A cabling formula for $\nu^+$ invariant

0.59 MB·English

by  Zhongtao Wu

A cabling formula for ν+ invariant
Zhongtao WU
5
1 Department of Mathematics, The Chinese Universiy of Hong Kong
0
Lady Shaw Building, Shatin, Hong Kong
2
Email: [email protected]
n
a
J
0 Abstract
2 We prove a cabling formula for the concordance invariant ν+, defined
by the author and Hom in [3]. This gives rise to a simple and effective
]
T 4-ball genus bound for many cable knots.
G
. 1 Introduction
h
t
a The invariant ν+, or equivalently ν−, is a concordance invariant defined
m
by the author and Hom [3], and by Ozsv´ath-Stipsicz-Szab´o [7] based on Ras-
[
mussen’s local h invariant [13]. It gives a lower bound on the 4-ball genus of

1 knots and can get arbitrarily better than the bounds from Ozsv´ath-Szab´o τ
v invariant. In this paper, we prove a cabling formula for ν+. The main result is:
9
4 Theorem 1.1. For p,q >0 and the cable knot K , we have
p,q
7
4 (p−1)(q−1)
ν+(K )=pν+(K)+
0 p,q 2
.
1
when q ≥(2ν+(K)−1)p−1.
0
5
As an application of the cabling formula, one can use ν+ to bound the 4-ball
1
: genus of cable knots; in certain special cases, ν+ determines the 4-ball genus
v
precisely.
i
X
Corollary 1.2. Suppose K is a knot such that ν+(K)=g (K)=n. Then
4
r
a
(p−1)(q−1)
ν+(K )=g (K )=pn+
p,q 4 p,q 2
for q ≥(2n−1)p−1.
Take K = T #2T # − T , for example. It is known that g (K) =
2,5 2,3 2,3;2,5 4
ν+(K)=2. UsingCorollary1.2,wecandeterminethe4-ballgenusofanycable
knots K when q ≥3p−1. This generalizes [3, Proposition 3.5].
p,q
Regarding the behavior of τ invariant under knot cabling, the question was
well-studied[1][12][14],culminatinginHom’sexplicitformulain[2]. Incontrast
1
totheratherexplicitcomputationalapproachusedinthesepapers,ourmethod
ofstudyisbasedonaspecialrelationshipbetweenν+ andsurgeryofknots,and
thus avoids the potential difficulty associated to the computation of the knot
Floer complex CFK∞(K ).
p,q
In order to carry out our proposed method, we need to compute the correc-
tion terms onboth sidesof the reducible surgery(4), which we shall describein
Section3. Themosttechnicalpartoftheargumentistoidentifytheprojection
map of the Spinc structures in the reducible surgery, and this is discussed in
Section 4. The proof of the main theorem follows in Section 5.
Acknowledgements. We like to thank Yi Ni for a helpful discussion. The
author was partially supported by a grant from the Research Grants Council
of the Hong Kong Special Administrative Region, China (Project No. CUHK
2191056).
2 The invariant ν+
In this section, we review the definition and properties of the ν+ invariant
from [3] and relevant backgrounds in Heegaard Floer theory. Heegaard Floer
homology is a collection of invariants for closed three-manifolds Y in the form
of homology theories HF∞(Y), HF+(Y), HF−(Y), H(cid:100)F(Y) and HFred(Y). In
Ozsv´ath-Szab´o[10]andRasmussen[13],acloselyrelatedinvariantisdefinedfor
null-homologous knots K ⊂ Y, taking the form of an induced filtration on the
Heegaard Floer complex of Y. In particular, let CFK∞(K) denote the knot
Floer complex of K ∈S3. Consider the quotient complexes
A+ =C{max{i,j−k}≥0} and B+ =C{i≥0}
k
where i and j refer to the two filtrations. The complex B+ is isomorphic to
CF+(S3). Associated to each k, there is a graded, module map
v+ :A+ →B+
k k
defined by projection and another map
h+ :A+ →B+
k k
defined by projection to C{j ≥ k}, followed by shifting to C{j ≥ 0} via the
U-action,andconcludingwithachainhomotopyequivalencebetweenC{j ≥0}
and C{i≥0}. Finally, the ν+ invariant is defined as
ν+(K):=min{k ∈Z|v+ :A+ →CF+(S3), v+(1)=1}. (1)
k k k
Here, 1 denotes the lowest graded generator of the non-torsion class in the
homology of the complex, and we abuse our notations by identifying A+ and
k
CF+(S3) with their homologies.
Recall that in the large N surgery, v+ corresponds to the maps induced
k
on HF+ by the two handle cobordism from S3 (K) to S3 [10, Theorem 4.4].
N
2
This allows one to extract 4-ball genus bound from functorial properties of the
cobordism map. We list below some additional properties of ν+, all of which
can be found in [3].
(a) ν+(K) is a smooth concordance invariant, taking nonnegative integer
value.
(b) τ(K)≤ν+(K)≤g (K). (See [9] for the definition of τ)
4
(cid:26)
0 if σ(K)≥0,
(c) For a quasi-alternating knot K, ν+(K)=
−σ(K) if σ(K)<0.
2
(d) For a strongly quasi-positive knot K,
ν+(K)=τ(K)=g (K)=g(K).
4
For a rational homology 3–sphere Y with a Spinc structure s, HF+(Y,s)
is the direct sum of two groups: the first group is the image of HF∞(Y,s) ∼=
F[U,U−1] in HF+(Y,s), which is isomorphic to T+ = F[U,U−1]/UF[U], and
its minimal absolute Q–grading is an invariant of (Y,s), denoted by d(Y,s), the
correction term [8]; the second group is the quotient modulo the above image
and is denoted by HF (Y,s). Altogether, we have
red
HF+(Y,s)=T+⊕HF (Y,s).
red
Using this splitting, we can associate for each integer k and the knot K a non-
negative integer V (K) that equals the U-exponent of v+ restricted to1 T+ ∈
k k
A+. Thissequenceof{V }isnon-increasing,i.e.,V ≥V ,andstabilizesat0
k k k k+1
forlargek. Observethattheminimumk forwhichV =0isthesameasν+(K)
k
defined in (1). This enables us to reinterpret the ν+ invariant in the following
more concise way.
ν+(K)=min{k ∈Z|V =0}, (2)
k
In addition, the sequence {V } completely determines the correction terms
k
of manifolds obtained from knot surgery. This can be seen from the surgery
formula [6, Proposition 1.6].
d(S3 (K),i)=d(L(p,q),i)−2max{V ,V }. (3)
p/q (cid:98)i(cid:99) (cid:98)p+q−1−i(cid:99)
q q
for p,q >0 and 0≤i≤p−1. We will explain this formula in greater detail in
Section 4.
We conclude this section by mentioning that an invariant equivalent to ν+,
denotedν−byOzsv´athandSzab´o,wasformulatedintermsofthechaincomplex
CFK− in [7]. That invariant played an important role to establish a 4-ball
genus bound for a one-parameter concordance invariant Υ (t) defined in the
K
same reference. For our purpose in the rest of the paper, we will not elaborate
on that definition.
1Again,weabusethenotationsbyidentifyingA+ withitshomology
K
3
3 Reducible surgery on cable knots
Recall that the (p,q) cable of a knot K, denoted K , is a knot supported
p,q
on the boundary of a tubular neighborhood of K with slope p/q with respect
to the standard framing of this torus. A well-known fact in low-dimensional
topology states that the pq-surgery on K results in a reducible 3-manifold.
p,q
Proposition 3.1.
S3 (K )∼=S3 (K)#L(p,q) (4)
pq p,q q/p
Theabovehomeomorphismisexhibitedinmanyreferences(cf[1]). Forself-
containedness, we include a proof of Proposition 3.1 below. Not only is this
reducible surgery a key ingredient of establishing our main result Theorem 1.1,
the geometric description of the homeomorphism is also crucial for justifying
Lemma 4.1.
Proof of Proposition 3.1. Denote N(K) the tubular neighborhood of K and
E(K) = S3 − N(K) its complement, and let T(K) be the boundary torus
of N(K). The cable K is embedded in T(K) as a curve of slope p/q. Con-
p,q
sider the tubular neighborhood N(K ) of the cable. The solid torus N(K )
p,q p,q
intersectsT atanannularneighborhoodA=N(K )∩T(K),andthebound-
K p,q
ary of this annulus consists of two parallel copies of K , denoted by λ and λ(cid:48),
p,q
each of which have linking number pq with K . Therefore, the surgery slope
p,q
of coefficient pq is given by λ (or equivalently, λ(cid:48)), and the pq-surgery on K
p,q
is performed by gluing a solid torus to the knot complement E(K ) in such a
p,q
way that the meridian is identified with a curve isotopic to λ.
On the other hand, one can think of the above gluing as attaching a pair
of 2-handles H , H to E(K ). See Figure 1. Since the exterior of K is
1 2 p,q p,q
homeomorphic to
E(K )=E(K)∪ N(K),
p,q T(K)−A
its pq-surgery can be decomposed as
S3 (K )=[E(K)∪H ]∪[N(K)∪H ].
pq p,q 1 2
As the 2-handles are attached along essential curves on T(K) (isotopic to λ),
E(K)∪H and N(K)∪H end up having a common boundary homeomorphic
1 2
to S2. This proves that S3 (K ) is a reducible manifold.
pq p,q
To further identify the two pieces of the reducible manifold, note that the
attaching curve is isotopic to λ, which has slope p/q on T(K). It follows that
N(K)∪H ∼=L(p,q)−D3.
2
FromtheperspectiveofE(K), thecurveλhasslopeq/p. Thus, theotherpiece
is
E(K)∪H ∼=S3 (K)−D3.
1 q/p
This completes the proof.
4
𝐻𝐻1
𝑝𝑝,𝑞𝑞 λ λ λ’ λ’
𝜆𝜆≈𝐾𝐾
T(K) λ’ 2 1
𝐻𝐻1≅𝐷𝐷 ×𝐷𝐷
𝐸𝐸(𝐾𝐾)
2 1
𝐻𝐻2≅𝐷𝐷 ×𝐷𝐷
Figure1: Attachthe2-handleH toE(K)alongK . Thetwodisk-endsofH
1 p,q 1
are identified with the corresponding disk ends of the other 2-handle H that is
2
attached to N(K).
4 Spinc structures in reducible surgery
Let us take a closer look at the surgery formula (3), in which there is an
implicit identification of Spinc structure
σ :Z/pZ→Spinc(S3 (K))
p/q
For simplicity, we use an integer 0 ≤ i ≤ p−1 to denote the Spinc structure
σ([i]), when[i]∈Z/pZisthecongruenceclassofimodulop. Theidentification
can be made explicit by the procedure in Ozsv´ath and Szab´o [11, Section 4,7].
In particular, it is independent of the knot K on which the surgery is applied2;
and it is affine:
σ[i+1]−σ[i]=[K(cid:48)]∈H (S3 (K))∼=Spinc(S3 (K))
1 p/q p/q
whereK(cid:48) isthedualknotofthesurgeryonK,andSpinc structuresareaffinely
identified with the first homology. Moreover, the conjugation map J on Spinc
structures can be expressed as
J(σ([i]))=σ([p+q−1−i]) (5)
(cf [5, Lemma 2.2]). We will use these identifications throughout this paper.
In [8, Proposition 4.8], Ozsv´ath and Szab´o made an identification of Spinc
structures on lens spaces through their standard genus 1 Heegaard diagram,
2Thus, formula (3) may be interpreted as comparing the correction terms of the “same”
Spinc structureofsurgeryondifferentknots.
5
which coincide with the above identification through surgery (on the unknot).
Theyalsoprovedthefollowingrecursiveformulaforthecorrectiontermsoflens
spaces
(2i+1−p−q)2−pq
d(L(p,q),i)= −d(L(q,r),j) (6)
4pq
for positive integers p > q and 0 ≤ i < p+q, where r and j are the reduction
module q of p and i, respectively. Substituting in q =1, one sees:
(2i−p)2−p
d(L(p,1),i)= (7)
4p
For a reducible manifold Y = Y #Y , there are projections from Spinc(Y)
1 2
to the Spinc structure of the two factors Spinc(Y ) and Spinc(Y ). Particularly,
1 2
this applies to the case of the reducible surgery S3 (K ) ∼= S3 (K)#L(p,q).
pq p,q q/p
Intermsofthecanonicalidentificationabove,wewriteφ :Z/pqZ→Z/qZand
1
φ : Z/pqZ → Z/pZ for the two projections. These two maps are independent
2
of the knot K, which we determine explicitly in the next lemma.
With the above notations and identifications of Spinc structures under-
stood, we apply the surgery formula (3) to both sides of the reducible manifold
S3 (K )∼=S3 (K)#L(p,q) and deduce
pq p,q q/p
d(L(pq,1),i)−2V (K )=d(L(q,p),φ (i))+d(L(p,q),φ (i))
i p,q 1 2
−2max{V (K),V (K)} (8)
(cid:98)φ1(i)(cid:99) (cid:98)p+q−1−φ1(i)(cid:99)
p p
for all i≤ pq. Here we used the fact that V ≥V when i≤ pq, as {V } is a
2 i pq−i 2 k
non-increasing sequence. When K is the unknot, (8) simplifies to
d(L(pq,1),i)−2V (T )=d(L(q,p),φ (i))+d(L(p,q),φ (i)) (9)
i p,q 1 2
as all V ’s are 0 for the unknot.
i
For the rest of the section, assume Y = S3 (K ), Y = S3 (K) and Y =
pq p,q 1 q/p 2
L(p,q), and denote K(cid:48) ⊂ Y = S3 (K) and K(cid:48) ⊂ Y = S3 (K ) the dual
1 q/p p,q pq p,q
knots of the surgery on K and K , respectively.
p,q
Lemma 4.1. The projection maps of the Spinc structure φ and φ are given
1 2
by:
(p−1)(q−1)
φ (i)=i− (mod q);
1 2
(p−1)(q−1)
φ (i)=i− (mod p).
2 2
Proof. Since the projection maps are affine, we assume
φ (i)=a ·i+b (mod q),
1 1 1
φ (i)=a ·i+b (mod p).
2 2 2
6
Note that the maps φ , φ are generally not homomorphisms. Nevertheless, we
1 2
claim that φ (i+1)−φ (i) = 1 (mod q). Under Ozsv´ath-Szab´o’s canonical
1 1
identificationσ :Z/pqZ→Spinc(Y),wehaveσ[i+1]−σ[i]=[K(cid:48) ]∈H (Y)∼=
p,q 1
Z/pqZ. Thus, it amounts to show that φ [K(cid:48) ]=[K(cid:48)] under the projection of
1 p,q
Y into the first factor Y .
1
This can be seen from the geometric description of the reducible surgery in
last section: The dual knot K(cid:48) , isotopic to the closed black curve on the right
p,q
of Figure 1, projects to an arc in E(K)∪H on the left of Figure 1. This arc
1
is closed up in Y by connecting it to a simple arc in D3 = Y −(E(K)∪H ).
1 1 1
Since the curve intersects λ once, it must represent3 [K(cid:48)]∈H (Y ). Hence
1 1
1=φ (i+1)−φ (i)=(a ·(i+1)+b )−(a ·i+b ) (mod q)
1 1 1 1 1 1
from which we see a =1. A similar argument proves a =1.
1 2
Todetermineb ,notethattheprojectionφ commuteswiththeconjugation
1 1
J as operations on Spinc structures. After substituting the equation φ (i) =
1
i+b (mod q) into φ ◦J =J ◦φ and applying (5), we get
1 1 1
(pq−i)+b =p+q−1−(i+b ) (mod q).
1 1
So
(cid:40)
−(p−1)(q−1) if q is odd
b = 2
1 −(p−1)(q−1) or − (p−1)(q−1) + q if q is even
2 2 2
where the identity is understood modulo q as before. Similar arguments also
imply:
(cid:40)
−(p−1)(q−1) if p is odd
b = 2
2 −(p−1)(q−1) or − (p−1)(q−1) + p if p is even
2 2 2
We argue that b = b = −(p−1)(q−1). This is evidently true when both p
1 2 2
and q are odd integers. When p is even and q is odd, we want to exclude the
possibilityb =−(p−1)(q−1) andb =−(p−1)(q−1)+p usingthemethodofproof
1 2 2 2 2
by contradiction. A similar argument will address the case for which p is odd
and q is even, and thus completes the proof.
We derive a contradiction by comparing the correction terms computed in
two ways. From equation (6) and (7), we have
(p−1)(q−1) (2j+1−p−q)2−pq
d(L(pq,1),j+ ) =
2 4pq
= d(L(q,p),j)+d(L(p,q),j)
for 0≤j <p+q.
On the other hand, it follows from (9) that
(p−1)(q−1) p
d(L(pq,1),j+ )=d(L(q,p),j)+d(L(p,q),j+ )
2 2
3Tobeaccurate,thisistrueuptoaproperchoiceoforientationofK(cid:48).
7
for 0 ≤ j < p + q − 1. Here, we used the fact that V (T ) = 0 for all
i p,q
i > (p−1)(q−1) (since g (T ) = (p−1)(q−1)) and the assumptions φ (i) =
2 4 p,q 2 1
i−(p−1)(q−1) andφ (i)=i−(p−1)(q−1)+p. Comparingtheabovetwoidentities,
2 2 2 2
we obtain
p
d(L(p,q),j)=d(L(p,q),j+ ) (10)
2
Recall from Lee-Lipshitz [4, Corollary 5.2] that correction terms of lens
spaces also satisfy the identity
p−1−2j
d(L(p,q),j+q)−d(L(p,q),j)=
p
for 0≤j <p. It follows
p p p−1−2(j+ p) −1−2j
d(L(p,q),j+ +q)−d(L(p,q),j+ )= 2 =
2 2 p p
Yet, according to (10),
p p p−1−2j
d(L(p,q),j+ +q)−d(L(p,q),j+ )=d(L(p,q),j+q)−d(L(p,q),j)=
2 2 p
We reach a contradiction! This completes the proof of the lemma.
As a quick check of Lemma 4.1, let us look at the surgery S3 (T ) ∼=
15 3,5
L(5,3)#L(3,5). ThecorrectiontermsofthethreelensspaceswithSpinc struc-
ture i are computed using (6) and summarized in Table 1 below.
(cid:80)
(cid:80)
(cid:80)(cid:80) i
(cid:80) 0 1 2 3 4 5 6 ···
lens space(cid:80)(cid:80)(cid:80)
L(15,1) 7/2 77/30 53/30 11/10 17/30 1/6 −1/10 ···
L(5,3) 2/5 0 2/5 −2/5 −2/5 2/5 2/5 ···
L(3,5) 1/6 1/6 −1/2 1/6 1/6 −1/2 1/6 ···
Table 1: Table of correction terms for lens spaces
Meanwhile, we compute the projection functions φ (i), φ (i) (according to
1 2
the formula in Lemma 4.1) and V (T ), and summarize the results in Table
i 3,5
2. We can then verify identity (9) using the values of correction termprovided
in Table 1. In particular, note that at i = (p−1)(q−1) = 4, there is the column
2
φ = φ = V = 0, as expected from Lemma 4.1, and there is the identity
1 2
17/30=2/5+1/6 that we can read off.
5 Proof of cabling formula
In this section, we prove Theorem 1.1. First, note the following relationship
between the sequences V (K ) and V (K) if we compare (8) and (9).
i p,q i
8
(cid:80)
(cid:80)
(cid:80)(cid:80) i
(cid:80) 0 1 2 3 4 5 6 ···
Function (cid:80)(cid:80)(cid:80)
φ 1 2 3 4 0 1 2 ···
1
φ 2 0 1 2 0 1 2 ···
2
V 2 1 1 1 0 0 0 ···
Table 2: projections φ , φ are given by Lemma 4.1
1 2
.
Lemma 5.1. Given p,q >0 and i≤ pq, the sequence of non-negative integers
2
V (K ) and V (K) satisfy the relation
i p,q i
V (K )=V (T )+2max{V (K),V (K)} (11)
i p,q i p,q (cid:98)φ1(i)(cid:99) (cid:98)p+q−1−φ1(i)(cid:99)
p p
Here, φ (i)=i− (p−1)(q−1) as above.
1 2
In order to evaluate ν+(K ) from equation (2), it is enough to determine
p,q
the minimum i such that V (K )=0. Since V (T )>0 when i< (p−1)(q−1),
i p,q i p,q 2
we only need to consider i≥ (p−1)(q−1) by (11).
2
Proof of Theorem 1.1. Whenq ≥(2ν+(K)−1)p+1,wehavepν+(K)+(p−1)(q−1) ≤
2
pq, so the condition i ≤ pq in Lemma 5.1 is satisfied for all i in the range
2 2
(p−1)(q−1) ≤i≤pν+(K)+ (p−1)(q−1). Equation (11) simplifies to
2 2
V (K )=2V (K) (12)
i p,q (cid:98)φ1(i)(cid:99)
p
as V (T ) = 0 and (cid:98)φ1(i)(cid:99) ≤ (cid:98)p+q−1−φ1(i)(cid:99) for φ (i) = i − (p−1)(q−1). As
i p,q p p 1 2
V (K) = 0 if and only if i ≥ ν+(K), it is easy to see from (12) and Lemma
i
4.1 that the minimum i such that V (K )=0 is pν+(K)+ (p−1)(q−1). Hence,
i p,q 2
ν+(K)=pν+(K)+ (p−1)(q−1).
2
When q < (2ν+(K)−1)p+1, the cabling formula for ν+(K ) is still un-
p,q
known. Nevertheless, the preceding argument gives the following lower bound.
Proposition 5.2. For p,q >0 and the cable knot K ,
p,q
pq
ν+(K )≥
p,q 2
when q <(2ν+(K)−1)p+1.
Theorem1.1areusefulwhenoneaimstodeterminethe4-ballgenusofsome
cable knots (e.g. Corollary 1.2).
9
Proof of Corollary 1.2. The 4-ball genus of the cable knot
(p−1)(q−1)
g (K )≤pg (K)+
4 p,q 4 2
since on can construct a slice surface for K from p parallel copies of a slice
p,q
surfaceforK togetherwith(p−1)q half-twistedbands. UsingTheorem1.1, we
see
(p−1)(q−1) (p−1)(q−1)
pn+ =ν+(K )≤g (K )≤pn+ ,
2 p,q 4 p,q 2
from which Corollary 1.2 follows immediately.
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