The Big Dehn Surgery Graph and the link of
Dedicated to Bill Thurston
Abstract
In a talk at the Cornell Topology Festival in 2004, W. Thurston discussed a graph which we call тАЬThe Big Dehn Surgery GraphтАЭ, Here we explore this graph, particularly the link of . and prove facts about the geometry and topology of , We also investigate some interesting subgraphs and pose what we believe are important questions about . .
1. Introduction
In unpublished work, W. Thurston described a graph that had a vertex for each closed, orientable, 3-manifold and an edge between two distinct vertices and if there exists a Dehn surgery between , and That is, there is a knot . and is obtained by non-trivial Dehn surgery along in The edges are unoriented since . is also obtained from via Dehn surgery. Roughly following W. Thurston, we will call this graph the Big Dehn Surgery Graph, denoted by We will sometimes denote the vertex . by If . and are obtained from one another via Dehn surgery along two distinct knots, we do not make two distinct edges, although this would also make an interesting graph. We first record some basic properties of These follow from just some of the extensive work that has been done in the field of Dehn surgery. .
The graph has the following basic properties: (i) is connected; (ii) has infinite valence; and (iii) has infinite diameter.
The graph is connected by the beautiful work of Lickorish Reference 21 and Wallace Reference 35 who independently showed that all closed, orientable can be obtained by surgery along a link in -manifolds That every vertex . in has infinite valence can be seen, amongst other ways, by constructing a hyperbolic knot in via the work of Myers in Reference 26. Then by work of Thurston Reference 33 all but finitely many fillings are hyperbolic, and the volumes of the filled manifolds approach the volume of the cusped manifold. The graph has infinite diameter since the rank of can change by at most one via drilling and filling, and there are 3-manifolds with arbitrarily high rank.
The Lickorish proof explicitly constructs a link, and therefore allows us to describe a natural notion of distance. A shortest path from to in counts the minimum number of components needed for a link in to admit as a surgery. We will refer to the number of edges in a shortest edge path between and as the Lickorish path length and denote this function by For example, if . denotes the Poincar├й homology sphere, then See ┬з .3 for more on Lickorish path length appears in the literature as surgery distance (see .Reference 3, Reference 18). This gives us a metric on which we assume throughout the paper. ,
The Big Dehn Surgery Graph is very big. In order to get a handle on it, we will study some useful subgraphs. We denote the subgraph of a graph generated by the vertices by The link of a vertex . is the subgraph If there is an automorphism of . taking a vertex to a vertex then the links of , and are isomorphic as graphs. We study the links of vertices and a possible characterization of the link of in 2. Associated to any knot in a manifold is a the complete graph on infinitely many vertices (to distinguish between knots and complete graphs we will use , and to denote complete graphs). When we want to describe a subgraph of we will use , See .2 for notation conventions used in this paper.
Interestingly, not every arises this way. We prove this in 4 and make some further observations about these subgraphs. In 6 we study the subgraph The vertices of the subgraph . are closed hyperbolic 3-manifolds and there is an edge between two vertices and if there is a cusped hyperbolic 3-manifold with two fillings homeomorphic to and We also study the geometry of . and showing that neither is , in -hyperbolic7. In 7 we also construct flats of arbitrarily large dimension in An infinite family of hyperbolic 3-manifolds with weight one fundamental group which are not obtained via surgery on a knot in . is given in 3. This shows that a characterization of the vertices in the link of remains open. Bounded subgraphs whose vertices correspond to other geometries are detailed in 5.
2. The link of
We now set notation which we will use for the remainder of the paper. A slope on the boundary of a 3-manifold is an isotopy class of unoriented, simple closed curves on We denote the result of Dehn surgery on . along a knot with filling slope by We denote Dehn filling along a link . by or with a dash denoting an unfilled component. Thus the exterior of , in is denoted and the complement is denoted by We will say that . or is hyperbolic if its interior admits a complete hyperbolic metric of finite volume. For knot and link exteriors in we will frame the boundary tori homologically, unless otherwise noted.
Here we study the links of vertices in particularly the link of , As above, the link of a vertex in . is the subgraph If . is associated to the manifold the vertices in this subgraph correspond to distinct manifolds which can be obtained via Dehn surgery on knots in , We refer to this subgraph as the link of . in , or just the link of .
The link of in is connected. There are several proofs of this fact. Perhaps the most intuitive is to use that a crossing change on a knot in can be realized as a Dehn surgery along an unknotted circle; see Reference 31. One must be careful to ensure that none of these surgeries results in .
The proof we give here arose from conversations with Luisa Pauoluzzi, and the path shows that the link of has bounded diameter.
The link of in is connected and of bounded diameter.
We show any surgery on a knot in , is at most distance three in the link from a lens space. Let , denote a cable of Then there is a surgery slope . and a lens space such that see ;Reference 10. Thus is distance one from a surgery along and distance one from a lens space.
тЦаOne might hope to distinguish the links of vertices combinatorially in For example, is the link of any vertex in . connected? of bounded diameter? A negative answer would lead to an obstruction to automorphisms of the graph that do not fix More generally, an answer to the following question would lead to a better understanding of how the Dehn surgery structure of a manifold relates to the homeomorphism type. .
Does the graph admit a non-trivial automorphism?
Given our results below in ┬з3, we do not know of a conjectured answer to the following problem, which amounts to characterizing manifolds obtained via surgery on a knot in .
Characterize the vertices in the link of .
3. Hyperbolic examples with weight one fundamental groups
A group is weight if it can be normally generated by elements and no normal generating set with fewer elements exists. Recall that all knot groups are weight one and hence all manifolds obtained by surgery along a knot in have weight one fundamental groups. It is a folklore question if a manifold which admits a geometric structure and has a weight one fundamental group can always be realized as surgery along a knot in (see Reference 1, Question 9.23). The restriction to geometric manifolds is necessary since the fundamental group of is weight one, where is the Poincar├й homology sphere. This cannot be surgery along a knot in since if a reducible manifold is surgery along a non-trivial knot in one of the factors is a lens space ,Reference 11.
In Theorem 3.4 we show that there are infinitely many hyperbolic 3-manifolds whose fundamental groups are normally generated by one element but which are not in the link of (Theorem 3.4). Our technique is a generalization to the hyperbolic setting of a method of Margaret Doig, who in Reference 8 first came up with examples that could not be obtained via surgery on a knot in using the Boyer and Lines -invariant.Reference 5 exhibited a different set of small Seifert fibered spaces which are weight one but not surgery along a knot in .
Before describing the hyperbolic examples, we make a few remarks regarding the weight one condition. We have the following obstruction to surgery due to James Howie:
Every one relator product of three cyclic groups is non-trivial.
This implies, for example, that is not obtained via surgery on a knot in , since its fundamental group is not weight one. However, when the , are pairwise relatively prime, its homology is cyclic.
The following proposition extends this consequence of HowieтАЩs result to hyperbolic manifolds.
There are hyperbolic -manifolds with cyclic homology such that for each , is weight at least two.
Just as above, with all the pairwise relatively prime. By Reference 26, Theorem 1.1, there exists a knot such that bounds an immersed disk in and is hyperbolic. Denote by and where , and are chosen such that and , bounds an immersed disk when considered as a curve in .
Let denote the normal closure of in If . is a curve in representing the isotopy class then , Observe that . as since bounds an immersed disk in Thus, there exists a surjective homomorphism . In particular, . is weight at least two.
If we let then , is cyclic of order and by ThurstonтАЩs Hyperbolic Dehn Surgery Theorem Reference 33, Theorem 5.8.2, is hyperbolic for sufficiently large .
тЦаOver the two papers Reference 2Reference 3, Dave Auckly exhibited hyperbolic integral homology spheres that could not be surgery along a knot in However, it is unknown if these examples have weight one since his construction involves a 4-dimensional cobordism that preserves homology, but not necessarily group weight. .
Margaret Doig has recently exhibited examples of manifolds admitting a Thurston geometry, but which cannot be obtained by surgery along a knot in As part of a larger result, she shows: .
Of the infinite family of elliptic manifolds with and non-cyclic fundamental group, only one (up to homeomorphism) can be realized as surgery on a knot in and that is , surgery on the right-handed trefoil.
Although not explicitly stated in her result, for a finite group the weight of , is determined by the weight (see Reference 20), and so the above elliptic manifolds have weight one fundamental groups.
Using similar techniques and the work of Greene and Watson in Reference 13, we are able to exhibit hyperbolic manifolds that have weight one fundamental groups but are never surgery along a knot in As in Greene and Watson, our examples are the double branched covers of the knots . (see Figure 1a) where , and , We denote these knots by . and their corresponding double branched covers by The techniques of the proof may require us to omit finitely many of these double branched covers from the statement of the theorem. We will use . to denote the manifolds in this (possibly) pared down set.
There is an infinite family of hyperbolic manifolds, none of which can be realized as surgery on a knot in , Furthermore, these manifolds have weight one fundamental groups. .
In the following proof, we require two standard definitions from Heegaard-Floer homology (see Reference 28, Reference 8). First, a rational homology sphere is an L-space if the hat version of its Heegaard-Floer homology is as simple as possible, namely for each Spin structure of the hat version of , has a single generator and no cancellation. The d-invariant, is an invariant assigned to each ,Spin structure of which records the minimal degree of any non-torsion class of coming from see section 4 of ;Reference 28. Crudely, the can be thought of as a way of measuring how far from -invariant a manifold is. This mentality is motivated by the argument in the proof below.
For this proof, we use notation from Reference 30. As noted above, Greene and Watson Reference 13 study the family of knots and their double branched covers The manifolds . have the following properties:
Each is an ( -spaceReference 13, Proposition 11).
The defined in -invariant,Reference 28 of the satisfies the following relation: ,
for all
As they observe, (1) and (2) above imply:
Let
In particular, the
Since the manifolds
If a knot
For the remainder of this section, we will consider
If a knot surgery
where
Again, we are using notation from Reference 30 and in particular
Now suppose that
Next, we establish that all but at most finitely many
Indeed, the Kanenobu knots
The coding is as follows: given a row
We have that at most finitely many of the
Finally, we establish that
We claim
For all
Since we can produce the unknot by switching two crossing regions of the diagram for
In Reference 22, Marengon extends the techniques given here to exhibit an infinite four parameter family of double branched covers of knots given by a Kaneobu like construction.
4. Complete infinite subgraphs
Here we discuss an interesting property which may allow one to тАЬseeтАЭ knots in the graph