Content uploaded by Mariel Vazquez

Author content

All content in this area was uploaded by Mariel Vazquez on Dec 05, 2017

Content may be subject to copyright.

arXiv:1710.07418v1 [math.GT] 20 Oct 2017

DISTANCE ONE LENS SPACE FILLINGS AND BAND SURGERY

ON THE TREFOIL KNOT

TYE LIDMAN AND ALLISON H. MOORE AND MARIEL VAZQUEZ

Abstract. We study lens spaces that are related by distance one Dehn ﬁllings. More

precisely, we prove that if the lens space L(n, 1) is obtained by a surgery along a knot

in the lens space L(3,1) that is distance one from the meridional slope, then nis in

{−6,±1,±2,3,4,7}. This is proved by studying the behavior of the Heegaard Floer

d-invariants under integral surgery along knots in L(3,1). The main result yields a

classiﬁcation of the coherent and non-coherent band surgeries from the trefoil to T(2, n)

torus knots and links. This classiﬁcation result is motivated by local reconnection

processes in nature, which are modeled as band surgeries. Of particular interest is the

study of recombination on circular DNA molecules.

1. Introduction

The question of whether Dehn surgery along a knot Kin the three-sphere yields a three-

manifold with ﬁnite fundamental group is a topic of long-standing interest, particularly

the case of cyclic surgeries. The problem remains open, although substantial progress

has been made towards classifying the knots in the three-sphere admitting lens space

surgeries [BL89, Ber90, GT00, Ras04, OS05, Bak08a, Bak08b, Hed11]. When the exterior

of the knot is Seifert ﬁbered, there may be inﬁnitely many cyclic surgery slopes, such as

for a torus knot in the three-sphere [Mos71]. In contrast, the celebrated cyclic surgery

theorem [CGLS87] implies that if a compact, connected, orientable, irreducible three-

manifold with torus boundary is not Seifert ﬁbered, then any pair of ﬁllings with cyclic

fundamental group has distance at most one. Here, the distance between two surgery

slopes refers to their minimal geometric intersection number, and a slope refers to the

isotopy class of an unoriented simple closed curve on the bounding torus. Dehn ﬁllings

that are distance one from the ﬁber slope of a cable space are especially prominent

in surgeries yielding prism manifolds [BH96]. Fillings distance one from the meridional

slope were also exploited in [Bak15] to construct cyclic surgeries on knots in the Poincar´e

homology sphere.

In this paper, we are particularly interested in Dehn surgeries along knots in L(3,1)

which yield other lens spaces. The speciﬁc interest in L(3,1) is motivated by the study

of local reconnection in nature, such as DNA recombination (discussed below). Note

that by taking the knot Kto be a core of a genus one Heegaard splitting for L(3,1), one

may obtain L(p, q) for all p, q. More generally, since the Seifert structures on L(3,1) are

classiﬁed [GL17], one could enumerate the Seifert knots in L(3,1) and use this along with

the cyclic surgery theorem to characterize lens space ﬁllings when the surgery slopes are

1

2 TYE LIDMAN AND ALLISON H. MOORE AND MARIEL VAZQUEZ

Figure 1. The links L1and L2diﬀer in a three ball in which a rational

tangle replacement is made. Reconnection sites are schematically indi-

cated in red. (Left) A coherent band surgery. (Right) A non-coherent

band surgery.

of distance greater than one. This strategy does not cover the case where the surgery

slopes intersect the meridian of Kexactly once. We will refer to these slopes as distance

one surgeries, also called integral surgeries. In this article we are speciﬁcally concerned

with distance one Dehn surgeries along Kin L(3,1) yielding L(n, 1). We prove:

Theorem 1.1. The lens space L(n, 1) is obtained by a distance one surgery along a knot

in the lens space L(3,1) if and only if nis one of ±1,±2,3,4,−6or 7.

This result was motivated by the study of reconnection events in nature. Reconnection

events are observed in a variety of natural settings at many diﬀerent scales, for example

large-scale magnetic reconnection of solar coronal loops, reconnection of ﬂuid vortices,

and microscopic recombination on DNA molecules (e.g. [LZP+16, KI13, SIG+13]). Links

of special interest in the physical setting are four-plats, or equivalently two-bridge links,

where the branched double covers are lens spaces. In particular, the trefoil T(2,3)

is the most probable link formed by any random knotting process [RCV93, SW93],

and T(2, n) torus links appear naturally when circular DNA is copied within the cell

[ASZ+92]. During a reconnection event, two short chain segments, the reconnection sites,

are brought together, cleaved, and the ends are reconnected. When acting on knotted

or linked chains, reconnection may change the link type. Reconnection is understood

as a band surgery between a pair of links (L1, L2) in the three-sphere and is modeled

locally by a tangle replacement, where the tangle encloses two reconnection sites as

illustrated in Figure 1. Site orientation is important, especially in the physical setting,

as explained in Section 5.2. Depending on the relative orientation of the sites, the tangle

replacement realizes either a coherent (respectively non-coherent) band surgery, as the

links are related by attaching a band (see Figure 1). More details on the connection to

band surgery are included in Section 5.

We are therefore interested in studying the connection between the trefoil and other

torus links by coherent and non-coherent band surgery. The Montesinos trick implies

that the branched double covers of two links related by a band surgery are obtained by

distance one Dehn ﬁllings of a three-manifold with torus boundary. Because L(n, 1) is

the branched double cover of the torus link T(2, n), Theorem 1.1 yields a classiﬁcation

of the coherent and non-coherent band surgeries from the trefoil T(2,3) to T(2, n) for

all n.

DISTANCE ONE LENS SPACE FILLINGS AND BAND SURGERY ON THE TREFOIL KNOT 3

n n

Figure 2. Non-coherent bandings: (Left) T(2, n −2) to T(2, n + 2).

(Center) T(2, n) to itself. (Right) T(2,3) to the unknot.

Figure 3. Coherent bandings: (Left) T(2,3) to T(2,2). (Center) T(2,3)

to T(2,4). (Right) T(2,3) to T(2,−6) (see also [DIMS12, Theorem 5.10]).

Corollary 1.2. The torus knot T(2, n)is obtained from T(2,3) by a non-coherent band-

ing if and only if nis ±1, 3 or 7. The torus link T(2, n)is obtained from T(2,3) by a

coherent banding if and only if nis ±2, 4 or -6.

Proof. Theorem 1.1 obstructs the existence of any coherent or non-coherent banding from

T(2,3) to T(2, n) when nis not one of the integers listed in the statement. Bandings

illustrating the remaining cases are shown in Figures 2 and 3.

In our convention T(2,3) denotes the right-handed trefoil. The statement for the left-

handed trefoil is analogous after mirroring. Note that Corollary 1.2 certiﬁes that each

of the lens spaces listed in Theorem 1.1 is indeed obtained by a distance one surgery

from L(3,1). We remark that a priori, a knot in L(3,1) admitting a distance one lens

space surgery to L(n, 1) does not necessarily descend to a band move on T(2,3) under

the covering involution.

When nis even, if the linking number of T(2, n) is +n/2, Corollary 1.2 follows as a

consequence of the behavior of the signature of a link [Mur65]. If the linking number is

instead −n/2, Corollary 1.2 follows from the characterization of coherent band surgeries

between T(2, n) torus links and certain two-bridge knots in [DIMS12, Theorem 3.1].

While both coherent and non-coherent band surgeries have biological relevance, more

attention in the literature has been paid to the coherent band surgery model (see for

example [IS11, DIMS12, SIG+13, ISV14, BI15, BIRS16, SYB+17]). This is due in part

4 TYE LIDMAN AND ALLISON H. MOORE AND MARIEL VAZQUEZ

to the relative diﬃculty in working with non-orientable surfaces, as is the case with

non-coherent band surgery on knots.

Overview of main result. The key ingredients in the proof of Theorem 1.1 are a

set of formulas, namely [NW15, Proposition 1.6] and its generalizations in Propositions

4.1 and 4.2, which describe the behavior of d-invariants under certain Dehn surgeries.

Recall that a d-invariant or correction term is an invariant of the pair (Y, t), where Y

is an oriented rational homology sphere and tis an element of Spinc(Y)∼

=H2(Y;Z).

More generally, each d-invariant is a Spincrational homology cobordism invariant. This

invariant takes the form of a rational number given by the minimal grading of an element

in a distinguished submodule of the Heegaard Floer homology, H F +(Y, t) [OS03]. Work

of Ni-Wu [NW15] relates the d-invariants of surgeries along a knot Kin S3, or more

generally a null-homologous knot in an L-space, with a sequence of non-negative integer-

valued invariants Vi, due to Rasmussen (see for reference the local h-invariants in [Ras03]

or [NW15]).

With this we now outline the proof of Theorem 1.1. Suppose that L(n, 1) is obtained

by surgery along a knot Kin L(3,1). As is explained in Lemma 2.1, the class of |n|

modulo 3 determines whether or not Kis homologically essential. When n≡0 (mod 3),

we have that Kis null-homologous. In this case, we take advantage of the Dehn surgery

formula due to Ni-Wu mentioned above and a result of Rasmussen [Ras03, Proposition

7.6] which bounds the diﬀerence in the integers Vias ivaries. Then by comparing this

to a direct computation of the correction terms for the lens spaces of current interest,

we obstruct a surgery from L(3,1) to L(n, 1) for n6= 3 or −6.

When |n| ≡ ±1 (mod 3), we must generalize the correction term surgery formula of Ni-

Wu to a setting where Kis homologically essential. The technical work related to this

generalization makes use of the mapping cone formula for rationally null-homologous

knots [OS11], and is contained in Section 4. This surgery formula is summarized in

Propositions 4.1 and 4.2, which we then use in a similar manner as in the null-homologous

case. We ﬁnd that among the oriented lens spaces of order ±1 modulo 3, ±L(2,1), L(4,1)

and L(7,1) are the only nontrivial lens spaces with a distance one surgery from L(3,1),

completing the proof of Theorem 1.1.

Outline. In Section 2, we establish some preliminary homological information that will

be used throughout and study the Spincstructures on the two-handle cobordisms arising

from distance one surgeries. Section 3 contains the proof of Theorem 1.1, separated into

the three cases as described above. Section 4 contains the technical arguments pertaining

to Propositions 4.1 and 4.2, which compute d-invariants of certain surgeries along a

homologically essential knot in L(3,1). Lastly, in Section 5 we present the biological

motivation for the problem in relation with DNA topology and discuss coherent and

non-coherent band surgeries more precisely.

2. Preliminaries

2.1. Homological preliminaries. We begin with some basic homological preliminaries

on surgery on knots in L(3,1). This will give some immediate obstructions to obtaining

DISTANCE ONE LENS SPACE FILLINGS AND BAND SURGERY ON THE TREFOIL KNOT 5

certain lens spaces by distance one surgeries. Here we will also set some notation.

All singular homology groups will be taken with Z-coeﬃcients except when speciﬁed

otherwise.

Let Ydenote a rational homology sphere. First, we will use the torsion linking form on

homology:

ℓk :H1(Y)×H1(Y)→Q/Z.

See [CFH17] for a thorough exposition on this invariant.

In the case that H1(Y) is a cyclic group, it is enough to specify the linking form by

determining the value ℓk(x, x) for a generator xof H1(Y) and extending by bilinearity.

Consequently, if two rational homology spheres Y1and Y2have cyclic ﬁrst homology with

linking forms given by n

pand m

p, where p > 0, then the two forms are equivalent if and

only if n≡ma2(mod p) for some integer awith gcd(a, p) = 1. We take the convention

that L(p, q) is obtained by p/q-surgery on the unknot, and that the linking form is given

by q/p.1Following these conventions, p/q-surgery on any knot in an arbitrary integer

homology sphere has linking form q/p as well.

Let Kbe any knot in Y=L(3,1). The ﬁrst homology class of Kis either trivial

or it generates H1(Y) = Z/3, in which case we say that Kis homologically essential.

When Kis null-homologous, then the surgered manifold Yp/q (K) is well-deﬁned and

H1(Yp/q(K)) = Z/3⊕Z/p. When Kis homologically essential, there is a unique such

homology class up to a choice of an orientation on K. The exterior of Kis denoted

M=Y− N (K) and because Kis homologically essential, H1(M) = Z. Recall that

the rational longitude ℓis the unique slope on ∂M which is torsion in H1(M). In our

case, the rational longitude ℓis null-homologous in M. We write mfor a choice of dual

peripheral curve to ℓand take (m, ℓ) as a basis for H1(∂M ). Let M(pm +qℓ) denote

the Dehn ﬁlling of Malong the curve pm +qℓ, where gcd(p, q) = 1. It follows that

H1(M(pm +qℓ)) = Z/p and that the linking form of M(pm +qℓ) is equivalent to q/p

when p6= 0. Indeed, M(pm +qℓ) is obtained by p/q-surgery on a knot in an integer

homology sphere, namely the core of the Dehn ﬁlling M(m).

Recall that we are interested in the distance one surgeries to lens spaces of the form

L(n, 1). Therefore, we ﬁrst study when distance one surgery results in a three-manifold

with cyclic ﬁrst homology. We begin with an elementary homological lemma.

Lemma 2.1. Fix a non-zero integer n. Suppose that Y′is obtained from Y=L(3,1)

by a distance one surgery on a knot Kand that H1(Y′) = Z/n.

(i) If n= 3k±1, then Kis homologically essential.

(ii) If Kis homologically essential, the slope of the meridian on Mis 3m+ (3r+ 1)ℓ

for some integer r. Furthermore, there is a choice of msuch that r= 0.

1We choose this convention to minimize confusion with signs. The deviation from −q/p to q/p is

irrelevant for our purposes, since this change will uniformly switch the sign of each linking form computed

in this section. Because ℓk1and ℓk2are equivalent if and only if −ℓk1and −ℓk2are equivalent, this will

not aﬀect the results.

6 TYE LIDMAN AND ALLISON H. MOORE AND MARIEL VAZQUEZ

(iii) With the meridian on Mgiven by 3m+ℓas above, then if n= 3k+1 (respectively

n= 3k−1), the slope inducing Y′on Mis (3k+ 1)m+kℓ (respectively (3k−

1)m+kℓ).

(iv) If n= 3k, then Kis null-homologous and the surgery coeﬃcient is ±k. Further-

more, gcd(k, 3) = 1.

Proof. (i) This follows since surgery on a null-homologous knot in Yhas H1(Yp/q (K)) =

Z/3⊕Z/p.

(ii) By the discussion preceding the lemma, we have that the desired slope must be

3m+qℓ for some qrelatively prime to 3. In this case, M(3m+qℓ) has linking form

equivalent to 1/3 or 2/3, depending on whether q≡1 or 2 (mod 3). Since 2 is not a

square mod 3, we see that the linking form 2/3 is not equivalent to that of 1/3, which is

the linking form of L(3,1). Therefore, q≡1 (mod 3) and the meridian is 3m+ (3r+ 1)ℓ

for some r. By instead using the peripheral curve m′=m+rℓ, which is still dual to ℓ,

we see that the meridian is given by 3m′+ℓ.

(iii) By the previous item, we may choose msuch that the meridional slope of Kon M

is given by 3m+ℓ. Now write the slope on Myielding Y′as (3k±1)m+qℓ. In order

for this slope to be distance one from 3m+ℓ, we must have that q=k.

(iv) Note that if Kis null-homologous, then the other two conclusions easily hold since

H1(Y′) = Z/3⊕Z/k. Therefore, we must show that Kcannot be homologically essential.

If Kwas essential, then the slope on the exterior would be of the form 3km +sℓ for some

integer s. The distance from the meridian is then divisible by 3, which is a contradiction.

In this next lemma, we use the linking form to obtain a surgery obstruction.

Lemma 2.2. Fix a non-zero odd integer n. Let Kbe a knot in Y=L(3,1) with a

distance one surgery to Y′having H1(Y′) = Z/n and linking form equivalent to sgn(n)

|n|.

If n≡1 (mod 3), then n > 0.

Proof. Suppose that n < 0. Write n= 1 −3jwith j > 0. By assumption, the linking

form of Y′is −1/(3j−1). By Lemma 2.1(iii), the linking form of Y′is also given by

j/(3j−1). Consequently, −jis a square modulo 3j−1 or equivalently, −3 is a square

modulo 3j−1, as −3 is the inverse of −j. Because nis odd, the law of quadratic

reciprocity implies that for any prime pdividing 3j−1, we have that p≡1 (mod 3).

This contradicts the fact that 3j−1≡ −1 (mod 3).

Remark 2.3. By an argument analogous to Lemma 2.2, one can prove that if n= 3k−1

is odd, then n≡1or 11 (mod 12).

Lemma 2.2 does not hold if nis even. This can be seen since −L(2,1) ∼

=L(2,1) is

obtained from a distance one surgery on a core of the genus one Heegaard splitting of

L(3,1). In Section 2.3 we will be able to obtain a similar obstruction in the case that n

is even.

DISTANCE ONE LENS SPACE FILLINGS AND BAND SURGERY ON THE TREFOIL KNOT 7

2.2. The four-dimensional perspective. Given a distance one surgery between two

three-manifolds, we let Wdenote the associated two-handle cobordism. For details on

the framed surgery diagrams and associated four-manifold invariants used below, see

[GS99].

Lemma 2.4. Suppose that Y′is obtained from a distance one surgery on L(3,1).

(i) If |H1(Y′)|= 3k−1, then Wis positive-deﬁnite, whereas if |H1(Y′)|= 3k+ 1,

then Wis negative-deﬁnite.

(ii) The order of H1(Y′)is even if and only if Wis Spin.

Proof. (i) In either case, Lemma 2.1 implies that Y′is obtained by integral surgery on

a homologically essential knot Kin L(3,1). First, L(3,1) is the boundary of a four-

manifold N, which is a +3-framed two-handle attached to B4along an unknot. Let

Zdenote N∪W. Since b±

2(Z) = b±

2(N) + b±

2(W), we see that Wis positive-deﬁnite

(respectively negative-deﬁnite) if and only if b+

2(Z) is equal to 2 (respectively 1).

Since Kis homologically essential, after possibly reversing the orientation of Kand

handlesliding Kover the unknot, we may present Y′by surgery on a two-component

link with linking matrix

Q=3 1

1c,

which implies that the order of H1(Y′) is |3c−1|. Since the intersection form of Z

is presented by Q, we see that b+

2(Z) equals 2 (respectively 1) if and only if c > 0

(respectively c≤0). The claim now follows.

(ii) We will use the fact that an oriented four-manifold whose ﬁrst homology has no

2-torsion is Spin if and only if its intersection form is even. First, note that H1(W) is

a quotient of Z/3, so H1(W;Z/2) = 0. Next, view L(3,1) as the boundary of the Spin

four-manifold Xobtained from attaching −2-framed two-handles to B4along the Hopf

link. This is indeed Spin, because Xis simply-connected and has even intersection form.

After attaching Wto X, we obtain a presentation for the intersection form of W∪X:

QW∪X=

−2 1 a

1−2b

a b c

.

Since this matrix presents H1(Y′), we compute that |H1(Y′)|is even if and only if c

is even if and only if the intersection form of W∪Xis even. Since Xis Spin and we

are attaching Walong a Z/2-homology sphere, we see that the simply-connected four-

manifold W∪Xis Spin if and only if Wis Spin. Consequently, |H1(Y′)|is even if and

only if Wis Spin.

2.3. d-invariants, lens spaces, and Spin manifolds. As mentioned in the introduc-

tion, the main invariant that we will use is the d-invariant, d(Y, t), of a Spincrational ho-

mology sphere (Y, t). These invariants are intrinsically related with the intersection form

of any smooth, deﬁnite four-manifold bounding Y[OS03]. In some sense, the d-invariants

8 TYE LIDMAN AND ALLISON H. MOORE AND MARIEL VAZQUEZ

can be seen as a reﬁnement of the torsion linking form on homology. For homology lens

spaces, this notion can be made more precise as in [LS17, Lemma 2.2].

We assume familiarity with the Heegaard Floer package and the d-invariants of rational

homology spheres, referring the reader to [OS03] for details. We will heavily rely on the

following recursive formula for the d-invariants of a lens space.

Theorem 2.5 (Ozsv´ath-Szab´o, Proposition 4.8 in [OS03]).Let p > q > 0be relatively

prime integers. Then, there exists an identiﬁcation Spinc(L(p, q)) ∼

=Z/p such that

(1) d(L(p, q), i) = −1

4+(2i+ 1 −p−q)2

4pq −d(L(q, r), j )

for 0≤i < p +q. Here, rand jare the reductions of pand i(mod q)respectively.

Under the identiﬁcation in Theorem 2.5, it is well-known that the self-conjugate Spinc

structures on L(p, q) correspond to the integers among

(2) p+q−1

2and q−1

2.

(See for instance [DW15, Equation (3)].)

For reference, following (1), we give the values of d(L(n, 1), i), including d(L(n, 1),0),

for n > 0:

d(L(n, 1), i) = −1

4+(2i−n)2

4n

(3)

d(L(n, 1),0) = n−1

4.

It is useful to point out that d-invariants change sign under orientation-reversal [OS03].

Using the work of this section, we are now able to heavily constrain distance one surgeries

from L(3,1) to L(n, 1) in the case that nis even.

Proposition 2.6. Suppose that there is a distance one surgery between L(3,1) and

L(n, 1) where nis an even integer. Unless n= 2 or 4, we have n < 0. In the case

that n < 0, the two-handle cobordism from L(3,1) to L(n, 1) is positive-deﬁnite and

the unique Spin structure on L(n, 1) which extends over this cobordism corresponds to

i=|n

2|.

A technical result that we need is established ﬁrst, which makes use of Lin’s Pin(2)-

equivariant monopole Floer homology [Lin16].

Lemma 2.7. Let (W, s) : (Y, t)→(Y′,t′)be a Spin cobordism between L-spaces satisfying

b+

2(W) = 1 and b−

2(W) = 0. Then

(4) d(Y′,t′)−d(Y, t) = −1

4.

Proof. By [Lin17, Theorem 5], we have that

α(Y′,t′)−β(Y, t)≥ − 1

8,

DISTANCE ONE LENS SPACE FILLINGS AND BAND SURGERY ON THE TREFOIL KNOT 9

where αand βare Lin’s adaptation of the Manolescu invariants for Pin(2)-equivariant

monopole Floer homology. Conveniently, for L-spaces, α=β=d

2[CG13, Ram14, Lin16,

HR17]. Thus, we have

(5) d(Y′,t′)−d(Y, t)≥ − 1

4.

On the other hand, we may reverse orientation on Wto obtain a negative-deﬁnite Spin

cobordism (−W, s) : (−Y, t)→(−Y′,t′). Therefore, we have from [OS03, Theorem 9.6]

that

d(−Y′,t′)−d(−Y, t)≥c1(s)2+b2(−W)

4=1

4.

Combined with (5), this completes the proof.

Proof of Proposition 2.6. For completeness, we begin by dispensing with the case of

n= 0, i.e., S2×S1. This is obstructed by Lemma 2.1, since no surgery on a null-

homologous knot in L(3,1) has torsion-free homology.

Therefore, assume that n6= 0. The two-handle cobordism Wis Spin by Lemma 2.4.

First, suppose that b+

2(W) = 1 (and consequently b−

2(W) = 0), so that we may apply

Lemma 2.7. Because son Wrestricts to self-conjugate Spincstructures tand t′on Y

and Y′, (2) and (4) imply that

(6) d(L(n, 1), i)−d(L(3,1),0) = −1

4,

where imust be one of 0 or |n

2|. Applying Equation (3) to L(3,1), we conclude that

d(L(n, 1), i) = 1

4.

If i= 0, Equation (3) applied to L(n, 1) implies that d(L(n, 1),0) = |n|−1

4for n > 0

and 1−|n|

4for n < 0. The only solution agreeing with (6) is when n= +2. If i=|n|

2,

Equation (3) implies that d(L(n, 1), i) is −1

4for n > 0 and 1

4for n < 0, and so (6) holds

whenever n < 0. Note that in this case, Wis positive-deﬁnite.

Now, suppose that b+

2(W) = 0. Therefore, we apply Lemma 2.7 instead to −Wto see

that

−d(L(n, 1), i) + d(L(3,1),0) = −1

4,

where again, i= 0 or |n

2|. In this case, there is a unique solution given by n= +4 when

i= 0. This completes the proof.

2.4. d-invariants and surgery on null-homologous knots. Throughout the rest of

the section, we assume that Kis a null-homologous knot in a rational homology sphere

Y. By Lemma 2.1, this will be relevant when we study surgeries to L(n, 1) with n≡0

(mod 3). Recall that associated to K, there exist non-negative integers Vt,i for each

i∈Zand t∈Spinc(Y) satisfying the following property:

Property 2.8 (Proposition 7.6 in [Ras03]).

Vt,i ≥Vt,i+1 ≥Vt,i −1.

10 TYE LIDMAN AND ALLISON H. MOORE AND MARIEL VAZQUEZ

When Kis null-homologous in Y, the set of Spincstructures Spinc(Yp(K)) is in one-

to-one correspondence with Spinc(Y)⊕Z/p. The projection to the ﬁrst factor comes

from considering the unique Spincstructure on Ywhich extends over the two-handle

cobordism Wp(K) : Y→Yp(K) to agree with the chosen Spincstructure on Yp(K).

With this in mind, we may compute the d-invariants of Yp(K) as follows. The result

below was proved for knots in S3, but the argument immediately generalizes to the

situation considered here.

Proposition 2.9 (Proposition 1.6 in [NW15]).Fix an integer p > 0and a self-conjugate

Spincstructure ton an L-space Y. Let Kbe a null-homologus knot in Y. Then,

there exists a bijective correspondence i↔tibetween Z/pZand the Spincstructures

on Spinc(Yp(K)) that extend tover Wp(K)such that

(7) d(Yp(K),ti) = d(Y, t) + d(L(p, 1), i)−2Nt,i

where Nt,i = max{Vt,i, Vt,p−i}. Here, we assume that 0≤i < p.

In order to apply Proposition 2.9, we must understand the identiﬁcations of the Spinc

structures precisely. In particular, the correspondence between iand tiis given in [OS08,

Theorem 4.2]. Let sbe a Spincstructure on Wp(K) which extends tand let tibe the

restriction to Yp(K). Then, we have from [OS08, Theorem 4.2] that iis determined

by

(8) hc1(s),[b

F]i+p≡2i(mod 2p),

where [ b

F] is the surface in Wp(K) coming from capping oﬀ a Seifert surface for K. For

this to be well-deﬁned, we must initially choose an orientation on K, but the choice will

not aﬀect the end result.

Before stating the next lemma, we note that if Yis a Z/2-homology sphere, then

H1(Wp(K); Z/2) = 0, and thus there is at most one Spin structure on Wp(K). If p

is even, Wp(K) is Spin, since the intersection form is even and H1(Wp(K); Z/2) = 0.

Further, Yp(K) admits exactly two Spin structures, and thus exactly one extends over

Wp(K).

Lemma 2.10. Let Kbe a null-homologous knot in a Z/2-homology sphere Y. Let t

be the self-conjugate Spincstructure on Y, and let t0be the Spincstructure on Yp(K)

described in Proposition 2.9 above.

(i) Then, t0is self-conjugate on Yp(K).

(ii) The Spin structure t0does not extend to a Spin structure over Wp(K).

Proof. (i) By (8), we see that if sextends t0over Wp(K),

hc1(s),[b

F]i ≡ −p(mod 2p).

Note that sextends t0over Wp(K) and restricts to ton Y, since tis self-conjugate. The

above equation now implies that

hc1(s),[b

F]i ≡ p≡ −p(mod 2p).

DISTANCE ONE LENS SPACE FILLINGS AND BAND SURGERY ON THE TREFOIL KNOT 11

In the context of (8), i= 0. Consequently, we must have that salso restricts to t0on

Yp(K). Of course, this implies that t0is self-conjugate.

(ii) By (8), we deduce that for a Spin structure that extends tiover Wp(K), p≡2i

(mod 2p). Since we consider 0 ≤i≤p, we have that i=p

26= 0. Consequently, t0cannot

extend to a Spin structure on Wp(K).

3. The proof of Theorem 1.1

We now prove Theorem 1.1 through a case analysis depending on the order of the

purported lens space surgery modulo 3.

3.1. From L(3,1) to L(n, 1) where |n| ≡ 0(mod 3). The goal of this section is to

prove:

Proposition 3.1. There is no distance one surgery from L(3,1) to L(n, 1), where |n|=

3k, except when n= 3 or −6.

Proof. Let Kbe a knot in L(3,1) with a distance one surgery to L(n, 1) where |n|= 3k.

By Lemma 2.1(iv), we know that Kis null-homologous and the surgery coeﬃcient is

±k/1, and by Proposition 2.6, k6= 0.

The proof now follows from the four cases addressed in Propositions 3.2, 3.3, 3.4 and 3.5

below, which depend on the sign of nand the sign of the surgery on L(3,1). We obtain

a contradiction in each case, except when n= 3 or −6. These exceptional cases can be

realized through the band surgeries in Figures 2 and 3 respectively.

We now proceed through the case analysis described in the proof of Proposition 3.1.

Proposition 3.2. If k≥2, then L(3k, 1) cannot be obtained by +k/1-surgery on a

null-homologous knot in L(3,1).

Proof. By Proposition 2.6, 3kcannot be even, so we may assume that L(3k, 1) is obtained

by k-surgery on a null-homologous knot Kin Y=L(3,1) for kodd. Consequently,

there are unique self-conjugate Spincstructures on L(3k, 1), L(3,1),and L(k, 1). By (2),

Proposition 2.9, and Lemma 2.10,

(9) d(L(3k, 1),0) ≤d(L(3,1),0) + d(L(k, 1),0).

Using the d-invariant formula (3), when k≥2, we have

d(L(3k, 1),0) −d(L(3,1),0) −d(L(k, 1),0) = −1 + 3k

4−−1 + 3

4−−1 + k

4>0,

which contradicts (9).

Proposition 3.3. If k≥1, then −L(3k, 1) cannot be obtained by −k/1-surgery on a

null-homologous knot in L(3,1).

12 TYE LIDMAN AND ALLISON H. MOORE AND MARIEL VAZQUEZ

Proof. Suppose that −L(3k, 1) is obtained by −k/1-surgery on a null-homologous knot

in L(3,1). By Proposition 2.6, we cannot have that 3kis even. Indeed, in the current

case, the associated two-handle cobordism is negative-deﬁnite. Therefore, 3kis odd, and

we have unique self-conjugate Spincstructures on L(3k, 1) and L(k, 1).

By reversing orientation, L(3k, 1) is obtained by +k-surgery on a null-homologous knot

in −L(3,1). We may now repeat the arguments of Proposition 3.2 with a slight change.

We obtain that

d(L(3k, 1),0) ≤ −d(L(3,1),0) + d(L(k, 1),0).

By direct computation,

d(L(3k, 1),0) + d(L(3,1),0) −d(L(k, 1),0) = −1 + 3k

4+1

2−−1 + k

4>0.

Again, we obtain a contradiction.

Proposition 3.4. If k≥2, then L(3k, 1) cannot be obtained by −k/1-surgery on a

null-homologous knot in L(3,1).

Proof. As in the previous two propositions, Proposition 2.6 implies that kcannot be

even. Therefore, we assume that kis odd. We will equivalently show that if k≥3 is

odd, then −L(3k, 1) cannot be obtained by +k/1-surgery on a null-homologous knot in

−L(3,1).

Again, consider the statement of Proposition 2.9 in the case of the unique self-conjugate

Spincstructure on L(3k, 1). Writing tfor the self-conjugate Spincstructure on −L(3,1),

Equations (3) and (7) yield

2Nt,0=d(L(3k, 1),0) −d(L(3,1),0) + d(L(k, 1),0) = −1

4+3k

4−1

2+−1

4+k

4,

and so Nt,0=k−1

2. Since Vt,0≥Vt,k by Property 2.8, we have that Nt,0=Vt,0.

Next we consider Proposition 2.9 in the case that tis self-conjugate on −L(3,1) and

i= 1. From Property 2.8, we have that Vt,1must be either k−1

2or k−3

2. Since Nt,1=

max{Vt,1, Vt,k−1}=Vt,1, the same conclusion applies to Nt,1.

We claim that there is no Spincstructure on −L(3k, 1) compatible with (7) and Nt,1=

k−1

2or k−3

2. Suppose for contradiction that such a Spincstructure exists. Denote the

corresponding value in Z/3kby j. Of course, j6= 0, since j= 0 is induced by i= 0 on

L(k, 1).

First, consider the case that Nt,1=k−1

2. Applying (7) with i= 1 yields

k−1 = −1

4+(2j−3k)2

12k−1

2+−1

4+(2 −k)2

4k,

for some 0 < j < 3k. This simpliﬁes to the expression

k(3j+ 3) = j2+ 3.

Thus jis a positive integral root of the quadratic equation

f(j) = j2−3kj −(3k−3).

DISTANCE ONE LENS SPACE FILLINGS AND BAND SURGERY ON THE TREFOIL KNOT 13

For k > 0, there are no integral roots with 0 < j < 3k.

Suppose next that Nt,1=k−3

2. Equation (7) now yields

k−3 = −1

4+(2j−3k)2

12k−1

2+−1

4+(2 −k)2

4k

which simpliﬁes to the expression

k(3j−3) = j2+ 3.

Thus jis an integral root of the quadratic equation

f(j) = j2−3kj + (3k+ 3).

However, the only integral roots of this equation for k > 0 occur when k= 2 and j= 3,

and we have determined that kis odd. Thus, we have completed the proof.

Proposition 3.5. If k= 1 or k > 2, then L(−3k, 1) cannot be obtained by +k/1-surgery

on a null-homologous knot in L(3,1).

Proof. As a warning to the reader, this is the unique case where Proposition 2.6 does

not apply, and we must also allow for the case of keven. Other than this, the argument

mirrors the proof of Proposition 3.4 with some extra care to identify the appropriate

self-conjugate Spincstructures.

Consider the statement of Proposition 2.9 in the case that tis self-conjugate on L(3,1)

and i= 0 on L(k, 1). We would like to determine which Spincstructure on L(−3k, 1) is

induced by (7). As in the previous cases, when kis odd, t0is the unique self-conjugate

Spincstructure on L(−3k, 1), which corresponds to 0. We now establish the same con-

clusion if kis even. In this case, the proof of Lemma 2.10 shows that the Spincstruc-

tures t0and tk

2, as in Proposition 2.9, give the two self-conjugate Spincstructures on

L(−3k, 1). On the other hand, (2) asserts that the numbers 0 and 3k/2 also correspond

to the two self-conjugate Spincstructures on L(−3k, 1). Proposition 2.6 shows that

3k/2 corresponds to the Spin structure that extends over the two-handle cobordism,

while Lemma 2.10(ii) tells us that t0is the Spin structure that does not extend. In other

words, t0corresponds to 0 on L(−3k, 1).

Equations (3) and (7) now yield

2Nt,0=d(L(3k, 1),0) + d(L(3,1),0) + d(L(k, 1),0) = −1

4+3k

4+1

2+−1

4+k

4,

and so Nt,0=k

2. Since Vt,0≥Vt,k , we have that Nt,0=Vt,0.

Next we consider Proposition 2.9 in the case that tis self-conjugate on L(3,1) and

i= 1. From Property 2.8, we have that Vt,1must be either k

2or k−2

2. Since Nt,1=

max{Vt,1, Vt,k−1}=Vt,1, we also have Nt,1=k

2or k−2

2.

We claim that there is no Spincstructure on −L(3k, 1) compatible with Nt,1=k

2or k−2

2

in (7). Suppose for the contrary such a Spincstructure exists corresponding to j∈Z/3k.

Again, j6= 0.

14 TYE LIDMAN AND ALLISON H. MOORE AND MARIEL VAZQUEZ

In the case that Nt,1=k

2, then (7) yields

k=−1

4+(2j−3k)2

12k+1

2+−1

4+(2 −k)2

4k,

which simpliﬁes to the expression

k(3j+ 3) = j2+ 3.

As discussed in the proof of Proposition 3.4, there are no integral solutions with k≥1

and 0 < j < 3k.

In the case that Nt,1=k−2

2, then Equations (3) and (7) now yield

k−2 = −1

4+(2j−3k)2

12k+1

2+−1

4+(2 −k)2

4k,

which simpliﬁes to the expression

k(3j−3) = j2+ 3.

As discussed in the proof of Proposition 3.4, there is a unique integral root corresponding

to k= 2 and j= 3. This exceptional case arises due to the distance one lens space

surgery from L(3,1) to −L(6,1) described in [Bak11, Corollary 1.4]2(see also [MP06,

Table A.5]).

3.2. From L(3,1) to L(n, 1) where |n| ≡ 1(mod 3). The goal of this section is to

prove the following.

Proposition 3.6. There is no distance one surgery from L(3,1) to L(n, 1) where |n|=

3k+ 1, except when n=±1,4or 7.

As a preliminary, we use (1) to explicitly compute the d-invariant formulas that will be

relevant here. For k≥0,

d(L(3k+ 1,1), j) = −1

4+(−1 + 2j−3k)2

4(3k+ 1)

(10)

d(L(3k+ 1,1),0) = 3k

4

(11)

d(L(3k+ 1,3),1) = k

4

(12)

d(L(3k+ 1,3),4) = 8−11k+ 3k2

4(3k+ 1) .(13)

We will also need the following proposition about the d-invariants of surgery, proved in

Proposition 4.2 in Section 4. This can be seen as a partial analogue of Proposition 2.9

for homologically essential knots.

2While this is written as L(6,1) in [Bak11], Baker was working in the unoriented category.

DISTANCE ONE LENS SPACE FILLINGS AND BAND SURGERY ON THE TREFOIL KNOT 15

Proposition 3.7. Let Kbe a knot in L(3,1). Suppose that a distance one surgery on K

produces an L-space Y′where |H1(Y′)|= 3k+1 is even. Then there exists a non-negative

integer N0satisfying

(14) d(Y′,t) + d(L(3k+ 1,3),1) = 2N0,

where tis the unique self-conjugate Spincstructure on Y′.

Furthermore, if N0≥2, then there exists t′∈Spinc(Y′)and an integer N1equal to N0

or N0−1satisfying

(15) d(Y′,t′) + d(L(3k+ 1,3),4) = 2N1.

With the above technical result assumed, the proof of Proposition 3.6 will now follow

quickly. The strategy of proof is similar to that used in the case of L(3k, 1).

Proof of Proposition 3.6. By Lemma 2.4 and Proposition 2.6, we see that nmust be

odd or n= 4. In the latter case, we construct a coherent band surgery from the torus

knot T(2,3) to T(2,4) in Figure 3, which lifts to a distance one surgery from L(3,1) to

L(4,1). Therefore, for the remainder of the proof, we assume that nis odd. We also

directly construct a non-coherent band surgery from T(2,3) to T(2,7) and the unknot

in Figure 2, so we now focus on ruling out all even values of k≥4.

We begin by ruling out distance one surgeries to +L(3k+ 1,1) with k≥4. Since

n= 3k+ 1 is odd, there is a unique self-conjugate Spincstructure on L(3k+ 1,1). By

Equations (11), (12) and (14), we have N0=k

2. Since kis at least 4, we have N0≥2.

We claim that there is no solution to Equation (15) with N1=k

2or k−2

2. This will

complete the proof for the case of +L(3k+ 1,1).

First, consider the case of N1=k

2. Simplifying Equation (15) as in the proof of Propo-

sition 3.4 we obtain

j2−(1 + 3k)j+ (2 −3k) = 0.

It is straightforward to see that there are no non-negative integral roots of the quadratic

equation for positive k.

Next, we consider N1=k−2

2. In this case, (15) implies

j2−(1 + 3k)j+ (3k+ 4) = 0.

The roots are of the form

j=1

2(1 + 3k±p9k2−6k−15).

It is straightforward to verify that for k≥4, the lesser root is always strictly between 1

and 2, while the greater root is strictly between 3k−1 and 3k. Therefore, there are no

integer solutions. This completes the proof for the case of +L(3k+ 1,1).

To complete the proof of Proposition 3.6, it remains to show that −L(3k+ 1,1), with

k > 0 even, cannot be obtained from a distance one surgery along a homologically

essential knot in L(3,1). Proposition 3.7 establishes

(16) d(L(3k+ 1,1),0) = d(L(3k+ 1,3),1) −2N0

16 TYE LIDMAN AND ALLISON H. MOORE AND MARIEL VAZQUEZ

for some non-negative integer N0. However, from Equations (11) and (12), we have that

d(L(3k+ 1,1),0) = 3k

4>k

4=d(L(3k+ 1,3),1),

which contradicts (16).

3.3. From L(3,1) to L(n, 1) where |n| ≡ −1(mod 3). In this section, we handle the

ﬁnal case in the proof of Theorem 1.1:

Proposition 3.8. There is no distance one surgery from L(3,1) to L(n, 1), where |n|=

3k−1>0, except when n=±2.

As before, we state the d-invariant formulas that will be relevant for proving this theorem

ﬁrst.

d(L(3k−1,1), i) = −1

4+(2i−3k+ 1)2

4(3k−1)

(17)

d(L(3k−1,3),1) = k−2

4

(18)

d(L(3k−1,3),4) = 3k2−19k+ 18

4(3k−1) .(19)

The above follow easily from (1).

Next, we state a technical result about the d-invariants of surgery, similar to Proposi-

tion 3.7 above, that we will also prove in Proposition 4.1.

Proposition 3.9. Let Kbe a knot in L(3,1). Suppose that a distance one surgery on K

produces an L-space Y′where |H1(Y′)|= 3k−1>0. Then, there exists a non-negative

integer N0and a self-conjugate Spincstructure ton Y′such that

(20) d(Y′,t) = d(L(3k−1,3),1) −2N0.

In the case that kis odd, if t6=˜

tfor some self-conjugate ˜

t, then d(Y′,˜

t) = 1

4.

Furthermore, if N0≥2, then there exists another Spincstructure t′on Y′and an integer

N1equal to N0or N0−1satisfying

(21) d(Y′,t′) = d(L(3k−1,3),4) −2N1.

With this, the proof of Proposition 3.8 will be similar to the previous two cases.

Proof of Proposition 3.8. In the case that n=±2, we may construct a non-coherent

banding from T(2,3) to the Hopf link, as shown in Figure 3, which lifts to a distance

one surgery from L(3,1) to L(2,1) ∼

=L(−2,1). Therefore, we must rule out the case of

n=±(3k−1) with k≥2.

The proof will now be handled in two cases, based on the sign of n. First, we suppose

that +L(3k−1,1), with k≥2, is obtained by a distance one surgery on L(3,1). By

DISTANCE ONE LENS SPACE FILLINGS AND BAND SURGERY ON THE TREFOIL KNOT 17

Proposition 2.6, we only need to consider the case that 3k−1 is odd. Using (18), we

compute

d(L(3k−1,1),0) = 3k−2

4>k−2

4=d(L(3k−1,3),1).

This contradicts Proposition 3.9.

Now, we suppose there is a distance one surgery from L(3,1) to −L(3k−1,1) with

k≥2. By Lemma 2.2, we may assume that nis even. We begin with the case of k= 3.

Lemma 2.1 implies that if −L(8,1) was obtained by a distance one surgery, then the

linking form of −L(8,1) must be equivalent to 3

8. This is impossible since 5 is not a

square mod 8. Thus, we restrict to the case of k≥5 for the rest of the proof.

Proposition 3.9 and the fact that d(−L(3k−1,1),0) 6=1

4imply that

−d(L(3k−1,1),0) = d(L(3k−1,3),1) −2N0

for some non-negative integer N0. We compute from (17) and (18) that

N0=k−1

2.

Since we are in the case of k≥5, we may apply (21). Combined with (19), this yields

1

4−(2j−3k+ 1)2

4(3k−1) =3k2−19k+ 18

4(3k−1) −2N1,

for some 0 < j < 3k−1. Equivalently,

N1=5 + j+j2−7k−3jk + 3k2

2(3k−1) .

Here N1=k−1

2or k−3

2.

In the case of k−1

2, we are looking for integral roots of the quadratic equation

f(j) = j2+j(1 −3k) + (4 −3k).

For k≥5, there are no roots between 0 and 3k−1. For the case of k−3

2, we are instead

looking for integral roots of the quadratic

f(j) = j2+j(1 −3k) + (3k+ 2).

There are no integral roots in this case for k≥5. This completes the proof.

4. The mapping cone formula and d-invariants

In this section, we prove the following two key technical statements which were used

above in the proof of Theorem 1.1 in the cases of |n| ≡ ±1 (mod 3). These provide

analogues of Proposition 2.9 for certain surgeries on homologically essential knots in

L(3,1).

18 TYE LIDMAN AND ALLISON H. MOORE AND MARIEL VAZQUEZ

Proposition 4.1. Let Y=L(3,1) and suppose that Y′is an L-space obtained from a

distance one surgery on a knot in Y, where |H1(Y′)|= 3k−1with k≥1. Then there

exists a non-negative integer N0and a self-conjugate Spincstructure ton Y′satisfying

(22) d(Y′,t) = d(L(3k−1,3),1) −2N0.

Furthermore, if N0≥2, then there exists an integer N1satisfying N0≥N1≥N0−1

and

(23) d(Y′,t+P D[µ]) = d(L(3k−1,3),4) −2N1.

Here, [µ]represents the class in H1(Y′)induced by the meridian of the knot.

Moreover, if ˜

t6=tfor a self-conjugate Spincstructure ˜

t, then d(Y′,˜

t) = 1

4.

Proposition 4.2. Let Y=L(3,1) and suppose that Y′is an L-space obtained from a

distance one surgery on a knot in Y, where |H1(Y′)|= 3k+ 1 with k≥0. Then there

exists a non-negative integer N0and a self-conjugate Spincstructure ton Y′satisfying

(24) d(Y′,t) + d(L(3k+ 1,3),1) = 2N0.

Furthermore, if N0≥2, then there exists an integer N1satisfying N0≥N1≥N0−1

and

(25) d(Y′,t+P D[µ]) + d(L(3k+ 1,3),4) = 2N1.

Here, [µ]represents the class in H1(Y′)induced by the meridian of the knot.

Moreover, if ˜

t6=tfor a self-conjugate Spincstructure ˜

t, then d(Y′,˜

t) = 3

4.

Remark 4.3. We expect that the conclusions of these two propositions hold indepen-

dently of Y′being an L-space and the value of N0.

The general argument for the above propositions is now standard and is well-known to

experts. The strategy is to study the d-invariants using the mapping cone formula for

rationally null-homologous knots due to Ozsv´ath-Szab´o [OS11]. In Section 4.1, we review

the mapping cone formula. In Sections 4.2, 4.3 and 4.4 we establish certain technical

results about the mapping cone formula analogous to properties well-known for knots in

S3. Finally, in Section 4.5, we prove Propositions 4.1 and 4.2.

4.1. The mapping cone for rationally nullhomologous knots. In this subsection,

we review the mapping cone formula from [OS11], which will allow us to compute the

Heegaard Floer homology of distance one surgeries on knots in a rational homology

sphere. We assume the reader is familiar with the knot Floer complex for knots in

S3; we will use standard notation from that realm. For simplicity, we work in the

setting of a rational homology sphere Y. (As a warning, Ywill be −L(3,1) when

proving Proposition 4.2.) All Heegaard Floer homology computations will be done with

coeﬃcients in F=Z/2. As mentioned previously, singular homology groups are assumed

to have coeﬃcients in Z, unless otherwise noted.

Choose an oriented knot K⊂Ywith meridian µand a framing curve λ, i.e. a slope λ

on the boundary of a tubular neighborhood of Kwhich intersects the meridian µonce

DISTANCE ONE LENS SPACE FILLINGS AND BAND SURGERY ON THE TREFOIL KNOT 19

transversely. Here, λnaturally inherits an orientation from K. Let Y′denote the result

of λ-surgery.

We write Spinc(Y, K) for the relative Spincstructures on (M, ∂M ), which has an aﬃne

identiﬁcation with H2(Y, K ) = H2(M, ∂M ). Here, M=Y− N (K). If Kgenerates

H1(Y), then Spinc(Y, K ) is aﬃnely isomorphic to Z. In our applications, this will be the

case.

There exist maps GY,±K: Spinc(Y, K )→Spinc(Y) satisfying

(26) GY,±K(ξ+κ) = GY,±K(ξ) + i∗κ,

where κ∈H2(Y, K ) and i: (Y, pt)→(Y , K) is inclusion. Here, −Kdenotes Kwith the

opposite orientation. We have

GY,−K(ξ) = GY ,K (ξ) + P D[λ].

If Y′=Yλ(K) is obtained by surgery on K, we will write K′or Kλfor the core of

surgery.

Associated to ξ∈Spinc(Y, K ) is the Z⊕Z-ﬁltered knot Floer complex Cξ=C F K∞(Y, K, ξ).

Here, the biﬁltration is written (algebraic,Alexander). We have Cξ+P D[µ]=Cξ[(0,−1)],

i.e. we shift the Alexander ﬁltration on Cξby one. Note that not every relative Spinc

structure is necessarily related by a multiple of P D[µ], so we are not able to use this

to directly compare the knot Floer complexes for an arbitrary pair of relative Spinc

structures.

For each ξ∈Spinc(Y, K), we deﬁne the complexes A+

ξ=Cξ{max{i, j} ≥ 0}and B+

ξ=

Cξ{i≥0}. The complex B+

ξis simply C F +(Y, GY,K (ξ)), while A+

ξrepresents the

Heegaard Floer homology of a large surgery on Kin a certain Spincstructure, described

in slightly more detail below.

The complexes A+

ξand B+

ξare related by grading homogenous maps

v+

ξ:A+

ξ→B+

ξ, h+

ξ:A+

ξ→B+

ξ+P D[λ].

Rather than deﬁning these maps explicitly, we explain how these can be identiﬁed with

certain cobordism maps as follows. Fix n≫0 and consider the three-manifold Ynµ+λ(K)

and the induced cobordism from Ynµ+λ(K) to Yobtained by attaching a two-handle to

Y, reversing orientation, and turning the cobordism upside down. We call this cobordism

W′

n, which is negative-deﬁnite. Fix a generator [F]∈H2(W′

n, Y ) such that P D[F]|Y=

P D[K]. Equip Ynµ+λ(K) with a Spincstructure t. It is shown in [OS11, Theorem 4.1]

that there exist two particular Spincstructures vand h=v+P D[F] on W′

nwhich

extend tover W′

nand an association Ξ : Spinc(Ynµ+λ(K)) →Spinc(Y, K ) satisfying

commutative squares:

(27) CF +(Ynµ+λ(K),t)

fW′

n,v

≃//A+

ξ

v+

ξ

C F +(Ynµ+λ(K),t)

fW′

n,h

≃//A+

ξ

h+

ξ

C F +(Y, GY,K (ξ)) ≃//B+

ξC F +(Y, GY,−K(ξ)) ≃//B+

ξ+P D[λ],

20 TYE LIDMAN AND ALLISON H. MOORE AND MARIEL VAZQUEZ

where ξ= Ξ(t). Here, fW′

n,sdenotes the Spinccobordism map in Heegaard Floer ho-

mology, as deﬁned in [OS06].

More generally, there exists a map EK,n,λ : Spinc(W′

n)→Spinc(Y, K ) such that if vand

hare as above, then

(28) EK,n,λ(v) = ξ, EK,n,λ(h) = ξ+nP D[µ] + P D [λ].

To make the notation more suggestive, we will write vξand hξfor the associated Spinc

structures on W′

nappearing in (27).

Recall that for any Spincrational homology sphere, the Heegaard Floer homology con-

tains a distinguished submodule isomorphic to T+=F[U, U −1]/U ·F[U], called the

tower. Since W′

nis negative-deﬁnite, on the level of homology, v+

ξinduces a grading

homogeneous non-zero map between the towers, which is necessarily multiplication by

UNfor some integer N≥0. We denote this integer by Vξ. The integer Hξis deﬁned

similarly. These numbers Vξare also known as the local h-invariants, originally due to

Rasmussen [Ras03]. A direct analogue of [Ras03, Proposition 7.6] (Property 2.8), using

Cξ+P D[µ]=Cξ[(0,−1)], shows that for each ξ∈Spinc(Y, K ),

(29) Vξ≥Vξ+P D[µ]≥Vξ−1.

We are now ready to deﬁne the mapping cone formula. Deﬁne the map

(30) Φ : M

ξ

A+

ξ→M

ξ

B+

ξ,(ξ, a)7→ (ξ, v+

ξ(a)) + (ξ+P D[λ], h+

ξ(a)),

where the ﬁrst component of (ξ, a) simply indicates the summand in which the element

lives. Notice that the mapping cone of Φ splits over equivalence classes of relative Spinc

structures, where two relative Spincstructures are equivalent if they diﬀer by an integral

multiple of P D[λ]. We let the summand of the cone of Φ corresponding to the equivalence

class of ξbe written X+

ξ. Ozsv´ath and Szab´o show that there exist grading shifts on

the complexes A+

ξand B+

ξsuch that X+

ξcan be given a consistent relative Z-grading

[OS11]. In fact, these shifts can be done to X+

ξwith an absolute Q-grading. While we

do not describe the grading shifts explicitly at the present moment, it is important to

point out that these shifts only depend on the homology class of the knot. With this,

we are ready to state the connection between the mapping cone formula and surgeries

on K.

Theorem 4.4 (Ozsv´ath-Szab´o, [OS11]).Let ξ∈Spinc(Y, K). Then there exists a quasi-

isomorphism of absolutely-graded F[U]-modules,

(31) X+

ξ≃C F +(Yλ(K), GYλ(K),Kλ(ξ)).

Finally, we remark that the entire story above has an analogue for the hat ﬂavor of

Heegaard Floer homology. We denote the objects in the hat ﬂavor by b

Aξ,b

Xξ,bvξ, etc.

The analogue of (31) is then a quasi-isomorphism

(32) b

Xξ≃d

C F (Yλ(K), GYλ(K),Kλ(ξ)).

DISTANCE ONE LENS SPACE FILLINGS AND BAND SURGERY ON THE TREFOIL KNOT 21

...

''

❖

❖

❖

❖

❖

❖

❖

❖

❖

❖

❖

❖

❖

❖

❖A+

ξ−5P D[m]

v+

ξ−5P D[m]

h+

ξ−5P D[m]

''

❖

❖

❖

❖

❖

❖

❖

❖

❖

❖

❖

❖

❖

❖

A+

ξ

v+

ξ

h+

ξ

''

❖

❖

❖

❖

❖

❖

❖

❖

❖

❖

❖

❖

❖

❖A+

ξ+5P D[m]

v+

ξ+5P D[m]

''

❖

❖

❖

❖

❖

❖

❖

❖

❖

❖

❖

❖

❖

❖

❖

...

. . . B+

ξ−5P D[m]B+

ξB+

ξ+5P D[m]...

Figure 4. The mapping cone formula for surgery on a knot in L(3,1)

resulting in a three-manifold Y′with |H1(Y′)|= 5 corresponding to the

Spincstructure GY′,K′(ξ).

A+

ξ−N·P D[λ]

''

❖

❖

❖

❖

❖

❖

❖

❖

❖

❖

❖

A+

ξ−(N−1)·P D[λ]

%%

❑

❑

❑

❑

❑

❑

❑

❑

❑

❑

❑

❑

...

❃

❃

❃

❃

❃

❃

❃

❃

❃A+

ξ

##

●

●

●

●

●

●

●

●

●

●A+

ξ+P D[λ]

""

❋

❋

❋

❋

❋

❋

❋

❋

❋

❋

❋

...

##

❍

❍

❍

❍

❍

❍

❍

❍

❍

❍

❍A+

ξ+N·P D[λ]

B+

ξ−(N−1)·P D[λ]. . . B+

ξB+

ξ+P D[λ]. . . B+

ξ+N·P D[λ]

Figure 5. The truncated mapping cone X+,N

ξcomputing

C F +(Yλ(K), GYλ(K),Kλ(ξ)) in the case that |H1(Yλ(K))| ≡ −1 (mod 3).

4.2. Preliminaries speciﬁc to knots in L(3,1).Through Sections 4.2-4.4, Kwill

denote a homologically essential knot in Y=L(3,1) and λwill denote a framing such

that Y′=Yλ(K) is an L-space with |H1(Y′)|= 3k−1 for some k > 0. The case of

|H1(Y′)|= 3k+ 1 is dealt with similarly, and the necessary changes are described in

Section 4.5 below. Recall that we give λthe orientation induced by K.

The mapping cone formula for any homologically essential knot in L(3,1) is easier to

describe than in generality. We have that Spinc(Y , K)∼

=Z. Write [m] for the generator of

H1(M) such that [µ] = 3[m] (instead of −3[m]). Consequently, since [µ]·[λ] = 1, we have

that [λ] = (3k−1)[m] by Lemma 2.1(iii). Therefore, for ﬁxed ξ∈Spinc(Y , K), we see

that the mapping cone X+

ξconsists of the Aξ′and Bξ′where ξ′−ξ= (3k−1)j·P D[m] for

some j∈Z. For a more pictorial representation, see Figure 4 for the case of k= 2.

Ozsv´ath and Szab´o show that for ﬁxed ξ, there exists Nsuch that v+

ξ+j·P D[µ]and

h+

ξ−j·P D[µ]are quasi-isomorphisms for j > N. Using this, the mapping cone formula

is quasi-isomorphic (via projection) to the quotient complex depicted in Figure 5. We

will denote the truncated complex by X+,N

ξ, which now depends on ξ, even though the

homology does not. Note that the shape of the truncation is special to the case that

H1(Yλ(K)) has order 3k−1. Were the order to be 3k+ 1, there would instead be one

more B+

ξthan A+

ξand h+

ξwould translate by −(3k+ 1)P D[m]. This issue will be dealt

with in Proposition 4.2 by reversing orientations and performing surgery on knots in

−L(3,1) instead.

22 TYE LIDMAN AND ALLISON H. MOORE AND MARIEL VAZQUEZ

By [BBCW12, Lemma 6.7], since Yλ(K) is an L-space obtained by a distance one surgery

in an L-space, we have that

(33) H∗(b

Aξ)∼

=F, H∗(A+

ξ)∼

=T+for all ξ∈Spinc(Y, K ).

Indeed, the orientation conventions from [BBCW12, Lemma 6.7] are speciﬁed by the

condition that [µ] and [λ] are positive multiples of the same homology class, which is

the setting we are in. Of course, since Y=L(3,1) is an L-space, we also have that

H∗(b

Bξ)∼

=Fand H∗(B+

ξ)∼

=T+for all ξ. Equation (33) implies that the Heegaard

Floer homology of Yλ(K) is completely determined by the numbers Vξand Hξfor each

ξ∈Spinc(Y, K ).

4.3. Spincstructures. In order to understand the Heegaard Floer homology of surgery

using the mapping cone, we must understand the various Spincand relative Spincstruc-

tures that appear. These are well-understood in the setting of a nullhomologous knot,

and are likely known to experts, but we include them here for completeness. As in the

previous subsection, Kwill denote a homologically essential knot in Y=L(3,1) and

λis a framing such that Y′=Yλ(K) is an L-space with |H1(Y′)|= 3k−1 for some

k > 0.

Fix n≫0 throughout. By ﬁxing the appropriate parity of n, we can compute from

Lemma 2.1 that “large positive surgery”, i.e. Ynµ+λ(K), has a unique self-conjugate

Spincstructure. We denote this by t0. Further, let ξ0= Ξ(t0) be the induced relative

Spincstructure as in (27). Recall that for ξ∈Spinc(Y, K ), we write vξand hξto be the

Spincstructures on W′

ndeﬁned above (27).

Proposition 4.5. Let [γ]∈H1(Y). Then, Vξ0+P D[γ]=Hξ0−P D [γ].

This is the analogue of the more familiar formula Vs=H−sfor knots in S3.

Proof. We will use an observation of Ni and Vafaee from [NV16, Proof of Lemma 2.6].

Consider the pair (W′

n, H), where His the 2-handle attached to Y×I. Note that H

is contractible, so we see that H2(W′

n)∼

=H2(W′

n, H)∼

=H2(Y, K ). By excision, we

now see that H2(W′

n) is naturally identiﬁed with H2(M, ∂ M)∼

=Z. We deﬁne ǫto be

this identiﬁcation. The assignment EK,n,λ : Spinc(W′

n)→Spinc(Y, K ) discussed above

(28) is aﬃne over ǫ, i.e., EK,n,λ(s)−EK,n,λ(s′) = ǫ(s−s′). It follows from (28) that

ǫ(P D[F]) = nP D[µ] + P D [λ]. For shorthand, we write Efor EK,n,λ.

By the conjugation invariance of Spinccobordism maps in Floer homology [OS06, The-

orem 3.6], it suﬃces to show that vξ0+P D[γ]and hξ0−P D[γ]are conjugate Spincstructures

on W′

n. Because W′

nis deﬁnite and H2(W′

n)∼

=Z, the Spinc-conjugation classes are

completely determined by c2

1. First, we will establish that vξ0=hξ0, i.e., the case of

[γ] = 0.

It follows from [OS11, Proof of Proposition 4.2] that vξ0and hξ0are characterized as the

two Spincstructures on the negative-deﬁnite cobordism W′

nextending t0which have the

largest values of c2

1. Indeed, there it is shown that every Spincstructure extending t0is

of the form vξ0+n·P D[F] and that one of vξ0,hξ0maximizes the quadratic function

DISTANCE ONE LENS SPACE FILLINGS AND BAND SURGERY ON THE TREFOIL KNOT 23

L1

L2J

−3

2

k−1

J

3k−1

3

Figure 6. Surgery on the link L=L1∪L2⊂S3is equivalent by a

slam-dunk move to surgery along the knot J⊂S3. (Left) The surgery

diagram also shows integral surgery on the knot KJin L(3,1) yielding a

manifold with |H1|= 3k−1.

c1(vξ0+n·P D[F])2. If there was an additional Spincstructure sharing the same value of

c2

1with one of vξ0or hξ0, this would imply that the ﬁrst Chern class of the maximizing

Spincstructure would be 0, forcing W′

nto be Spin. By Lemma 2.4, this implies that

|H1(Ynµ+λ(K))|is even, contradicting the choice of nmade at the beginning of this

subsection.

Of course c1(vξ0)2=c1(vξ0)2and similarly for hξ0. Because t0is self-conjugate on

Ynµ+λ(K), we deduce that either vξ0=hξ0and hξ0=vξ0or vξ0=vξ0and hξ0=hξ0.

Since hξ0=vξ0+P D[F], it must be that vξ0=hξ0, proving the desired claim for

P D[γ] = 0.

Now, ﬁx an arbitrary [γ]∈H1(Y). We see that

E(hξ0−P D[γ]) = E(vξ0−P D[γ]) + nP D [µ] + P D[λ]

=ξ0−P D[γ] + nP D[µ] + P D[λ]

=E(hξ0)−P D[γ]

=E(hξ0−ǫ−1(P D[γ]))

=E(vξ0+ǫ−1(P D[γ])),

where the ﬁrst three lines follow from (28), the fourth is the aﬃne action of H2(W′

n)

on Spinc(W′

n), and the ﬁfth is because vξ0=hξ0. Since Eis injective, we see that

hξ0−P D[γ]=vξ0+ǫ−1(P D[γ]). On the other hand, E(vξ0+ǫ−1(P D[γ])) = E(vξ0+P D[γ])

because Eis aﬃne over ǫ, and thus vξ0+ǫ−1(P D[γ]) = vξ0+P D[γ]. This establishes that

vξ0+P D[γ]and hξ0−P D[γ]are conjugate, which is what we needed to show.

Remark 4.6. It follows from the proof of Proposition 4.5 that if t+,t−∈Spinc(Ynµ+λ(K))

are such that Ξ(t±) = ξ0±P D[γ]for [γ]∈H1(Y), then t+and t−are conjugate.

In order to prove Proposition 4.1, we will need to identify self-conjugate Spincstructures

on Yλ(K) in the mapping cone formula. This will be done in Lemmas 4.7 and 4.11 below.

Before doing so, it will be useful to describe a particular example of Yλ(K) by a concrete

surgery diagram. (See Figure 6.) Let LJ=L1∪L2denote the Hopf link connect sum

with a knot J⊂S3at L1. We may consider Yas −3/2-surgery on L2, where Kis the

image of L1under the surgery. We will write this special knot in L(3,1) as KJ. In this

24 TYE LIDMAN AND ALLISON H. MOORE AND MARIEL VAZQUEZ

case, λis represented by the framing k−1 on L1, and after a slam-dunk move, we see

that the resulting manifold is S3

3k−1

3

(J). In general, to compute p/q-surgery on a knot

Jin S3using the mapping cone formula, we follow the recipe of Ozsv´ath-Szab´o. First,

deﬁne

r

q=p

q−p

q.

Then, consider surgery on the link LJwhere L2has coeﬃcient −q/r and L1has integral

surgery coeﬃcient ⌊p

q⌋.

In particular, KUwill be an important knot to understand later on, where Uis the unknot

in S3. This is the reason the d-invariants of L(3k−1,3) show up in Proposition 4.1.

Finally, we note that KUis a core of the genus one Heegaard splitting of L(3,1).

Lemma 4.7. Let ξ0be as above. Then, GYλ(K),Kλ(ξ0)is a self-conjugate Spincstructure

on Yλ(K).

Proof. By assumption, ξ0= Ξ(t0) is the relative Spincstructure induced by the unique

self-conjugate Spincstructure on the large positive surgery Ynµ+λ(K). Since the state-

ment is purely homological, it suﬃces to prove the lemma in the case of a particular

model knot, provided that this knot is homologically essential in L(3,1). Thus we con-

sider the model knot KUas described above. In our case, we are interested in the

+(k−1)-framed two-handle attachment along KU⊂ −L(3,2) illustrated in Figure 6.

The mapping cone formula in this case has been explicitly computed in [OS11] and can

be completely rephrased in terms of the knot Floer complex for the unknot in S3. More

precisely, this is the mapping cone formula for (3k−1)/3-surgery along the unknot in

S3.

Write A+

sand Vs,Hsfor the A+

s-complexes and numerical invariants Vs, Hscoming

from the mapping cone formula for integer surgeries along the unknot in S3, computed

in [OS08, Section 2.6]. The proof of [OS11, Theorem 1.1] shows that there exists an

aﬃne isomorphism g: Spinc(Y, KU)→Z, such that

(34) A+

ξ=A+

⌊g(ξ)

3⌋,

for each ξ∈Spinc(Y, KU). Furthermore, we have that Vξ=V⌊g(ξ)

3⌋and Hξ=H⌊g(ξ)

3⌋. In

this setting, the Spincstructure GYλ(K),Kλ(ξ) is, up to conjugation, the Spincstructure

on L(3k−1,3) corresponding to g(ξ) modulo 3k−1. We claim that g(ξ0) = 1, which is

suﬃcient since on L(3k−1,3), 1 corresponds to a self-conjugate Spincstructure by (2).

From [OS08, Section 2.6], we have

(35) Vs=(0 if s≥0

−sif s < 0,Hs=(sif s≥0

0 if s < 0.

In order for the Vsand Hsto be compatible with Proposition 4.5 and (29), since ξ7→

⌊g(ξ)

3⌋, we must have that g(ξ0) = 1, completing the proof.