwise linking numbers.

The sets and are clearly closed under connected sum. However, if then need not lie in but will certainly lie in . Thus neither nor is a subgroup of . However each of the sets in the following definition is a subgroup.

###### Definition 2.4.

The set of n-bipolar knots is .

Then, in summary, we have:

###### Proposition 2.5.

and induce filtrations of by submonoids

 {0}⊂⋯⊂Pn⊂⋯⊂P0⊂C, {0}⊂⋯⊂Nn⊂⋯⊂N0⊂C,

We call these the ()-positive filtration and ()-negative filtration of . Moreover both and induce filtrations of by subgroups

 {0}⊂⋯⊂Bn⊂⋯⊂B0⊂C, {0}⊂⋯⊂⟨Pn⟩⊂⋯⊂⟨P0⟩⊂C.

where denotes the subgroup of generated by the set . The former we call the ()-bipolar filtration.

Finally,

###### Definition 2.6.

Let denote the subgroup represented by knots that are topologically slice, and let denote

###### Proposition 2.7.

and are filtrations of by submonoids while is a filtration of by subgroups

 {0}⊂⋯⊂Tn⊂⋯⊂T0⊂T.

We close with several curiosities, the first of which is quite useful later in the paper.

###### Corollary 2.8.

If and then no non-zero multiple of lies in .

###### Proof.

For sake of contradiction, suppose for a non-zero integer . Since is a group we can assume that . Then in particular . Since , it follows that . Since , and so . Since the latter is closed under connected-sum,

 K=mK+(m−1)(−K)∈Nn,

Secondly, if we restrict to Tor(), by which we mean the torsion subgroup of , the -positive filtration is a filtration by subgroups.

###### Corollary 2.9.

and are filtrations of by subgroups, and in fact each equals .

###### Proof of Corollary 2.9.

Suppose in for some . If then so . But since , , so . Thus . This shows that

 Pn∩Tor(C)=Bn∩Tor(C)=Nn∩Tor(C),

and so each is a subgroup since is the intersection of two subgroups. ∎

## 3. Examples of Knots in Pn, Nn and Bn

In this section we give examples of knots lying deep in the various filtrations.

If can be transformed to by changing some set of positive crossings to negative crossings then , as can be seen by blowing-up at the singular points in the trace of the homotopies that accomplish the crossing changes  [11, Lemma 3.4][7, Prop. 2.2]. Thus it follows that:

###### Proposition 3.1 (Cochran-Lickorish).

Any knot that can be changed to a slice knot by changing positive crossings (between the same component) to negative crossings lies in .

###### Proof.

In  [11, Lemma 3.4] it is shown that if satisfies the hypothesis then is slice in a punctured connected-sum of copies of . Thus . ∎

###### Example 3.2.

Any knot that admits a positive projection lies in . The (twisted or untwisted) Whitehead double of any knot using a positive clasp lies in , since it can be unknotted by changing a single positive crossing. The figure knot lies in since it can be unknotted via a positive or a negative crossing.

Question: Does every strongly quasipositive knot lie in ?

It is easy to create knots and links lying in or using the satellite construction and generalizations of this. In particular suppose that is a solid torus embedded in in an unknotted fashion, and is the oriented meridian circle of . Suppose is a knot in which when viewed as a knot in will be denoted . Suppose that . If then is said to have winding number zero in . Suppose that is any knot. Then let denote the satellite knot with as pattern and as companion. This is also called the result of infection on by along . It is known that induces a well-defined operator on .

###### Proposition 3.3.

With notation in the preceding paragraph, suppose that (respectively , ), and . Then

 (3.1) R(η,Pn−k)⊂Pn;

respectively

 (3.2) R(η,Nn−k)⊂Nn;
 (3.3) R(η,Bn−k)⊂Bn.

This generalizes  [7, Proposition 2.7].

###### Proof.

In the next section we will give an equivalent definition of and . Using the equivalent definition, the proof of Proposition 3.3 is almost identical to that of  [6, Lemma 6.4] (which was done for the -solvable filtration). We include a different proof.

By symmetry it suffices to prove equation 3.1. Suppose and let . We will show that . Suppose has slice disk in the -positon . Suppose has slice disk in the -positon . We will describe a slice disk for in an -positon . Recall that lies in an unknotted solid torus whose exterior in we denote . The circle may be viewed as a meridian of or as a longitude of . Form a new -manifold, , as the union of and , identifying in the boundary of the former with in the boundary of the latter, in such a way that the meridian of is identified with . One first observes that is the union of with where the meridian of is identified with the meridian of . Thus is homeomorphic to and the image, under this identification, of the knot is the satellite knot . Thus is slice in (merely by letting be the image of the slice disk ).

Since is simply-connected, is normally generated by a meridian of , which has a representative in . Then, since is simply-connected, it follows from the Seifert-Van Kampen theorem that is simply-connected.

A Mayer-Vietoris sequence shows that

 H2(VK)≅H2(VR)⊕H2(VJ∖ΔJ)≅H2(VR))⊕H2(VJ).

By Definition 2.2 the latter two groups have bases, and , disjoint from and respectively. Thus the union of these bases is a basis for consisting of embedded surfaces disjoint from . The intersection form with respect to these bases is an identity matrix.

It remains only to show that these surfaces satisfy the -condition of Definition 2.2. This is clear for the . For the surfaces it suffices to show that:

 i∗(π1(VJ∖ΔJ)(n−k))⊂π1(VK∖ΔK)(n).

This follows from two facts. First recall that is normally generated by a meridian, , of , and that this meridian is identified with . Secondly, by hypothesis , so . Hence . ∎

###### Corollary 3.4.

Each of the submonoids discussed is closed under forming satellites in the sense that any satellite knot whose pattern knot and companion knot both lie in (respectively , ) itself lies in (respectively , ). If the winding number is zero then (respectively , ).

Since taking the untwisted Whitehead double of a knot is a satellite operator with winding number zero and with unknotted pattern, we have the following.

###### Example 3.5.

Let denote the untwisted positive Whitehead double operator (whose clasp has positive crossings). Let denote the untwisted negative Whitehead double operator. Now if , (respectively , ) then both and lie in (respectively , ). Since the figure eight knot, , lies in , both and lie in . Since the right-handed trefoil knot, , is a positive knot, (but not in as we shall see later). On the other hand lies in and also lies in since it can be unknotted by changing the negative crossing undoing the clasp.

###### Example 3.6.

Suppose is the ribbon knot . Let be the right-handed trefoil knot, be the left-handed trefoil knot and be the unknot. Then by Proposition 3.1. Note that, , the knot on the left-hand side of Figure 3.1 is also a ribbon knot, as is the knot (not pictured). Thus the knot on the right-hand side of Figure 3.1 may be viewed as a winding number zero satellite knot with ribbon pattern knot in two different ways and hence we can apply Proposition 3.3 in two different ways to conclude that

 K≡R(LHT,RHT)=(R(LHT,−))(RHT)∈P1,

and

 K≡R(LHT,RHT)=(R(−,RHT))(LHT)∈N1.

Thus .

Unfortunately the positive filtration is still not as discriminating as we would hope for certain kinds of topologically slice knots, namely those with Alexander polynomial .

###### Corollary 3.7.

Suppose is a winding number zero satellite knot whose pattern knot is a slice knot with Alexander polynomial one, and whose companion knot lies in . Then for all . Thus the Whitehead double (with either clasp) of a -positive knot is -positive for all .

###### Proof.

The winding number zero hypothesis means that . Since has Alexander polynomial one, its Alexander module is trivial, that is, . It follows that for all . Hence for each . Since is slice . Now Proposition 3.3 with implies the desired result.

## 4. Obstructions to lying in P0,N0 and T0

It is well-known that the signature of a positive knot is non-positive  [7, Corollary 3.4]. More generally we shall see that (the signs of) many concordance invariants obstruct being slice in a positive definite manifold. In this section we show that membership in is obstructed by the signs of classical signatures as well as the sign of the -invariant and -invariant. We also see that the slicing obstructions of Donaldson, Fintushel-Stern and Hedden-Kirk also obstruct membership in .

If is a knot in , is a Seifert matrix for and is a complex number of norm , then recall the Levine-Tristram -signature of , is the signature of

 (1−ω)V+(1−¯¯¯ω)VT.

However, for equal to a root of the Alexander polynomial of , we redefine to be the average of the two limits . The resulting function, , we shall call the Levine-Tristram signature function of . This function is a concordance invariant. If is a prime the signatures corresponding to where are called the (Tristram) -signatures of .

The following generalizes Theorem 3.16 and Lemma 4.3 of  [7]

###### Proposition 4.1.

If then the Levine-Tristram signature function of is non-positive. Moreover, for a prime power , if, in addition, the -signatures of are zero, then the -fold cyclic cover of branched over bounds a compact -manifold whose intersection form is positive definite and for which .

Thus if then its signature function is identically zero. Since it is known that knot signature function detects elements that are of infinite order in Levine’s algebraic concordance group we have:

###### Corollary 4.2.

If then has finite order in the algebraic concordance group. Moreover, for any prime power , the -fold cyclic cover of branched over bounds two compact -manifolds whose intersection forms are (respectively) -definite and for which .

###### Corollary 4.3.

The knot signatures corresponding to the different prime roots of unity yield an epimorphism

 CB0↠Z∞.
###### Proof of Proposition 4.1.

Since , bounds a slice disk in a manifold as in Definition 2.2. Now we mimic the proof of  [11, Theorem 3.7]. Let be a prime power and let denote the -fold cyclic cover of branched over , which is well known to be a -homology sphere  [2, Lemma 4.2]. Since is disjoint from a basis for , it represents zero in . It follows that , generated by the meridian. Thus the -fold cyclic cover of branched over , denoted , is defined and has boundary . Since , it follows from the proof of  [2, Lemma 4.2] that . Thus .

To compute the signature of we make into a closed -manifold and use the -signature theorem. Let be the -ball together with a Seifert surface for pushed into its interior. Let

 (Y,F)=(V,Δ)∪(−B4,−FK)

be the closed pair, let denote the -fold cyclic branched cover of , and let be the -fold cyclic branched cover of . Note that acts on , , and with , and respectively as quotient. Choose a generator for this action. Let , , denote the -eigenspace for the action of on ; let denote the rank of this eigenspace, and let denote the alternating sum of these ranks (similarly for and ). Let denote the signature of the -eigenspace of the isometry acting on (similarly for and ). By a lemma of Rochlin, using the -signature theorem  [50][2, Lemma 2.1], since is closed and ,

 σ(˜Y,j)=σ(Y).

Since glued along the rational homology sphere , this translates to

 σ(˜V,j)−σ(˜W,j)=σ(V)−σ(B4).

Since the intersection form of is, by assumption, positive definite, , so

 σ(˜V,j)−σ(˜W,j)=β2(V).

Hence

 (4.1) σ(˜W,j)=σ(˜V,j)−β2(V).

Consider the covering space . By  [25, Proposition 1.1], for any ,

 (4.2) χ(V−Δ)=χ(˜V−˜Δ,j).

Since acts by the identity on , and, if , . Since , . On the other hand, since , we must have generated by a meridian. Since acts by the identity on the first homology of this meridian, and, if , . Since is obtained from by adding a -handle along a circle of infinite homological order,

 β2(˜V−˜Δ,j)=β2(˜V,j).

For the same reason, since , . Similarly, and . Thus equation  (4.2) becomes

 (4.3) β2(V)=β2(˜V,j),  and so  dβ2(V)=β2(˜V).

Combining this with equation  (4.1) we have

 (4.4) σ(˜W,j)=σ(˜V,j)−β2(˜V,j).

Thus is non-positive.

But it is known that, if , then is a -signature of  [53][28, Chapter 12], that is

 σ(˜W,j)=σωj(K),

where . Since the roots of unity, as varies, are dense in the circle, this implies that the entire signature function of is non-positive.

Additionally, from  (4.4), we see also that is positive definite if and only if for each . Thus if all of the signatures of are zero, then is positive definite. ∎

Even for topologically slice knots, membership in is often obstructed by the theorems of Donaldson  [15], Fintushel-Stern  [18] and Ozsváth-Szabo  [45]. In this regard the following elementary observation is useful (this result is almost the same as  [7, Lemma 2.10]). Recall that if is an oriented knot and is non-zero then , the -framed Dehn surgery on , is a rational homology -sphere with via a canonical map sending the meridian to .

###### Proposition 4.4.

If then for any non-zero , bounds a compact -manifold with intersection form isomorphic to , for which there exist canonical isomorphisms . In particular if (respectively ) then both the -framed surgery on and the surgery on bound compact -manifolds with positive definite (respectively negative definite) diagonalizable intersection form and .

###### Proof.

Suppose via (as in Definition 2.1 for ). Then doing “Dehn surgery cross ” on the annulus in gives the desired manifold (more details are given in  [7, Lemma 2.10]). Here, since we have assumed , we get the extra -isomorphism property that was not present in  [7].

For the second statement, take and let . We remark that the -framed surgery on any knot bounds such a positive definite manifold, so really it is only the statement for that has content. ∎

As a consequence, we will show in Corollary 6.7 that the signs of the Ozsváth-Szabo d-invariants associated to the -surgeries on a knot obstruct membership in and .

###### Example 4.5.

Let where denotes the left-handed trefoil knot. Since has Alexander polynomial one it is topologically slice  [20]. Thus by Proposition 3.1. But because, by  [27, Corollary 2.5], -surgery on does not bound a -manifold with positive definite intersection form as required by Proposition 4.4. Hence

###### Example 4.6.

Suppose are odd and . Then the pretzel knot has Alexander polynomial and hence is topologically slice. As long as no product of two of is , it is of infinite order in the smooth concordance group  [11, Corollary 4.3]. Moreover the -fold branched cover, , is the Brieskorn sphere (with its orientation as the boundary of the canonical negative definite resolution) [11, 140-141]. Then, by  [18, Thms. 10.1, 10.4], cannot bound a positive definite -manifold as in the conclusion of Proposition 4.1. Thus . Alternatively, bounds its canonical -connected negative definite plumbing  [11, 137-141], , whose intersection form is not diagonalizable (this is proved in  [29, proof of Prop. 3.1,page 11-12]). Hence since would violate Donaldson’s theorem.

In  [17] Endo showed that a certain infinite subset of the family of knots in Example 4.6 is linearly independent in . In fact Endo’s argument (using the full strength of techniques of Furuta and Fintushel-Stern), together with our Proposition 4.1, shows the following:

###### Theorem 4.7.

The family of topologically slice pretzel knots considered by Endo generates a

 Z∞⊂T/T0.
###### Proof.

The claim is that Endo’s original proof (together with our Proposition 4.1 for ) proves this stronger result. Endo’s family is a specific sequence of pretzel knots, , , as in Example 4.6. Following Endo, we proceed by contradiction. Suppose that some non-trivial linear combination, , of such knots were to lie in . By taking the concordance inverse we may assume that if is the largest value for which occurs, then it occurs with a positive coefficient. Since , . The -fold branched cover, , is a connected sum of the corresponding Brieskorn homology spheres. By Proposition 4.1 for , is the boundary of a compact -manifold whose intersection form is positive definite and for which . Endo’s argument shows (using work of Furuta) that this is a contradiction in the case that , but it was known that the results of Furuta employed in the proof hold in our more general context (see  [21, Page 340],  [18, Thm. 1.1 and remarks below Thm. 1.2] [19, proof of Thm. 5.1]. ∎

More generally, for any knot with vanishing -signatures, by Proposition 4.1, the -fold cyclic cover of branched over bounds a compact -manifold whose intersection form is positive definite and for which . Therefore the obstruction of Hedden-Kirk (Fintushel-Stern, Furuta) vanishes.

Additionally, it follows immediately from work of Ozsváth-Szabo  [45, Thm. 1.1] that

###### Proposition 4.8.

(Ozsváth-Szabo) If then .

###### Corollary 4.9.

If then .

Then, regards Jennifer Hom’s -invariant we have:

If then .

###### Proof.

Note that both , the -cable of , and , the -cable of , may be viewed as satellites of with patterns and . Since the latter are trivial knots, by Corollary 3.4, both and lie in . By Corollary 4.9, . It then follows from  [35, Theorem 5.2] that . ∎

As regards the Rasmussen -invariant of knot concordance, it follows quickly from recent work of Kronheimer-Mrowka  [39, Cor. 1.1] that

If then .

###### Proof.

Suppose that bounds the slice disk in the -positon . We think of as representing a class in . Consider the exact sequence

 π2(S3,K)j∗→π2(V,K)i∗→π2(V,S3)≅H2(V,S3).

Since , by the relative Hurewicz theorem, . Since, by definition,