Abstract

In a talk at the Cornell Topology Festival in 2004, W. Thurston discussed a graph which we call “The Big Dehn Surgery Graph”, $\mathcal{B}$. Here we explore this graph, particularly the link of $S^3$, and prove facts about the geometry and topology of $\mathcal{B}$. We also investigate some interesting subgraphs and pose what we believe are important questions about $\mathcal{B}$.

1. Introduction

In unpublished work, W. Thurston described a graph that had a vertex $v_M$ for each closed, orientable, 3-manifold $M$ and an edge between two distinct vertices $v_M$ and $v_{M'}$, if there exists a Dehn surgery between $M$ and $M'$. That is, there is a knot $K \subset M$ and $M'$ is obtained by non-trivial Dehn surgery along $K$ in $M$. The edges are unoriented since $M$ is also obtained from $M'$ via Dehn surgery. Roughly following W. Thurston, we will call this graph the Big Dehn Surgery Graph, denoted by $\mathcal{B}$. We will sometimes denote the vertex $v_M$ by $M$. If $M$ and $M'$ 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 $\mathcal{B}$. These follow from just some of the extensive work that has been done in the field of Dehn surgery.

Proposition 1.1.

The graph $\mathcal{B}$ has the following basic properties: (i) $\mathcal{B}$ is connected; (ii) $\mathcal{B}$ has infinite valence; and (iii) $\mathcal{B}$ has infinite diameter.

The graph $\mathcal{B}$ is connected by the beautiful work of Lickorish 21 and Wallace 35 who independently showed that all closed, orientable $3$-manifolds can be obtained by surgery along a link in $S^3$. That every vertex $v_M$ in $\mathcal{B}$ has infinite valence can be seen, amongst other ways, by constructing a hyperbolic knot $K$ in $M$ via the work of Myers in 26. Then by work of Thurston 33 all but finitely many fillings are hyperbolic, and the volumes of the filled manifolds approach the volume of the cusped manifold. The graph $\mathcal{B}$ has infinite diameter since the rank of $H_1(M, \mathbb{R})$ 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 $v_{S^3}$ to $v_{M}$ in $\mathcal{B}$ counts the minimum number of components needed for a link in $S^3$ to admit $M$ as a surgery. We will refer to the number of edges in a shortest edge path between $v_{M_1}$ and $v_{M_2}$ as the Lickorish path length and denote this function by $p_L(v_{M_1},v_{M_2})$. For example, if $P^3$ denotes the Poincaré homology sphere, then $p_L(S^3, P^3 \# P^3) =2$. See §3 for more on $p_L$. Lickorish path length appears in the literature as surgery distance (see 3, 18). This gives us a metric on $\mathcal{B}$, 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 $\lbrace v_i \rbrace$ by $\langle \lbrace v_i \rbrace \rangle$. The link of a vertex $v$ is the subgraph $lk(v) = \langle w: p_L(v,w) =1 \rangle$. If there is an automorphism of $\mathcal{B}$ taking a vertex $v$ to a vertex $w$, then the links of $v$ and $w$ are isomorphic as graphs. We study the links of vertices and a possible characterization of the link of $S^3$ in $\S$2. Associated to any knot $K$ in a manifold $M$ is a $\mathbb{K}_\infty$, the complete graph on infinitely many vertices (to distinguish between knots and complete graphs we will use $\mathbb{K}_\infty$ and $\mathbb{K}_n$ to denote complete graphs). When we want to describe a $\mathbb{K}_\infty$ subgraph of $\mathcal{B}$, we will use $M_\infty ^{K} = \langle v_{M'}: M' = M(K;r) \rangle$. See $\S$2 for notation conventions used in this paper.

Interestingly, not every $\mathbb{K}_\infty$ arises this way. We prove this in $\S$4 and make some further observations about these subgraphs. In $\S$6 we study the subgraph $\mathcal{B}_H$. The vertices of the subgraph $\mathcal{B}_H$ are closed hyperbolic 3-manifolds and there is an edge between two vertices $v_M$ and $v_N$ if there is a cusped hyperbolic 3-manifold with two fillings homeomorphic to $M$ and $N$. We also study the geometry of $\mathcal{B}$ and $\mathcal{B}_H$, showing that neither is $\delta$-hyperbolic in $\S$7. In $\S$7 we also construct flats of arbitrarily large dimension in $\mathcal{B}$. An infinite family of hyperbolic 3-manifolds with weight one fundamental group which are not obtained via surgery on a knot in $S^3$ is given in $\S$3. This shows that a characterization of the vertices in the link of $S^3$ remains open. Bounded subgraphs whose vertices correspond to other geometries are detailed in $\S$5.

2. The link of $S^3$

We now set notation which we will use for the remainder of the paper. A slope on the boundary of a 3-manifold $M$ is an isotopy class of unoriented, simple closed curves on $\partial M$. We denote the result of Dehn surgery on $M$ along a knot $K \subset M$ with filling slope $r$ by $M(K;r)$. We denote Dehn filling along a link $L=K_1 \cup K_2\dots \cup K_n \subset M$ by $M(\lbrace K_1,\dots ,K_n \rbrace ;(r_1,\dots ,r_n))$ or $M(L; (r_1,\dots ,r_n))$, with a dash denoting an unfilled component. Thus the exterior of $K$ in $M$ is denoted $M(K;-)$ and the complement is denoted by $M\setminus K$. We will say that $M(K;-)$ or $M \setminus K$ is hyperbolic if its interior admits a complete hyperbolic metric of finite volume. For knot and link exteriors in $S^3$ we will frame the boundary tori homologically, unless otherwise noted.

Here we study the links of vertices in $\mathcal{B}$, particularly the link of $S^3$. As above, the link of a vertex in $\mathcal{B}$ is the subgraph $lk(v) = \langle w: p_L(v,w) =1\rangle$. If $v$ is associated to the manifold $M$, the vertices in this subgraph correspond to distinct manifolds which can be obtained via Dehn surgery on knots in $M$. We refer to this subgraph as the link of $M$ in $\mathcal{B}$, or just the link of $M$.

The link of $S^3$ in $\mathcal{B}$ is connected. There are several proofs of this fact. Perhaps the most intuitive is to use that a crossing change on a knot in $S^3$ can be realized as a Dehn surgery along an unknotted circle; see 31. One must be careful to ensure that none of these surgeries results in $S^3$.

The proof we give here arose from conversations with Luisa Pauoluzzi, and the path shows that the link of $S^3$ has bounded diameter.

Proposition 2.1.

The link of $S^3$ in $\mathcal{B}$ is connected and of bounded diameter.

Proof.

We show any surgery on a knot in $S^3$, $S^3(K;r)$, is at most distance three in the link from a lens space. Let $CK$ denote a cable of $K$. Then there is a surgery slope $pq$ and a lens space $L(p,q)$ such that $S^3(CK;pq) = S^3(K;p/q) \# L(p,q)$; see 10. Thus $S^3(CK;pq)$ is distance one from a surgery along $K$ and distance one from a lens space.

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One might hope to distinguish the links of vertices combinatorially in $\mathcal{B}$. For example, is the link of any vertex in $\mathcal{B}$ connected? of bounded diameter? A negative answer would lead to an obstruction to automorphisms of the graph that do not fix $S^3$. 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.

Question 2.2.

Does the graph $\mathcal{B}$ admit a non-trivial automorphism?

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 $S^3$.

Problem 2.3.

Characterize the vertices in the link of $S^3$.

3. Hyperbolic examples with weight one fundamental groups

A group is weight $n$ if it can be normally generated by $n$ 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 $S^3$ 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 $S^3$ (see 1, Question 9.23). The restriction to geometric manifolds is necessary since the fundamental group of $P^3 \# P^3$ is weight one, where $P^3$ is the Poincaré homology sphere. This cannot be surgery along a knot in $S^3$ since if a reducible manifold is surgery along a non-trivial knot in $S^3$, one of the factors is a lens space 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 $S^3$ (Theorem 3.4). Our technique is a generalization to the hyperbolic setting of a method of Margaret Doig, who in 8 first came up with examples that could not be obtained via surgery on a knot in $S^3$ using the $d$-invariant. Boyer and Lines 5 exhibited a different set of small Seifert fibered spaces which are weight one but not surgery along a knot in $S^3$.

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:

Theorem 3.1 (17, Corollary 4.2).

Every one relator product of three cyclic groups is non-trivial.

This implies, for example, that $M\cong L(p_1,q_1)\#L(p_2,q_2)\#L(p_3,q_3)$, is not obtained via surgery on a knot in $S^3$, since its fundamental group is not weight one. However, when the $p_i$ are pairwise relatively prime, its homology is cyclic.

The following proposition extends this consequence of Howie’s result to hyperbolic manifolds.

Proposition 3.2.

There are hyperbolic $3$-manifolds $\{N_j\}$ with cyclic homology such that for each $j$, $\pi _1(N_j)$ is weight at least two.

Proof.

Just as above, $M\cong L(p_1,q_1)\#L(p_2,q_2)\#L(p_3,q_3)$ with all the $p_i$ pairwise relatively prime. By 26, Theorem 1.1, there exists a knot $K \subset M$ such that $K$ bounds an immersed disk in $M$ and $M-K$ is hyperbolic. Denote by $\Gamma _K=\pi _1(M(K,-))$ and $\pi _1(\partial (M(K;-)))=\langle \mu , \lambda | [\mu ,\lambda ]=1\rangle$, where $\mu$ and $\lambda$ are chosen such that $M(K;\mu )=M$, and $\lambda$ bounds an immersed disk when considered as a curve in $M$.

Let $\langle \langle g \rangle \rangle _{G}$ denote the normal closure of $g$ in $G$. If $\gamma$ is a curve in $\partial (M(K;-))$ representing the isotopy class $\mu ^r\lambda ^s$, then $\pi _1(M(K;\gamma ))=\Gamma _K/\langle \langle \mu ^r\lambda ^s \rangle \rangle _{\Gamma _K}$. Observe that $\Gamma _K/\langle \langle \mu ^r\lambda ^s, \mu \rangle \rangle _{\Gamma _K} =\Gamma _K/\langle \langle \mu \rangle \rangle _{\Gamma _K} = \pi _1(M)$ as $\lambda \in \langle \langle \mu \rangle \rangle _{\Gamma _K}$ since $\lambda$ bounds an immersed disk in $M$. Thus, there exists a surjective homomorphism $f: \pi _1(M(K;\gamma )) \rightarrow \pi _1(M)$. In particular, $\pi _1(M(K;\gamma ))$ is weight at least two.

If we let $N_j = M(K;\mu \lambda ^j)$, then $H_1(N_j,Z)$ is cyclic of order $p_1p_2p_3$ and by Thurston’s Hyperbolic Dehn Surgery Theorem 33, Theorem 5.8.2, $N_j$ is hyperbolic for sufficiently large $j$.

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Over the two papers 23, Dave Auckly exhibited hyperbolic integral homology spheres that could not be surgery along a knot in $S^3$. 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 $S^3$. As part of a larger result, she shows:

Theorem 3.3 (8, Theorem 2).

Of the infinite family of elliptic manifolds with $H_1(Y) = \mathbb{Z}/4\mathbb{Z}$ and non-cyclic fundamental group, only one (up to homeomorphism) can be realized as surgery on a knot in $S^3$, and that is $(4,1)$ surgery on the right-handed trefoil.

Although not explicitly stated in her result, for a finite group $G$, the weight of $G$ is determined by the weight $G/G'$ (see 20), and so the above elliptic manifolds have weight one fundamental groups.

Using similar techniques and the work of Greene and Watson in 13, we are able to exhibit hyperbolic manifolds that have weight one fundamental groups but are never surgery along a knot in $S^3$. As in Greene and Watson, our examples are the double branched covers of the knots $K_{p,q}$ (see Figure 1a) where $p=-10n$, $q=10n+3$, and $n\geq 1$. We denote these knots by $K_n$ and their corresponding double branched covers by $M_n$. 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 $\lbrace X_n \rbrace$ to denote the manifolds in this (possibly) pared down set.

Theorem 3.4.

There is an infinite family of hyperbolic manifolds, $\lbrace X_n \rbrace$, none of which can be realized as surgery on a knot in $S^3$. Furthermore, these manifolds have weight one fundamental groups.

In the following proof, we require two standard definitions from Heegaard-Floer homology (see 28, 8). First, a rational homology sphere $M$ is an L-space if the hat version of its Heegaard-Floer homology is as simple as possible, namely for each Spin$^c$ structure $t$ of $M$, the hat version of $\widehat{HF}(M,t)$ has a single generator and no cancellation. The d-invariant, $d(M,t)$, is an invariant assigned to each Spin$^c$ structure $t$ of $M$ which records the minimal degree of any non-torsion class of $HF^+(M,t)$ coming from $HF^\infty (M,t)$; see section 4 of 28. Crudely, the $d$-invariant can be thought of as a way of measuring how far from $S^3$ a manifold is. This mentality is motivated by the argument in the proof below.

Proof.

For this proof, we use notation from 30. As noted above, Greene and Watson 13 study the family of knots $\lbrace K_n \rbrace$ and their double branched covers $M_n$. The manifolds $M_n$ have the following properties:

Each $M_n$ is an $L$-space (13, Proposition 11).

The $d$-invariant, defined in 28 of the $M_n$, satisfies the following relation:

$$\begin{equation} d(M_n, i) = 2 \tau (M_n, i) - \lambda \tag{1} \end{equation}$$

for all $n \geq 0$ and all $i \in \operatorname {Spin}^c(M_n)$. Here $\tau (M_n,i)$ is the Turaev torsion and $\lambda = \lambda (M_n)$ is the Casson-Walker invariant. That the Casson-Walker invariants are all identical follows from the work of Mullins 25, Theorem 7.1 and that the knots are ribbon and have identical Jones polynomials 13, Propositions 8 and 11. Furthermore, by 13, Proposition 14,

$$\begin{equation} \lim _{n \rightarrow \infty } \min \lbrace \tau (M_n, i) |i \in \operatorname {Spin}^c(M_n) \rbrace = -\infty . \tag{2} \end{equation}$$

As they observe, (1) and (2) above imply:

$$\begin{equation} \lim _{n \rightarrow \infty } \min \lbrace d(M_n, i) |i \in \operatorname {Spin}^c(M_n) \rbrace = -\infty . \tag{3}{} \end{equation}$$

Let $-M_n$ denote $M_n$ with the opposite orientation. Then since $d(-M_n, i) = -d(M_n,i)$, 28, Proposition 4.2, we have:

$$\begin{equation} \lim _{n \rightarrow \infty } \max \lbrace d(-M_n, i) |i \in \operatorname {Spin}^c(M_n) \rbrace = \infty . \tag{4}{} \end{equation}$$

In particular, the $d$-invariants are unbounded with either orientation.

Since the manifolds $M_n$ (and $-M_n$) are $L$-spaces, we may apply:

Theorem 3.5 (28, Theorem 1.2).

If a knot $K \subset S^3$ admits an $L$-space surgery, then the non-zero coefficients of $\Delta _K(T)$ are alternating $+1$s and $-1$s.

For the remainder of this section, we will consider $S^3_{p/q}(K)$ with $p/q\ne 1/0$. We assume that $p/q \geq 0$ by the following argument. Let $mK$ denote the mirror of $K$. Then, $S^3_{p/q}(K) =S^3_{-p/q}(mK)$ and we replace $K$ with $mK$ if necessary to fulfill the assumption that $p/q\geq 0$. The resulting manifold will be $S^3_{p/q}(K)$ with the opposite orientation. Note that a knot which admits an $L$-space surgery cannot be amphichiral 8, Corollary 12.

If a knot surgery $S^3_{p/q}(K)$ with $p/q \geq 0$ is an $L$-space, it is shown in 30, Theorem 1.2 that the correction terms $d(S^3_{p/q}(K), i)$ may be calculated as follows, for $|i|\leq p/2$, and $c = |\lfloor i/q \rfloor |$:

$$\begin{equation} d(S^3_{p/q}(K),i) -d(S^3_{p/q}(U),i) = -2 \sum _{j=1}^\infty ja_{c+j}\tag{5}{} \end{equation}$$

where $a_i$ is defined in terms of the normalized Alexander polynomial of $K$:

$$\Delta _K(T) = a_0 + \sum _{i=1}^n a_i(T^i + T^{-1}).$$

Again, we are using notation from 30 and in particular $S^3_{p/q}(K) = S^3(K;p/q)$. We note that Greene and Watson 13 establish that $H_1(M_n) = \mathbb{Z}/25\mathbb{Z}$. By homology considerations, if any $M_n$ is $p/q$ surgery on a (standard positively framed) knot $K$ in $S^3$, then $p=25$. The $L$-space condition implies $\frac{25}{q} \geq 2g(K)-1$ by 30, Corollary 1.4. We also know that such a $K$ is fibered by 1927 and that $g(K)$ is the degree of the symmetrized Alexander polynomial of $K$ by 29, bounding the number of terms on the right-hand side of equation (5).

Now suppose that $M_n = S^3_{p/q}(K)$. Then, since $M_n$ and $-M_n$ are $L$-spaces, the Alexander polynomials of such a $K$ have bounded coefficients by Theorem 3.5. Thus, the right-hand side of equation (5) is bounded. Since there are only finitely many $L(p,q)$ with $p=25$, $d(S^3_{p/q}(U),i)$ only can take on finitely many values. Therefore, $d(S^3_{p/q}(K),i)$ is globally bounded whenever $S^3_{p/q}(K) = M_n$. However, this contradicts the limit (3 or 4), and so at most finitely many of the $M_n$ can be surgery on any knot.

Next, we establish that all but at most finitely many $\{ M_n \}$ are hyperbolic.

Indeed, the Kanenobu knots $K_n$ are all obtained by tangle filling the two boundary components of the tangle $T$ in Figure 1b, and so the manifolds $\{ M_n \}$ are obtained by Dehn filling the double cover of $T$, which we denote by $M$. A triangulation for $M$ can be obtained by inputting $T$ labeled with cone angle $\frac{\pi }{2}$ into the computer software Orb (an orbifold version of the original Snappea) 16 to obtain an orbifold structure $Q$.⁠¹ Denote by $M$, the double cover of $Q$ corresponding to the unique index 2 torsion free subgroup $\pi _1^{orb}(Q)$. This computation shows that $M$ decomposes into 8 tetrahedra. In fact, SnapPy’s identify function 6 shows $M$ is homeomorphic to ‘t12060’ in the 8 tetrahedral census. Also, using SnapPy, a set of 8 gluing equations for $M$ are encoded by the following matrix:

¹

The file and instructions on how to use it are available on the arXiv version of this paper.

✖

$$\left( \begin{array}{rrrrrrrrrrrrrrrr|r} 2 & -2 & 2 & 1 & 2 & -1 & 1 & -1 & 2 & 0 & 0 & 2 & -1 & 1 & 2 & 0 & -4 \\0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 \\-1 & 2 & 0 & -1 & 0 & -1 & 0 & 0 & -2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 2 \\0 & -2 & -1 & 1 & -1 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & -2 \\0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 2 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & -2 \\0 & 0 & 0 & -1 & 0 & -1 & -1 & 1 & 0 & 0 & -1 & 1 & -1 & 1 & -1 & 1 & 2 \\0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & 1 & -1 & 1 & -1 & -1 & 1 & 0 \end{array} \right).$$

The coding is as follows: given a row $\begin{pmatrix} a_1 & b_1 & a_2 & b_2 & \dots & a_8 & b_8 &| c \end{pmatrix}$, we produce a log equation $a_1 \log (z_1) + b_1 \log (1-z_1)+ \dots +a_8 \log (z_8) + b_8 \log (1-z_8)- c \pi \cdot i = 0$. Given such an encoding $z = (2 i, \frac{1}{5} + \frac{3i}{5}, \frac{1}{5} + \frac{3i}{5}, \frac{1}{2} + \frac{i}{2}, 1 + 2i, \frac{1}{2} + i, \frac{1}{2} + \frac{i}{2}, \frac{1}{2} + i)$ is an exact solution and therefore $M$ and ‘t12060’ admit a complete hyperbolic structure. By Thurston’s Hyperbolic Dehn Surgery Theorem 33, Theorem 5.8.2, the manifolds $M_n$ limit to $M$. Thus, there are at most finitely many non-hyperbolic $M_n$.

We have that at most finitely many of the $M_n$ are surgery on a knot and at most finitely many are non-hyperbolic. We denote the subsequence of $M_n$ that are hyperbolic and cannot be surgery along a knot by $X_n$.

Finally, we establish that $\pi _1(M_n)$ is weight one and therefore $\pi _1(X_n)$ is weight one. As noted in 13, $\S$4.2,

$$\begin{align*} \pi _1(M_n)=\langle a_1,a_2,a_3,a_4 | b_1,b_2,b_3,b_4\rangle \quad \text{with}\quad b_1&=(a_1^{-1} a_2)^{10n}a_4^{-1}a_1^2,\\ b_2&=a_2^{-1}a_3(a_2^{-1}a_1)^{10n}a_2^{-1},\\ b_3&=(a_4^{-1}a_3)^{10n+3}a_3^{-1}a_2 a_3^{-2},\quad \text{and}\\ b_4&=a_1^{-1} a_4 (a_3^{-1}a_4)^{10n+3}a_4^2. \end{align*}$$

We claim $\pi _1(M_n)/ \langle \langle a_1 \rangle \rangle _{\pi _1(M_n)}$ is trivial. First, the relations $b_1$, $b_2$ become $a_2^{10n}=a_4$ and $a_2^{10n+2}=a_3$, respectively. Also, the relations $b_3$ and $b_4$ reduce to $a_2^{10n-1}=1$ and $a_2^{10n-6}=1$ respectively. The claim follows as $gcd(10n-1,10n-6)=1$.

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Corollary 3.6.

For all $n$, $p_L(M_n,S^3)\leq 2$ and for all but at most finitely many $n$, $p_L(M_n,S^3)=2$.

Proof.

Since we can produce the unknot by switching two crossing regions of the diagram for $K_n$ as in Figure 2, the Montesinos trick shows that $M_n$ can be obtained from surgery along a two component link in $S^3$. Hence, we see the upper bound $p_L(M_n,S^3)\leq 2$ and $p_L(M_n,S^3)\geq 2$ is established for all but at most finitely many $n$ by Theorem 3.4.

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Remark 3.7.

In 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 “see” knots in the graph $\mathcal{B}$. We also want to employ the notion of the set of neighbors of a vertex in a graph. More formally, for a graph