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# Distance one lens space fillings and band surgery on the trefoil knot

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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 $n$ is in $\{-6, \pm 1, \pm 2, 3, 4, 7\}$. 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.
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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)
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 n0 (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
pand m
p, where p > 0, then the two forms are equivalent if and
only if nma2(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/3Z/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 is the unique slope on M which is torsion in H1(M). In our
case, the rational longitude is null-homologous in M. We write mfor a choice of dual
peripheral curve to and 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 +q) is equivalent to q/p
when p6= 0. Indeed, M(pm +q) 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 Yis 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)
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 k2are 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+as above, then if n= 3k+1 (respectively
n= 3k1), the slope inducing Yon 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/3Z/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+q) has linking form
equivalent to 1/3 or 2/3, depending on whether q1 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, q1 (mod 3) and the meridian is 3m+ (3r+ 1)
for some r. By instead using the peripheral curve m=m+rℓ, which is still dual to ,
we see that the meridian is given by 3m+.
(iii) By the previous item, we may choose msuch that the meridional slope of Kon M
is given by 3m+. Now write the slope on Myielding Yas (3k±1)m+q. In order
for this slope to be distance one from 3m+, we must have that q=k.
(iv) Note that if Kis null-homologous, then the other two conclusions easily hold since
H1(Y) = Z/3Z/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 Yhaving H1(Y) = Z/n and linking form equivalent to sgn(n)
|n|.
If n1 (mod 3), then n > 0.
Proof. Suppose that n < 0. Write n= 1 3jwith j > 0. By assumption, the linking
form of Yis 1/(3j1). By Lemma 2.1(iii), the linking form of Yis also given by
j/(3j1). Consequently, jis a square modulo 3j1 or equivalently, 3 is a square
modulo 3j1, as 3 is the inverse of j. Because nis odd, the law of quadratic
reciprocity implies that for any prime pdividing 3j1, we have that p1 (mod 3).
This contradicts the fact that 3j1≡ −1 (mod 3).
Remark 2.3. By an argument analogous to Lemma 2.2, one can prove that if n= 3k1
is odd, then n1or 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 Yis obtained from a distance one surgery on L(3,1).
(i) If |H1(Y)|= 3k1, 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 Yis 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 NW. 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 Yby surgery on a two-component
Q=3 1
1c,
which implies that the order of H1(Y) is |3c1|. 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 c0). 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 WX:
QWX=
2 1 a
12b
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 WXis even. Since Xis Spin and we
are attaching Walong a Z/2-homology sphere, we see that the simply-connected four-
manifold WXis 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 pq)2
4pq d(L(q, r), j )
for 0i < 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+q1
2and q1
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+(2in)2
4n
(3)
d(L(n, 1),0) = n1
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 ton 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 n0
(mod 3). Recall that associated to K, there exist non-negative integers Vt,i for each
iZand tSpinc(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) : YYp(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 itibetween 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,pi}. Here, we assume that 0i < 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+p2i(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), p2i
(mod 2p). Since we consider 0 ip, 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 k2, 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 k2, we have
d(L(3k, 1),0) d(L(3,1),0) d(L(k, 1),0) = 1 + 3k
41 + 3
41 + k
4>0,
Proposition 3.3. If k1, 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
21 + k
4>0.
Proposition 3.4. If k2, 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 k3 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
41
2+1
4+k
4,
and so Nt,0=k1
2. Since Vt,0Vt,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 k1
2or k3
2. Since Nt,1=
max{Vt,1, Vt,k1}=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=
k1
2or k3
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=k1
2. Applying (7) with i= 1 yields
k1 = 1
4+(2j3k)2
12k1
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) = j23kj (3k3).
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=k3
2. Equation (7) now yields
k3 = 1
4+(2j3k)2
12k1
2+1
4+(2 k)2
4k
which simpliﬁes to the expression
k(3j3) = j2+ 3.
Thus jis an integral root of the quadratic equation
f(j) = j23kj + (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,0Vt,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 k2
2. Since Nt,1=
max{Vt,1, Vt,k1}=Vt,1, we also have Nt,1=k
2or k2
2.
We claim that there is no Spincstructure on L(3k, 1) compatible with Nt,1=k
2or k2
2
in (7). Suppose for the contrary such a Spincstructure exists corresponding to jZ/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+(2j3k)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 k1
and 0 < j < 3k.
In the case that Nt,1=k2
2, then Equations (3) and (7) now yield
k2 = 1
4+(2j3k)2
12k+1
2+1
4+(2 k)2
4k,
which simpliﬁes to the expression
k(3j3) = 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 k0,
d(L(3k+ 1,1), j) = 1
4+(1 + 2j3k)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) = 811k+ 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 Ywhere |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 N02, then there exists tSpinc(Y)and an integer N1equal to N0
or N01satisfying
(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 k4.
We begin by ruling out distance one surgeries to +L(3k+ 1,1) with k4. 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 N02.
We claim that there is no solution to Equation (15) with N1=k
2or k2
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=k2
2. In this case, (15) implies
j2(1 + 3k)j+ (3k+ 4) = 0.
The roots are of the form
j=1
2(1 + 3k±p9k26k15).
It is straightforward to verify that for k4, the lesser root is always strictly between 1
and 2, while the greater root is strictly between 3k1 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),
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|=
3k1>0, except when n=±2.
As before, we state the d-invariant formulas that will be relevant for proving this theorem
ﬁrst.
d(L(3k1,1), i) = 1
4+(2i3k+ 1)2
4(3k1)
(17)
d(L(3k1,3),1) = k2
4
(18)
d(L(3k1,3),4) = 3k219k+ 18
4(3k1) .(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 Ywhere |H1(Y)|= 3k1>0. Then, there exists a non-negative
integer N0and a self-conjugate Spincstructure ton Ysuch that
(20) d(Y,t) = d(L(3k1,3),1) 2N0.
In the case that kis odd, if t6=˜
tfor some self-conjugate ˜
t, then d(Y,˜
t) = 1
4.
Furthermore, if N02, then there exists another Spincstructure ton Yand an integer
N1equal to N0or N01satisfying
(21) d(Y,t) = d(L(3k1,3),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=±(3k1) with k2.
The proof will now be handled in two cases, based on the sign of n. First, we suppose
that +L(3k1,1), with k2, 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 3k1 is odd. Using (18), we
compute
d(L(3k1,1),0) = 3k2
4>k2
4=d(L(3k1,3),1).
Now, we suppose there is a distance one surgery from L(3,1) to L(3k1,1) with
k2. 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 k5 for the rest of the proof.
Proposition 3.9 and the fact that d(L(3k1,1),0) 6=1
4imply that
d(L(3k1,1),0) = d(L(3k1,3),1) 2N0
for some non-negative integer N0. We compute from (17) and (18) that
N0=k1
2.
Since we are in the case of k5, we may apply (21). Combined with (19), this yields
1
4(2j3k+ 1)2
4(3k1) =3k219k+ 18
4(3k1) 2N1,
for some 0 < j < 3k1. Equivalently,
N1=5 + j+j27k3jk + 3k2
2(3k1) .
Here N1=k1
2or k3
2.
In the case of k1
2, we are looking for integral roots of the quadratic equation
f(j) = j2+j(1 3k) + (4 3k).
For k5, there are no roots between 0 and 3k1. For the case of k3
looking for integral roots of the quadratic
f(j) = j2+j(1 3k) + (3k+ 2).
There are no integral roots in this case for k5. 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 Yis an L-space obtained from a
distance one surgery on a knot in Y, where |H1(Y)|= 3k1with k1. Then there
exists a non-negative integer N0and a self-conjugate Spincstructure ton Ysatisfying
(22) d(Y,t) = d(L(3k1,3),1) 2N0.
Furthermore, if N02, then there exists an integer N1satisfying N0N1N01
and
(23) d(Y,t+P D[µ]) = d(L(3k1,3),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 Yis an L-space obtained from a
distance one surgery on a knot in Y, where |H1(Y)|= 3k+ 1 with k0. Then there
exists a non-negative integer N0and a self-conjugate Spincstructure ton Ysatisfying
(24) d(Y,t) + d(L(3k+ 1,3),1) = 2N0.
Furthermore, if N02, then there exists an integer N1satisfying N0N1N01
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 Ybeing 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 KYwith 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 Ydenote 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) + iκ,
where κH2(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 Kor Kλfor the core of
surgery.
Associated to ξSpinc(Y, K ) is the ZZ-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 ξSpinc(Y, K), we deﬁne the complexes A+
ξ=Cξ{max{i, j} 0}and B+
ξ=
Cξ{i0}. 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+
ξB+
ξ, h+
ξ:A+
ξB+
ξ+P D[λ].
Rather than deﬁning these maps explicitly, we explain how these can be identiﬁed with
certain cobordism maps as follows. Fix n0 and consider the three-manifold Y+λ(K)
and the induced cobordism from Y+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 Y+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(Y+λ(K)) Spinc(Y, K ) satisfying
commutative squares:
(27) CF +(Y+λ(K),t)
fW
n,v
//A+
ξ
v+
ξ
C F +(Y+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+
homogeneous non-zero map between the towers, which is necessarily multiplication by
UNfor some integer N0. 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 ξSpinc(Y, K ),
(29) VξVξ+P D[µ]Vξ1.
We are now ready to deﬁne the mapping cone formula. Deﬁne the map
(30) Φ : M
ξ
A+
ξM
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 ξSpinc(Y, K). Then there exists a quasi-
(31) X+
ξC 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
Xξd
C F (Yλ(K), GYλ(K),Kλ(ξ)).
DISTANCE ONE LENS SPACE FILLINGS AND BAND SURGERY ON THE TREFOIL KNOT 21
...
''
A+
ξ5P D[m]
v+
ξ5P D[m]
h+
ξ5P D[m]
''
A+
ξ
v+
ξ
h+
ξ
''
A+
ξ+5P D[m]
v+
ξ+5P D[m]
''
...
. . . B+
ξ5P 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 Ywith |H1(Y)|= 5 corresponding to the
Spincstructure GY,K(ξ).
A+
ξN·P D[λ]
''
A+
ξ(N1)·P D[λ]
%%
...
A+
ξ
##
A+
ξ+P D[λ]
""
...
##
A+
ξ+N·P D[λ]
B+
ξ(N1)·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)|= 3k1 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] = (3k1)[m] by Lemma 2.1(iii). Therefore, for ﬁxed ξSpinc(Y , K), we see
that the mapping cone X+
ξconsists of the Aξand Bξwhere ξξ= (3k1)j·P D[m] for
some jZ. 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+
ξj·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 3k1. 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
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 ξSpinc(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
ξSpinc(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)|= 3k1 for some
k > 0.
Fix n0 throughout. By ﬁxing the appropriate parity of n, we can compute from
Lemma 2.1 that “large positive surgery”, i.e. Y+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 ξSpinc(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]=Hξ0P D [γ].
This is the analogue of the more familiar formula Vs=Hsfor 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(ss). 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 hξ0P 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
k1
J
3k1
3
Figure 6. Surgery on the link L=L1L2S3is equivalent by a
slam-dunk move to surgery along the knot JS3. (Left) The surgery
diagram also shows integral surgery on the knot KJin L(3,1) yielding a
manifold with |H1|= 3k1.
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(Y+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
Y+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(hξ0P D[γ]) = E(vξ0P D[γ]) + nP D [µ] + P D[λ]
=ξ0P D[γ] + nP D[µ] + P D[λ]
=E(hξ0)P D[γ]
=E(hξ0ǫ1(P D[γ]))
=E(vξ0+ǫ1(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
hξ0P D[γ]=vξ0+ǫ1(P D[γ]). On the other hand, E(vξ0+ǫ1(P D[γ])) = E(vξ0+P D[γ])
because Eis aﬃne over ǫ, and thus vξ0+ǫ1(P D[γ]) = vξ0+P D[γ]. This establishes that
vξ0+P D[γ]and hξ0P 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+,tSpinc(Y+λ(K))
are such that Ξ(t±) = ξ0±P D[γ]for [γ]H1(Y), then t+and tare 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=L1L2denote the Hopf link connect sum
with a knot JS3at 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 k1 on L1, and after a slam-dunk move, we see
that the resulting manifold is S3
3k1
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
qp
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(3k1,3) 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 Y+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
+(k1)-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 (3k1)/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 ξSpinc(Y, KU). Furthermore, we have that Vξ=Vg(ξ)
3and Hξ=Hg(ξ)
3. In
this setting, the Spincstructure GYλ(K),Kλ(ξ) is, up to conjugation, the Spincstructure
on L(3k1,3) corresponding to g(ξ) modulo 3k1. We claim that g(ξ0) = 1, which is
suﬃcient since on L(3k1,3), 1 corresponds to a self-conjugate Spincstructure by (2).
From [OS08, Section 2.6], we have
(35) Vs=(0 if s0
sif s < 0,Hs=(sif s0
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.