A note on cables and the involutive concordance invariants
Kristen Hendricks Abhishek Mallick

§ ABSTRACT 
 We prove a formula for the involutive concordance invariants of the cabled knots in terms of that of the companion knot and the pattern knot. As a consequence, we show that any iterated cable of a knot with parameters of the form (odd,1) is not smoothly slice as long as either of the involutive concordance invariants of the knot is nonzero. Our formula also gives new bounds for the unknotting number of a cabled knot, which are sometimes stronger than other known bounds coming from knot Floer homology.



Microwave-free imaging magnetometry with nitrogen-vacancy centers in nanodiamonds at near-zero field
    Adam M. Wojciechowski
    September 9, 2024
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§ INTRODUCTION

Cabling is a natural operation on a knot which acts on the smooth concordance group. There has been considerable interest in characterizing the behavior of various knot concordance invariants under cabling. Typically, one hopes to prove a formula which relates the values of some concordance invariant of a cabled knot to the value of the invariant on the companion knot and the value of the invariant on the pattern knot, in this case a torus knot. Some examples of invariants which are known to admit such formulas include the Levine-Tristram signatures <cit.>, the Heegaard Floer τ-invariant and ϵ-invariant  <cit.>, Rasmussen's V_0-invariant <cit.>, and the Heegaard Floer ν^+-invariant <cit.>, among others. 

Recently, concordance invariants stemming from the involutive variant of Heegaard Floer homology defined by the first author and Manolescu <cit.> have been shown to be fruitful in many applications. In this paper we will be interested in the concordance invariants (K) and (K), which can be thought of as the involutive analog of Rasmussen's V_0(K) invariant <cit.>. An early indication of the utility of these invariants was that the involutive concordance invariants are able to detect non-sliceness of certain rationally slice knots. Later, equivariant refinements of these invariants defined by the second author, Dai, and Stoffregen <cit.> were used in  to show that (2,1)-cable of the figure-eight knot is not smoothly slice <cit.>.

In this article we give the cabling formula for the involutive concordance invariants. Let K_p,q denote the (p,q)-cable of a knot K, and V_s represent the generalization <cit.> of Rasmussen's V_0-invariant <cit.>.


Let p and q be positive integers with (p,q)=1. Then the involutive concordance invariants satisfy the following relations:



  * If p is odd, we have
 
    (K_p,q)= (K) + V_0(T_p,q) ,   (K_p,q)= (K) + V_0(T_p,q).

 
 
  * If p is even,
 
    (K_p,q)= max{ V_⌊s/p⌋(K), V_⌊p+q -1-s/p⌋(K) } + V_0(T_p,q)  ,   (K_p,q)= V_0(T_p.q),

 
where s ≡p+q-1/2 (mod  q).


Theorem <ref> implies that the involutive concordance invariants are determined by the knot Floer homology  when the longitudinal winding parameter p is even. On the other hand, when p is odd, the involutive concordance invariants incorporate the corresponding invariants for the companion knot. 

Theorem <ref> implies the following corollary. Let K_p_1,q_1; p_2, q_2; p_3, q_3; ⋯ ; p_k,q_k
represent the iterated cable of the knot K. For example, K_p_1,q_1; p_2, q_2 is the (p_2,q_2)-cable of the (p_1,q_1)-cable of K.
 If the parameters p_i are all odd and positive, and either (K) or (K) is non-zero, then K_p_1,1; p_2,1; p_3,1; ⋯ ; p_k,1 is not smoothly slice.

Corollary <ref> is related to a famous open question by Miyazaki <cit.> which asks whether there are non-slice knots K for which the cable knot K_p,1 is slice.

 It is possible to produce examples of topologically (and therefore algebraically) slice knots which are trivial with respect to ordinary Heegaard Floer homology and involutively nontrivial, for which the corollary above detects nonsliceness of the cable; for example, one such family appears in <cit.>. The authors are, however, not presently aware of a family of examples which are not better elucidated by other methods.


We now explain the strategy for the proof of Theorem <ref>. A direct approach would be to determine the full involutive knot Floer invariant of the cabled knot, which takes the form of the knot Floer chain complex of Ozsváth-Szabó and Rasmussen 𝒞𝒦ℱ(𝒦) together with the knot conjugation symmetry ι_K on the complex. The involutive concordance invariants of the knot may then in principle be extracted from this data. However, it is challenging to compute the full knot Floer chain complex for a general cabled knot, and additionally challenging to determine the symmetry ι_K. In general, direct computations of ι_K have only been carried out for simple chain complexes in which the map is determined by its algebraic properties, and for tensor products of those complexes.

Instead, our proof takes an indirect approach, using certain consequences of the surgery formula in involutive Heegaard Floer homology proved by Hom-Stoffregen-Zemke and the first author <cit.> to deduce the formulas presented in Thoerem <ref>. This approach is similar to one used by Ni-Wu to compute the Heegaard Floer correction terms of three-manifolds using surgery formulas <cit.>. 



 §.§ Applications to the unknotting number

We now discuss an application of Theorem <ref> to the unknotting number. Recall that the unknotting number u(K) of a knot K is the minimum number of times the knot must be passed through itself to turn it into an unknot. There are many well-known bounds in the literature for the unknotting number of a knot. In one recent example, in <cit.>, Alishahi and Eftekhary gave a bound using the torsion order of the HFK^- flavor of the knot Floer homology. This bound was later used by Hom, Lidman, and Park <cit.> to give bounds for the unknotting number of cabled knots. In particular, the authors showed that

    u(K_p,q) ≥ p.

It is also well-known that the unknotting number is greater than or equal to the slice genus of the knot. Hence using the relationship of Rasmussen's V_0 invariant to the slice genus <cit.>, together with the cabling formula for the V_0-invariant <cit.>, one can also write down the bounds

    u(K_p,q) ≥ 2 V_0(K) + 2 V_0(T_p,q),  if  g_4(K_p,q)  is even

and

    u(K_p,q) ≥ 2 V_0(K) + 2 V_0(T_p,q) -1,  if  g_4(K_p,q)  is odd.


In particular, without knowing the slice genus of the knot, we have that


    u(K_p,q) ≥ 2 V_0(K) + 2 V_0(T_p,q) -1.


Using Theorem <ref>, we improve the above bounds:

Let K_p,q be a cabled knot with p odd. Then


    u(K_p,q) ≥ 2(K) + 2V_0(T_p,q) - 2  and  u(K_p,q) ≥ -2(K) - 2V_0(T_p,q) - 2.



More precisely, one has



  * u(K_p,q) ≥ 2(K) + 2V_0(T_p.q) - 1  and  u(K_p,q) ≥ -2(K) - 2V_0(T_p,q) - 1,    g_4(K_p,q)  odd,

  * u(K_p,q) ≥ 2(K) + 2V_0(T_p,q) - 2  and  u(K_p,q) ≥ -2(K) - 2V_0(T_p,q) - 2  ,   g_4(K_p,q)  even


although of course one in general one would like to apply Theorem <ref> in situations where the slice genus is unknown.

 A statement similar to Theorem <ref> could also be made for the 4-dimensional positive clasp number of a knot in place of the unknotting number. In another direction, because  and  are concordance invariants, the bound from Theorem <ref> is in fact a bound on the concordance unknotting number of K_p,q, which is to say the minimum unknotting number of a knot J concordant to K_p,q.



In Example <ref> we show that there are infinitely many knots for which the bound from Theorem <ref> is stronger than that from <cit.> or the aforementioned bound from the V_0-invariant. 


 §.§ Organization

This paper is organized as follows. In Section <ref> we review the definition of the involutive concordance invariants and structural features of the involutive surgery formula. We then prove Theorem <ref>, Corollary <ref>, and Theorem <ref> in Section <ref>. Finally, in Section <ref> we consider an example in which our formula improves on other bounds from knot Floer homology.



 §.§ Acknowledgments

The authors are grateful to Jen Hom, Sungkyung Kang, Tye Lidman, JungHwan Park, and Ian Zemke for helpful comments. Portions of this work were carried out while the second author was in attendance at a Simons Semester on “Knots, Homologies and Physics” at the Institute of Mathematics of the Polish Academy of Science; other portions were carried out while the first author was present at the BIRS workshop “What's your Trick?: A nontraditional conference in low-dimensional topology.” We are grateful to both institutions for their hospitality.




§ BACKGROUND ON INVOLUTIVE HEEGAARD FLOER HOMOLOGY
 
In this section, we briefly recall the definition of the involutive knot concordance invariants of a knot and the involutive correction terms of a three-manifold from <cit.> and their relationship to surgeries on the knot <cit.>. We begin by recalling the algebraic setup.

  An iota-complex is a pair (C,ι) of the following form:


  * C is an absolutely ℚ-graded and relatively ℤ-graded finitely-generated chain complex over 𝔽_2[U] with the property that U^-1H_*(C) ≃𝔽_2[U,U^-1] with some grading shift.

  * ι is a grading-perserving chain map such that ι^2 is chain homotopic to the identity.



To a rational homology sphere Y together with a  structure 𝔰 on Y, Ozsváth and Szabó's Heegaard Floer homology associates a chain complex (Y,𝔰) <cit.>; if 𝔰 is conjugation-invariant, involutive Floer homology appends to this data a chain map ι. The pair ((Y,𝔰), ι) is an iota-complex in the sense of Definition <ref>.

The involutive Heegaard Floer invariants satisfy a tidy connected sum formula, as follows. 


    (Y_1 # Y_2, 𝔰_1 #𝔰_2, ι) ≃ ((Y_1, 𝔰_1) ⊗(Y_2, 𝔰_2), ι_1 ⊗ι_2).


That is, the two complexes are chain homotopy equivalent via maps which commute up to homotopy with the involutions in their pairs. This relationship is sometimes called a strong equivalence. The statement for the chain complexes is due to <cit.> and the statement for the involution is due to <cit.>.


In ordinary Heegaard Floer homology, the d-invariant of a chain complex C satisfying the criterion above is the maximum grading of an element a in C such that [U^n a] ≠ 0 for all positive n, and the d-invariant or correction term of (Y,𝔰) is d((Y,s)) <cit.>. From this, one may extract Rasmussen's concordance invariant V_0 as V_0(K) = -1/2 d((S^3_+1(K)) <cit.>. The d-invariant is additive under connected sum; that is, 


    d((Y_1#Y_2), 𝔰_1#𝔰_2) = d(Y_1, 𝔰_1) + d(Y_2, 𝔰_2).


An important special case is that of the lens space L(p,q). The iota-complex associated to L(p,q) in any  structure 𝔰 is a copy of the pair (𝔽_2[U], Id), with the grading of the element 1 in 𝔽_2[U] being d(L(p,q),𝔰). Therefore, taking the connected sum with a lens space has the sole effect of imposing a grading shift on ((Y), 𝔰).

We may now define the involutive variants of the d-invariant. We follow <cit.>, which is a reformulation of the original definitions from <cit.>.


Let (C,ι) be an iota-complex. Then (C,ι) is the maximum grading of a homogeneous cycle a ∈ C such that [U^na] ≠ 0 for all positive n and furthermore there exists an element b such that ∂ b= (id + ι)a.



Let (C,ι) be an iota-complex. Consider triples (x,y,z) consisting of elements of C, with at least one of x or y nonzero such that ∂ y = (id + ι)x, ∂ z = U^m x, for some m ≥ 0 and [U^n(U^my + (id + ι)z)] ≠ 0 for all n ≥ 0. If x ≠ 0, assign this triple the value (x)+1; if x=0, assign this triple the value (y). Then (C,ι) is defined as the maximum of these grading values across all valid triples (x,y,z).


Given a three-manifold Y with conjugation-invariant -structure 𝔰, we say that (Y, 𝔰) = ((Y,𝔰)) and likewise (Y, 𝔰) = ((Y, 𝔰)). The invariants  and  are not additive under connected sum, but instead, for 𝔰 = 𝔰_1 #𝔰_2, satisfy the following <cit.>:


    (Y_1,𝔰_1) + (Y_2, 𝔰_2) ≤(Y_1#Y_2,𝔰) ≤(Y_1,𝔰_1) + (Y_2, 𝔰_2) ≤(Y_1#Y_2,𝔰) ≤(Y_1,𝔰_1) + (Y_2, 𝔰_2).



We may now define the invariants  and .


Let K be a knot in S^3. The involutive concordance invariants of K are

    (K):= -1/2(S^3_+1(K)),     (K):= -1/2(S^3_+1(K)).

 
These invariants also admit a more general relationship to surgeries, which is an extension of a formula for the non-involutive d-invariant proved by Ni and Wu <cit.>. The set of -structures on p/q-surgery on K may be identified with ℤ/pℤ; in this paper we follow the convention used by <cit.> and <cit.> for this identification. Ni and Wu show that

    d(S^3_p/q(K), [s])= d(L(p,q),[s]) - 2max{ V_⌊s/p⌋(K), V_⌊p+q -1-s/p⌋(K) }.

Here V_i represents the ith concordance invariant in the sequence defined in <cit.>, which sequence generalizes V_0. In <cit.>, the first author, Hom, Stoffregen, and Zemke and the first author prove a surgery formula in involutive Heegaard Floer homology, one of whose consequences is an involutive analog of the relationship above, which we now recall. With respect to the convention above, the conjugation action on -structures is


    J([i])=[p+q-1-i],  where  [i] ∈ℤ_p.


In particular, if p and q are both odd, then [q-1/2] ∈ℤ_p represents the unique self-conjugate -structure on the manifold S^3_p/q(K). When p is even and q is odd, there are two self-conjugate -structures on the manifold S^3_p/q(K), to wit [q-1/2] and [p+q-1/2]. Finally, when p is odd and q is even, there is a unique -structure [p+q-1/2]. With this in mind, we have the following.
<cit.> 
Suppose that q is odd, then we have

    (S^3_p/q(K), [ q-1/2])= d(L(p,q),[q-1/2]) - 2(K)
 
    (S^3_p/q(K), [q-1/2])= d(L(p,q),[ q-1/2]) - 2(K).
 
If one of p or q is even, then

    (S^3_p/q(K), [p+q-1/2])= d(S^3_p/q(K),[p+q-1/2])
 
    (S^3_p/q(K), [p+q-1/2])= d(L(p,q),[p+q-1/2]).



Note that the original version of the theorem is stated using a different indexing convention for the  structures, which we have translated into the Ozsváth-Szabó convention here; for more on this, see <cit.>. For the statement of the full surgery formula of which this relationship is a consequence, see <cit.>.






§ PROOFS OF THE MAIN THEOREMS
 


 §.§ A cabling formula for the involutive concordance invariants

In this subsection, we prove Theorems <ref> and  <ref>. A key ingredient of our proof will be the following well-known diffeomorphism (see for example <cit.> or <cit.> for a proof):

    S^3_pq(K_p,q) = S^3_q/p(K) # L(p,q).

Let us now denote the projections of a  -structure to the two summands S^3_q/p(K) and L(p,q) as π_1 and π_2 respectively. We may treat these as functions

    π_1 ℤ_pq→ℤ_q,     π_2 ℤ_pq →ℤ_p.


We record the following lemma, which is a direct consequence of <cit.>.


The self-conjugate -structure [0] ∈ℤ_pq projects to the summands as follows:



  * If both p and q are odd, we have
 
    π_1(0) ≡p-1/2  (mod  q);     π_2(0) ≡q-1/2  (mod  p).

 
 
  * If p is odd and q is even, 
 
    π_1(0) ≡p-1/2  (mod  q);     π_2(0) ≡p+q-1/2  (mod  p).

 
 
  * If p is even and q is odd,
 
    π_1(0) ≡p+q-1/2  (mod  q);     π_2(0) ≡q-1/2  (mod  p).

 


In <cit.> it was shown that the projections maps satisfy the following relations

    π_1(0) ≡ - (p-1)(q-1)/2  (mod  q);     π_2(0) ≡ - (p-1)(q-1)/2  (mod  p).

The desired result now follows from a direct calculation.


We may now prove Theorem <ref>.


We break the proof into three cases, depending on the parity of p and q.

Case 1. We first consider the case when p and q are both odd. We will give the proof for . The proof for  is identical. 



Per Lemma <ref>, the projection of the [0]  structure on S^3_pq(K) to L(p,q) has image the unique self-conjugate -structure, represented by

    [q-1/2]   (mod  p).

Similarly, for q/p-Dehn surgery with q odd, the projection of the [0]  structure on S^3_pq(K) to S^3_q/p(K) has image the unique self-conjugate -structure, namely

    [p-1/2]   (mod  q).

Now applying Lemma <ref> and (<ref>) to the case that both p and q are odd, we we see that the iota-complex ((S^3_pq(K_p,q),[0]), ι) is strongly equivalent to the iota-complex ((S^3_q/p(K),[p-1/2]),ι) shifted in grading by d(L(p,q),[q-1/2]).  It therefore follows that 

    (S^3_pq(K_p,q),[0])= (S^3_q/p(K), [ p-1/2]) + d(L(p,q),[q-1/2]).

Applying Theorem <ref>, we get that 


    (S^3_pq(K_p,q),[0])= d(L(pq,1),[0]) -2(K_p,q)

and also that

    (S^3_q/p(K),[p-1/2]) = d(L(q,p),[p-1/2]) - 2(K).


Combining Equations (<ref>) and (<ref>) with Equation (<ref>) we get 

    d(L(pq,1),[0]) -2(K_p,q)=d(L(q,p),[p-1/2]) - 2(K) + d(L(p,q),[q-1/2]).

Now by plugging in the unknot for K in Equation <ref>, we get

    d(L(pq,1),[0]) -2(T_p,q) = d(L(q,p),[p-1/2]) +  d(L(p,q),[q-1/2]).

Substituting Equation (<ref>) into Equation (<ref>) and observing that (T_p,q)= V_0(T_p,q) <cit.> we get the desired equality. The proof for  is similar.

Case 2. Let us now consider the case when p is odd and q is even. We first consider . The two self-conjugate -structures on S^3_q/p(K) are given by

    [p-1/2]   (mod  q)   and  [p+q-1/2]   (mod  q).

On the other hand, on L(p,q) there is only one self-conjugate  structure, to wit

    [p+q-1/2] (mod  p).

Now, in this case, Lemma <ref> implies

    (S^3_pq(K_p,q),[0])= (S^3_q/p(K), [ p-1/2]) + d(L(p,q),[p+q-1/2]).

Again applying Theorem <ref>, we also have

    (S^3_pq(K_p,q),[0])= d(L(pq,1),[0]) -2(K_p,q)

and

    (S^3_q/p(K),[p-1/2]) = d(L(q,p),[p-1/2]) - 2(K).

The rest of the proof is identical to that for the case when p and q are both odd. The proof for  is also similar.

Case 3. We are left to consider the case when p is even. In this case by Lemma <ref> we have


    (CF(S^3_pq(K_p,q),[0])) = (S^3_q/p(K), [ p+q-1/2]) + d(L(p,q),[q-1/2]).


and from Theorem <ref> we still have


    (CF(S^3_pq(K_p,q),[0])) = d(L(pq,1),[0]) - 2(K).


However, in this case our second consequence of Theorem <ref> is of the form

    (S^3_q/p(K),[p+q-1/2]) = d(S^3_q/p(K),[p+q-1/2]).

Now let us choose s ≤ q-1 such that

    s ≡p+q-1/2  (mod  q).

Applying the Ni-Wu formula (<ref>) , we get

    d(S^3_q/p(K),[p+q-1/2]) = d(L(q,p),[p+q-1/2]) - 2  max{ V_⌊s/p⌋(K), V_⌊p+q -1-s/p⌋(K) }.

Combining Equations (<ref>), (<ref>), (<ref>) and (<ref>) and following the similar steps as above gives us the desired equality for .

As  and  have significantly different expressions in this final case, we also comment briefly on the argument for . The analog of Equation (<ref>) is

    (S^3_q/p(K),[p+q-1/2]) = d(L(q,p),[p+q-1/2]).

Therefore following similar steps as for the  case, we get

    (K_p,q) = V_0(T_p.q).

This completes the proof.

The proof of Corollary <ref> now follows.

We may apply Theorem <ref> recursively.





 §.§ Unknotting number bound

In this subsection, we produce a bound for the unknotting number, using Theorem <ref>.


In <cit.> Juhász-Zemke proved a bound for the slice genus using the involutive concordance invariants. Specifically, they showed

    -⌈g_4(K) + 1/2⌉≤(K) ≤(K) ≤⌈g_4(K) + 1/2⌉.

Now observe that

    g_4(K) ≤ u(K).

Hence, we immediately get the desired inequality by replacing K with K_p,q and applying Theorem <ref> for p odd.




§ EXAMPLES
 
We now discuss an example for which our cabling formula improves on other bounds from knot Floer homology.

 
We consider the knot K:= -2 T_6,7# T_6,13. It was shown in <cit.> that the knot Floer chain complex of K splits into equivariant summands one of which is the tensor product of 𝔽[U,U^-1] with the complex C in Figure <ref> in such a way that (K) and (K) is the involutive concordance invariant of the complex in Figure <ref> with the action of ι shown; in the language of the literature, the two complexes are locally equivalent. Indeed, one may compute  and  of the knot K by computing  and  of a suitable subcomplex of C ⊗𝔽[U,U^-1]; for more on this computation, see <cit.> and <cit.>.



In particular, computation shows that (K)=3. Now the unknotting number bound for K_3,2  from Theorem <ref> gives

    u(K_3,2) ≥ 2 (K) + 2 - 2 = 6

Note that slice genus, and hence unknotting number, bounds for K_3,2 coming from the Ozsváth-Szabó τ-invariant, from the V_0-invariant, and from the ν^+ invariant <cit.> are all 1. Finally, observe that the unknotting number bound from <cit.> shows that u(K_3,2) ≥ 3. 

A similar argument applies to the general situation of the (n,2)-cables of the knots K_n = -2T_2n,2n+1#T_2n,4n+1 for n odd considered in <cit.>, which have chain complexes locally equivalent to the analog of the complex in Figure <ref> with length n arrows in the box, and have (K_n) = n.







amsalpha




