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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)
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