In a talk at the Cornell Topology Festival in 2004, W. Thurston discussed a graph which we call “The Big Dehn Surgery Graph”, 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 ..
In unpublished work, W. Thurston described a graph that had a vertex for each closed, orientable, 3-manifold and an edge between two distinct vertices and if there exists a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 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 Poincaré homology sphere, then See § .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.
One 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. .
Given our results below in §3, we do not know of a conjectured answer to the following problem, which amounts to characterizing manifolds obtained via surgery on a knot in .
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 Poincaré 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:
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 Howie’s result to hyperbolic manifolds.
Over 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: .
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.
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.
4. Complete infinite subgraphs
Here we discuss an interesting property which may allow one to “see” knots in the graph We also want to employ the notion of the set of neighbors of a vertex in a graph. More formally, for a graph .