2. Surgery diagrams and Rolfsen twists

The two main ingredients in the proof of Theorem2 are the so-called Rolfsen twist [Reference Rolfsen15] and Garside’s theory of braids [Reference Garside7]. To make this note self-contained, we will explain Rolfsen twists in this section and a direct consequence of Garside’s work needed in our proof in Section 3. Readers who are already familiar with these concepts can skip ahead to Section 4 for the proof of Theorem2.

First, let us briefly review the basics of Dehn surgery. For more details, the reader is referred, for example, to [Reference Rolfsen14]. Let $L = L_1 \cup \ldots \cup L_m$ be a link in  $S^3$ of $m$ components. The boundary of $S^3 \setminus \nu L$ is orientation-preservingly diffeomorphic to a disjoint union of $m$ tori $\bigcup _{j=1}^m S^1 \times S^1$ . The diffeomorphism type of the 3-manifold obtained by Dehn surgery along $L$ , that is, by gluing $m$ solid tori $\bigcup _{j=1}^m S^1 \times D^2$ to $S^3 \setminus \nu L$ , is determined by the image of $\bigcup _{j=1}^m \{1\} \times \partial D^2$ under an orientation-preserving diffeomorphism

\begin{equation*}\bigcup _{j=1}^m \partial (S^1 \times D^2 )\longrightarrow S^3 \setminus \nu L.\end{equation*}

Given a meridian $\mu _j$ and a Seifert longitude $\lambda _j$ of the link component $L_j$ , the image curves in $\partial (\overline {\nu L_j})$ can be expressed as $p_j \mu _j + q_j \lambda _j$ for coprime integers $p_j, q_j$ for every $j \in \{1, \ldots , m\}$ . The quotient

\begin{equation*}p_j/q_j \in \mathbb{Q} \cup \{\infty \}\end{equation*}

is called the surgery coefficient of the link component $L_j$ . By the above discussion, the diffeomorphism type of the 3-manifold obtained by Dehn surgery on the link $L$ is determined by a diagram of $L$ where each component $L_j$ of $L$ is labelled by $p_j/q_j$ . We call $L$ together with the labellings $p_j/q_j$ , $i\in \{1,\ldots ,m\}$ , a surgery diagram.

Let us remark that a link component $L_j$ with surgery coefficient $\infty =1/0$ means that we identify the meridian of the attached solid torus with the meridian of $L_j$ . The corresponding surgery simply glues in a solid torus in exactly the same way as the removed tubular neighbourhood of $L_j$ . Therefore, a component with surgery coefficient $\infty =1/0$ can be removed from a surgery diagram without changing the diffeomorphism type of the surgered manifold.

Let us now turn to Rolfsen twists. To that end, let $L= L_1 \cup \ldots \cup L_m$ be a surgery diagram with an unknotted component, say $L_1$ , with surgery coefficient $p_1/q_1$ . For any integer $N\in \mathbb{Z}$ , an $N$ -fold Rolfsen twist along $L_1$ creates out of $L$ a new surgery diagram $L'$ as follows:

• • We cut the exterior of $L_1$ , which is a solid torus, along a Seifert disk of $L_1$ and re-glue it with an $N$ -fold twist (where $N=1$ corresponds to one right-handed full twist) to get a new link $L'= L'_1 \cup \ldots \cup L'_m$ (the components of which are in natural bijection with the components of $L$ ),

• • the surgery coefficient of $L'_1$ is $\frac {1}{N+\frac {q_1}{p_1}}$ and the surgery coefficients of the other components $L'_j$ are given by $\frac {p_j}{q_j}+N\operatorname {lk}(L_j,L_1)^2$ , where $\operatorname {lk}$ denotes the linking number for a choice of orientation of $L$ .

We refer to Figure 1 for a visualisation.

Proof. The process of cutting the exterior of $L_1$ along a Seifert disk of $L_1$ and re-glueing it with an $N$ -fold twist describes a diffeomorphism of the exterior of $L$ to the exterior of $L'$ . It is straightforward to check that this diffeomorphism acts on the boundary tori as

\begin{align*} \mu _1 &\longmapsto \mu '_1+N\lambda '_1\\[5pt] \lambda _1 &\longmapsto \lambda '_1\\[5pt] \mu _j &\longmapsto \mu '_j\\[5pt] \lambda _j &\longmapsto N\operatorname {lk}(L_j,L_1)^2\mu '_j+\lambda '_j \end{align*}

and thus the surgery coefficients transform as claimed.

3. Garside theory

The second key ingredient in our proof of Theorem2 is Garside’s theory of braids [Reference Garside7]. A braid on $k$ strands is a collection of $k$ disjoint, properly embedded arcs in the cylinder $[0,1]\times D^2$ that run monotonically in the $[0,1]$ -direction from $k$ fixed points in $\{0\} \times D^2$ to $\{1\} \times D^2$ . Braids up to ambient isotopy of $[0,1]\times D^2$ fixing $\{0,1\} \times D^2$ pointwise form a group called the braid group $B_k$ . The group operation is defined by stacking braids. Closing a braid by identifying $\{0\} \times D^2$ and $\{1\} \times D^2$ , connecting the endpoints of the arcs and identifying the resulting solid torus with the tubular neighbourhood of an unknot in $S^3$ produces an oriented link in $S^3$ ; see Figure 2.Footnote ² By Alexander’s theorem, every link arises in this way [Reference Alexander3]. We refer to [Reference Birman and Brendle6] for more details on braids and their closures.

Figure 1.

An $N$ -fold Rolfsen twist.

Figure 2.

A braid (left) and its closure (right).

The braid group $B_k$ is generated by the Artin generators [Reference Artin4] denoted $\sigma _1, \ldots , \sigma _{n-1}$ , where $\sigma _i$ corresponds to a positive half-twist exchanging two neighbouring strands with indices $i$ and $i+1$ ; see Figure 2. A braid on $k$ strands is positive if it can be represented by a word in $B_k$ that is a product of positive powers of the Artin generators. The closure of a positive braid is called a braid positive link. We denote by

\begin{equation*}T_k=(\sigma _1 \ldots \sigma _{k-1})^{k}\end{equation*}

a full twist in $B_k$ ; see, for example, Figure 3. The following lemma is a direct corollary from Garside’s work [Reference Garside7].Footnote ³ For completeness and to keep our arguments purely geometric, we present a proof.

Figure 3.

The first row shows that a full twist $T_k$ commutes with any other braid $\beta$ on $k$ strands. The second row visualises that a full twist $T_k$ followed by a negative Artin generator $\sigma _i^{-1}$ yields a positive braid.

Proof. We start with two simple observations. First, the full twist $T_k$ in $B_k$ commutes with any other element $\beta$ of $B_k$ , that is, $T_k\beta =\beta T_k$ . See Figure 3 for an illustration. Second, for any Artin generator $\sigma _i$ , the braid $T_k\sigma _i^{-1}$ is positive; see Figure 3 again.

If $\beta$ is a positive braid, we can set $N = 0$ . Otherwise, we write $\beta$ as a word in the Artin generators and distinguish two cases. In the first case, the first letter of $\beta$ is a positive generator, say $\beta =\sigma _i\beta '$ . Using the above observations, we can then write $T_k^N\beta =\sigma _iT_k^N\beta '$ . In the second case, if the first letter of $\beta$ is a negative Artin generator, say $\beta =\sigma _i^{-1}\beta '$ , then we write

\begin{equation*} T_k^N\beta = T_k^{N-1} T_k\sigma _i^{-1}\beta ' = T_k^{N-1} p \beta ' = p T_k^{N-1} \beta ', \end{equation*}

where $p$ is a positive braid. The claim follows using an induction on the length of the braid word $\beta$ .

4. Braid positive surgery diagrams

In this section, we finally prove Theorem2.

Proof of Theorem 2. Given a closed, oriented, connected 3-manifold $M$ , let $L$ be a link whose number of components realises $s(M)$ . We can represent $L$ as the closure of a braid $\beta$ on, say, $k$ strands. Figure 4(a) represents a braid diagram $\beta$ of $L$ . Inserting a new component with surgery coefficient $\infty =1/0$ and performing an $N$ -fold Rolfsen twist, we obtain the diagram shown in Figure 4(b). Figure 4(c) shows the result of an isotopy of the new link diagram performed on the new component. The link $L^\prime$ in Figure 4(c) arises as the closure of a braid $\beta ^\prime$ . Note that $\beta ^\prime$ is a positive braid whenever the braid $\alpha := T_k^{N} \beta$ on $k$ strands is a positive braid. The theorem thus follows from Lemma4, choosing $N$ sufficiently large.

Figure 4.

Transforming a braid into a positive braid by a single surgery.