Distance one lens space fillings and band surgery on the trefoil knot
We study lens spaces that are related by distance one Dehn fillings. More precisely, we prove that if the lens space is obtained by a surgery along a knot in the lens space that is distance one from the meridional slope, then is in . This is proved by studying the behavior of the Heegaard Floer -invariants under integral surgery along knots in . The main result yields a classification of the coherent and non-coherent band surgeries from the trefoil to 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.
The question of whether Dehn surgery along a knot in 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 [BleilerLitherland, berge, GodaTeragaito, RasmussenLens, OSlens, Baker:SurgeryI, Baker:SurgeryII, Hedden]. 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 [Moser]. In contrast, the celebrated cyclic surgery theorem [CGLS] 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 [BleilerHodgson:DehnFilling]. Fillings distance one from the meridional slope were also exploited in [baker:poincare] to construct cyclic surgeries on knots in the Poincaré homology sphere.
In this paper, we are particularly interested in Dehn surgeries along knots in which yield other lens spaces. The specific interest in is motivated by the study of local reconnection in nature, such as DNA recombination (discussed below). Note that by taking the knot to be a core of a genus one Heegaard splitting for , one may obtain for all . More generally, since the Seifert structures on are classified [GeigesLange], one could enumerate the Seifert knots in and use this along with the cyclic surgery theorem to characterize lens space fillings when the surgery slopes are of distance greater than one. This strategy does not cover the case where the surgery slopes intersect the meridian of exactly 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 in yielding . We prove:
The lens space is obtained by a distance one surgery along a knot in the lens space if and only if is one of or .
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. [Li2016, Kleckner2013aa, Shimokawa]). 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 is the most probable link formed by any random knotting process [Rybenkov93, Shaw93], and torus links appear naturally when circular DNA is copied within the cell [AdamsCozz92]. 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 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 is the branched double cover of the torus link , Theorem 1.1 yields a classification of the coherent and non-coherent band surgeries from the trefoil to for all .
The torus knot is obtained from by a non-coherent banding if and only if is , 3 or 7. The torus link is obtained from by a coherent banding if and only if is , 4 or -6.
In our convention 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 . We remark that a priori, a knot in admitting a distance one lens space surgery to does not necessarily descend to a band move on under the covering involution.
When is even, if the linking number of is , Corollary 1.2 follows as a consequence of the behavior of the signature of a link [Murasugi]. If the linking number is instead , Corollary 1.2 follows from the characterization of coherent band surgeries between torus links and certain two-bridge knots in [DIMS, 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 [IshiharaShimokawa, DIMS, Shimokawa, ISV, BuckIshihara2015, BIRS, Stolz2017]). This is due in part 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 [NiWu, Proposition 1.6] and its generalizations in Propositions 4.1 and 4.2, which describe the behavior of -invariants under certain Dehn surgeries. Recall that a -invariant or correction term is an invariant of the pair , where is an oriented rational homology sphere and is an element of . More generally, each -invariant is a rational 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, [OSAbs]. Work of Ni-Wu [NiWu] relates the -invariants of surgeries along a knot in , or more generally a null-homologous knot in an L-space, with a sequence of non-negative integer-valued invariants , due to Rasmussen (see for reference the local h-invariants in [RasmussenThesis] or [NiWu]).
With this we now outline the proof of Theorem 1.1. Suppose that is obtained by surgery along a knot in . As is explained in Lemma 2.1, the class of modulo 3 determines whether or not is homologically essential. When (mod 3), we have that is null-homologous. In this case, we take advantage of the Dehn surgery formula due to Ni-Wu mentioned above and a result of Rasmussen [RasmussenThesis, Proposition 7.6] which bounds the difference in the integers as varies. Then by comparing this to a direct computation of the correction terms for the lens spaces of current interest, we obstruct a surgery from to for or .
When , we must generalize the correction term surgery formula of Ni-Wu to a setting where is homologically essential. The technical work related to this generalization makes use of the mapping cone formula for rationally null-homologous knots [OSRational], 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 modulo 3, , and are the only nontrivial lens spaces with a distance one surgery from , completing the proof of Theorem 1.1.
In Section 2, we establish some preliminary homological information that will be used throughout and study the structures 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 -invariants of certain surgeries along a homologically essential knot in . 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.
2.1. Homological preliminaries
We begin with some basic homological preliminaries on surgery on knots in . This will give some immediate obstructions to obtaining certain lens spaces by distance one surgeries. Here we will also set some notation. All singular homology groups will be taken with -coefficients except when specified otherwise.
Let denote a rational homology sphere. First, we will use the torsion linking form on homology:
See [Friedl] for a thorough exposition on this invariant.
In the case that is a cyclic group, it is enough to specify the linking form by determining the value for a generator of and extending by bilinearity. Consequently, if two rational homology spheres and have cyclic first homology with linking forms given by and , where , then the two forms are equivalent if and only if for some integer with . We take the convention that is obtained by -surgery on the unknot, and that the linking form is given by .111We choose this convention to minimize confusion with signs. The deviation from to is irrelevant for our purposes, since this change will uniformly switch the sign of each linking form computed in this section. Because and are equivalent if and only if and are equivalent, this will not affect the results. Following these conventions, -surgery on any knot in an arbitrary integer homology sphere has linking form as well.
Let be any knot in . The first homology class of is either trivial or it generates , in which case we say that is homologically essential. When is null-homologous, then the surgered manifold is well-defined and . When is homologically essential, there is a unique such homology class up to a choice of an orientation on . The exterior of is denoted and because is homologically essential, . Recall that the rational longitude is the unique slope on which is torsion in . In our case, the rational longitude is null-homologous in . We write for a choice of dual peripheral curve to and take as a basis for . Let denote the Dehn filling of along the curve , where . It follows that and that the linking form of is equivalent to when . Indeed, is obtained by -surgery on a knot in an integer homology sphere, namely the core of the Dehn filling .
Recall that we are interested in the distance one surgeries to lens spaces of the form . Therefore, we first study when distance one surgery results in a three-manifold with cyclic first homology. We begin with an elementary homological lemma.
Fix a non-zero integer . Suppose that is obtained from by a distance one surgery on a knot and that .
If , then is homologically essential.
If is homologically essential, the slope of the meridian on is for some integer . Furthermore, there is a choice of such that .
With the meridian on given by as above, then if (respectively ), the slope inducing on is (respectively ).
If , then is null-homologous and the surgery coefficient is . Furthermore, .
(1) This follows since surgery on a null-homologous knot in has .
(2) By the discussion preceding the lemma, we have that the desired slope must be for some relatively prime to 3. In this case, has linking form equivalent to or , depending on whether or . Since 2 is not a square mod 3, we see that the linking form is not equivalent to that of , which is the linking form of . Therefore, and the meridian is for some . By instead using the peripheral curve , which is still dual to , we see that the meridian is given by .
(3) By the previous item, we may choose such that the meridional slope of on is given by . Now write the slope on yielding as . In order for this slope to be distance one from , we must have that .
(4) Note that if is null-homologous, then the other two conclusions easily hold since . Therefore, we must show that cannot be homologically essential. If was essential, then the slope on the exterior would be of the form for some integer . 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.
Fix a non-zero odd integer . Let be a knot in with a distance one surgery to having and linking form equivalent to . If , then .
Suppose that . Write with . By assumption, the linking form of is . By Lemma 2.1(3), the linking form of is also given by . Consequently, is a square modulo or equivalently, is a square modulo , as is the inverse of . Because is odd, the law of quadratic reciprocity implies that for any prime dividing , we have that . This contradicts the fact that . ∎
By an argument analogous to Lemma 2.2, one can prove that if is odd, then or .
2.2. The four-dimensional perspective
Given a distance one surgery between two three-manifolds, we let denote the associated two-handle cobordism. For details on the framed surgery diagrams and associated four-manifold invariants used below, see [GS].
Suppose that is obtained from a distance one surgery on .
If , then is positive-definite, whereas if , then is negative-definite.
The order of is even if and only if is .
(1) In either case, Lemma 2.1 implies that is obtained by integral surgery on a homologically essential knot in . First, is the boundary of a four-manifold , which is a -framed two-handle attached to along an unknot. Let denote . Since , we see that is positive-definite (respectively negative-definite) if and only if is equal to 2 (respectively 1).
Since is homologically essential, after possibly reversing the orientation of and handlesliding over the unknot, we may present by surgery on a two-component link with linking matrix
which implies that the order of is . Since the intersection form of is presented by , we see that equals 2 (respectively 1) if and only if (respectively ). The claim now follows.
(2) We will use the fact that an oriented four-manifold whose first homology has no 2-torsion is if and only if its intersection form is even. First, note that is a quotient of , so . Next, view as the boundary of the four-manifold obtained from attaching -framed two-handles to along the Hopf link. This is indeed , because is simply-connected and has even intersection form. After attaching to , we obtain a presentation for the intersection form of :
Since this matrix presents , we compute that is even if and only if is even if and only if the intersection form of is even. Since is and we are attaching along a -homology sphere, we see that the simply-connected four-manifold is if and only if is . Consequently, is even if and only if is . ∎
2.3. -invariants, lens spaces, and manifolds
As mentioned in the introduction, the main invariant that we will use is the -invariant, , of a rational homology sphere . These invariants are intrinsically related with the intersection form of any smooth, definite four-manifold bounding [OSAbs]. In some sense, the -invariants 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 [LidmanSivek, Lemma 2.2].
We assume familiarity with the Heegaard Floer package and the -invariants of rational homology spheres, referring the reader to [OSAbs] for details. We will heavily rely on the following recursive formula for the -invariants of a lens space.
Theorem 2.5 (Ozsváth-Szabó, Proposition 4.8 in [OSAbs]).
Let be relatively prime integers. Then, there exists an identification such that
for . Here, and are the reductions of and respectively.
Under the identification in Theorem 2.5, it is well-known that the self-conjugate structures on correspond to the integers among
(See for instance [DoigWehrli, Equation (3)].)
For reference, following (1), we give the values of , including , for :
It is useful to point out that -invariants change sign under orientation-reversal [OSAbs].
Using the work of this section, we are now able to heavily constrain distance one surgeries from to in the case that is even.
Suppose that there is a distance one surgery between and where is an even integer. Unless or , we have . In the case that , the two-handle cobordism from to is positive-definite and the unique structure on which extends over this cobordism corresponds to .
A technical result that we need is established first, which makes use of Lin’s -equivariant monopole Floer homology [Lin].
Let be a cobordism between L-spaces satisfying and . Then
By [LinSurgery, Theorem 5], we have that
where and are Lin’s adaptation of the Manolescu invariants for -equivariant monopole Floer homology. Conveniently, for L-spaces, [Cristofaro, Ramos, Lin, HuangRamos]. Thus, we have
On the other hand, we may reverse orientation on to obtain a negative-definite cobordism . Therefore, we have from [OSAbs, Theorem 9.6] that
Combined with (5), this completes the proof. ∎
Proof of Proposition 2.6.
For completeness, we begin by dispensing with the case of , i.e., . This is obstructed by Lemma 2.1, since no surgery on a null-homologous knot in has torsion-free homology.
Therefore, assume that . The two-handle cobordism is by Lemma 2.4. First, suppose that (and consequently ), so that we may apply Lemma 2.7. Because on restricts to self-conjugate structures and on and , (2) and (4) imply that
where must be one of or . Applying Equation (3) to , we conclude that .
If , Equation (3) applied to implies that for and for . The only solution agreeing with (6) is when . If , Equation (3) implies that is for and for , and so (6) holds whenever . Note that in this case, is positive-definite.
Now, suppose that . Therefore, we apply Lemma 2.7 instead to to see that
where again, or . In this case, there is a unique solution given by when . This completes the proof. ∎
2.4. -invariants and surgery on null-homologous knots
Throughout the rest of the section, we assume that is a null-homologous knot in a rational homology sphere . By Lemma 2.1, this will be relevant when we study surgeries to with . Recall that associated to , there exist non-negative integers for each and satisfying the following property:
Property 2.8 (Proposition 7.6 in [RasmussenThesis]).
When is null-homologous in , the set of structures is in one-to-one correspondence with . The projection to the first factor comes from considering the unique structure on which extends over the two-handle cobordism to agree with the chosen structure on . With this in mind, we may compute the -invariants of as follows. The result below was proved for knots in , but the argument immediately generalizes to the situation considered here.
Proposition 2.9 (Proposition 1.6 in [NiWu]).
Fix an integer and a self-conjugate structure on an L-space . Let be a null-homologus knot in . Then, there exists a bijective correspondence between and the structures on that extend over such that
where . Here, we assume that .
In order to apply Proposition 2.9, we must understand the identifications of the structures precisely. In particular, the correspondence between and is given in [OSInteger, Theorem 4.2]. Let be a structure on which extends and let be the restriction to . Then, we have from [OSInteger, Theorem 4.2] that is determined by
where is the surface in coming from capping off a Seifert surface for . For this to be well-defined, we must initially choose an orientation on , but the choice will not affect the end result.
Before stating the next lemma, we note that if is a -homology sphere, then , and thus there is at most one structure on . If is even, is , since the intersection form is even and . Further, admits exactly two structures, and thus exactly one extends over .
Let be a null-homologous knot in a -homology sphere . Let be the self-conjugate structure on , and let be the structure on described in Proposition 2.9 above.
Then, is self-conjugate on .
The structure does not extend to a structure over .
Note that extends over and restricts to on , since is self-conjugate. The above equation now implies that
In the context of (8), . Consequently, we must have that also restricts to on . Of course, this implies that is self-conjugate.
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 to where (mod 3)
The goal of this section is to prove:
There is no distance one surgery from to , where , except when or .
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 and the sign of the surgery on . We obtain a contradiction in each case, except when or . 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.
If , then cannot be obtained by -surgery on a null-homologous knot in .
By Proposition 2.6, cannot be even, so we may assume that is obtained by -surgery on a null-homologous knot in for odd. Consequently, there are unique self-conjugate structures on and . By (2), Proposition 2.9, and Lemma 2.10,
Using the -invariant formula (3), when , we have
which contradicts (9). ∎
If , then cannot be obtained by -surgery on a null-homologous knot in .
Suppose that is obtained by -surgery on a null-homologous knot in . By Proposition 2.6, we cannot have that is even. Indeed, in the current case, the associated two-handle cobordism is negative-definite. Therefore, is odd, and we have unique self-conjugate structures on and .
By reversing orientation, is obtained by -surgery on a null-homologous knot in . We may now repeat the arguments of Proposition 3.2 with a slight change. We obtain that
By direct computation,
Again, we obtain a contradiction. ∎
If , then cannot be obtained by -surgery on a null-homologous knot in .
As in the previous two propositions, Proposition 2.6 implies that cannot be even. Therefore, we assume that is odd. We will equivalently show that if is odd, then cannot be obtained by -surgery on a null-homologous knot in .
and so . Since by Property 2.8, we have that .
We claim that there is no structure on compatible with (7) and or . Suppose for contradiction that such a structure exists. Denote the corresponding value in by . Of course, , since is induced by on .
First, consider the case that . Applying (7) with yields
for some . This simplifies to the expression
Thus is a positive integral root of the quadratic equation
For , there are no integral roots with .
Suppose next that . Equation (7) now yields
which simplifies to the expression
Thus is an integral root of the quadratic equation
However, the only integral roots of this equation for occur when and , and we have determined that is odd. Thus, we have completed the proof. ∎
If or , then cannot be obtained by -surgery on a null-homologous knot in .
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 even. Other than this, the argument mirrors the proof of Proposition 3.4 with some extra care to identify the appropriate self-conjugate structures.
Consider the statement of Proposition 2.9 in the case that is self-conjugate on and on . We would like to determine which structure on is induced by (7). As in the previous cases, when is odd, is the unique self-conjugate structure on , which corresponds to . We now establish the same conclusion if is even. In this case, the proof of Lemma 2.10 shows that the structures and , as in Proposition 2.9, give the two self-conjugate structures on . On the other hand, (2) asserts that the numbers and also correspond to the two self-conjugate structures on . Proposition 2.6 shows that corresponds to the structure that extends over the two-handle cobordism, while Lemma 2.10(2) tells us that is the structure that does not extend. In other words, corresponds to on .
We claim that there is no structure on compatible with or in (7). Suppose for the contrary such a structure exists corresponding to . Again, .
In the case that , then (7) yields
which simplifies to the expression
As discussed in the proof of Proposition 3.4, there are no integral solutions with and .
which simplifies to the expression
As discussed in the proof of Proposition 3.4, there is a unique integral root corresponding to and . This exceptional case arises due to the distance one lens space surgery from to described in [BakerOPT, Corollary 1.4]222While this is written as in [BakerOPT], Baker was working in the unoriented category. (see also [MartelliPetronio, Table A.5]). ∎
3.2. From to where (mod 3)
The goal of this section is to prove the following.
There is no distance one surgery from to where , except when or .
As a preliminary, we use (1) to explicitly compute the -invariant formulas that will be relevant here. For ,
We will also need the following proposition about the -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.
Let be a knot in . Suppose that a distance one surgery on produces an L-space where is even. Then there exists a non-negative integer satisfying
where is the unique self-conjugate structure on .
Furthermore, if , then there exists and an integer equal to or satisfying
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 .
Proof of Proposition 3.6.
By Lemma 2.4 and Proposition 2.6, we see that must be odd or . In the latter case, we construct a coherent band surgery from the torus knot to in Figure 3, which lifts to a distance one surgery from to . Therefore, for the remainder of the proof, we assume that is odd. We also directly construct a non-coherent band surgery from to and the unknot in Figure 2, so we now focus on ruling out all even values of .
We begin by ruling out distance one surgeries to with . Since is odd, there is a unique self-conjugate structure on . By Equations (11), (12) and (14), we have . Since is at least 4, we have . We claim that there is no solution to Equation (15) with or . This will complete the proof for the case of .
It is straightforward to see that there are no non-negative integral roots of the quadratic equation for positive .
Next, we consider . In this case, (15) implies
The roots are of the form