Proof of Proposition 1.2.

$\Gamma $ has infinite diameter since a single surgery can change the rank of the first homology at most by one.

To show that the outdegree of $\Gamma $ is infinite, let $v_{(M,\xi )}$ be a vertex of $\Gamma $ . For an even integer n, choose a Legendrian knot L in a Darboux ball in $(M,\xi )$ with Thurston–Bennequin invariant $\operatorname {\mathrm {tb}}=1$ and rotation number $\operatorname {\mathrm {rot}}=n$ . Denote the contact manifold obtained by contact $(-1)$ -surgery along L by $L(-1)$ . A calculation, as for example in [Reference Ding, Geiges and Stipsicz6, Reference Durst and Kegel8], shows that the homology of $L(-1)$ is

$$ \begin{align*} H_{1}(L(-1))=H_{1}(M)\oplus \mathbb{Z}_{\mu_{L}}, \end{align*} $$

where the $\mathbb {Z}$ -summand is generated by a meridian $\mu _{L}$ of L, and that the Poincaré dual of the Euler class $\operatorname {\mathrm {e}}(L(-1))$ is given by

$$ \begin{align*} \operatorname{\mathrm{e}}(L(-1))=\operatorname{\mathrm{e}}(M,\xi)+n\mu_{L}, \end{align*} $$

where $\operatorname {\mathrm {e}}(M,\xi )$ denotes the Poincaré dual of the Euler class of $(M,\xi )$ . Thus, we get infinitely many different contact manifolds by contact $(-1)$ -surgery from M. The same construction with contact $(+1)$ -surgery along a Legendrian knot with Thurston–Bennequin invariant $\operatorname {\mathrm {tb}} = -1$ provides the fact that the indegree is also infinite.

Proof of Theorem 1.1.

Let E be a set consisting of k-different vertices in $\Gamma $ and let $(M_{0},\xi _{0})$ and $(M_{n},\xi _{n})$ be two contact $3$ -manifolds representing vertices in $\Gamma \!\setminus \!E$ . We consider a path

$$ \begin{align*} (M_{0},\xi_{0})\rightarrow (M_{1},\xi)\rightarrow (M_{2},\xi_{2})\rightarrow\cdots\rightarrow (M_{n},\xi_{n}) \end{align*} $$

in $\Gamma $ between $(M_{0},\xi _{0})$ and $(M_{n},\xi _{n})$ . We denote by $L_{0}$ a Legendrian surgery link in $(S^{3},\xi _{\mathrm {st}})$ representing $(M_{0},\xi _{0})$ . Let $K_{i+1}$ be a Legendrian knot in $(M_{i},\xi _{i})$ such that contact $(\pm 1)$ -surgery along $K_{i}$ is contactomorphic to $(M_{i+1},\xi _{i+1})$ . By Lemma 4.7.1 in [Reference Kegel22], we can represent $K_{1}$ as a Legendrian knot in the complement of $L_{0}$ and thus $L_{1}:=L\cup K_{1}$ is a surgery link in $(S^{3},\xi _{\mathrm {st}})$ for $(M_{1},\xi _{1})$ . By induction, we get surgery links $L_{i}$ in $(S^{3},\xi _{\mathrm {st}})$ representing $(M_{i},\xi _{i})$ describing the above path in $\Gamma $ . We will construct a path from $(M_{0},\xi _{0})$ to $(M_{n},\xi _{n})$ in $\Gamma \!\setminus \!E$ .

Let $p\geq 2$ be an integer such that no contact structure on $M_{i}\#L(p,1)$ is in E for all $i=0,\ldots ,n$ . (Such a p exists because E is finite.) We define a new sequence of Legendrian surgery links as follows. We set $L^{\prime }_{0}=L_{0}$ and $L^{\prime }_{i+1}=L_{i}\sqcup U_{p}(-1)$ , where $\sqcup $ denotes the disjoint union of knots and $U_{p}(-1)$ denotes the contact $(-1)$ -surgery along a Legendrian unknot with $\operatorname {\mathrm {tb}}=1-p$ . It follows that $L^{\prime }_{i+1}$ represents a contact structure on $M_{i}\#L(p,1)$ . Finally, we set $L^{\prime }_{n+2}$ to be the disjoint union of $L_{n}$ with $U_{p}(-1)$ together with a $(+1)$ -framed meridian $\mu _{U_{p}}$ of $U_{p}$ . Since $U_{p}(-1)\cup \mu _{U_{p}}(+1)$ yields $(S^{3},\xi _{\mathrm {st}})$ by [Reference Avdek1], we see that $L^{\prime }_{n+2}$ represents $(M_{n},\xi _{n})$ . Thus, a path in $\Gamma \!\setminus \!E$ between $(M_{0},\xi _{0})$ and $(M_{n},\xi _{n})$ is constructed. The same argument shows that $\Gamma $ is also k-edge connected.

Proof of Theorem 1.3.

Since $\Gamma $ is connected, k-connected and k-edge-connected, for any natural number k, and of infinite degree, the main results from [Reference Erdős, Grünwald and Vázsonyi13, Reference Erdős, Grünwald and Vázsonyi14, Reference Nash-Williams26] immediately imply that $\Gamma $ contains Eulerian and Hamiltonian paths.

Since contact $(-1)$ -surgery preserves tightness by [Reference Wand29], a Hamiltonian diwalk has to start at an overtwisted contact manifold and has to run first through all overtwisted contact manifolds before reaching a tight contact manifold. However, since there exists infinitely many overtwisted contact manifolds, this is not possible.

Proof of Corollary 2.2.

The main result of [Reference Nash-Williams25] says that an oriented graph admits an Eulerian dipath based at a vertex v if and only if the graph is countable, connected, $1$ -coherent, v-solenoidal and unbiased. We refer to [Reference Nash-Williams25] for the definitions. Since the contact surgery graph $\Gamma $ is connected and countable but admits no Eulerian dipath, it is enough to check that $\Gamma $ is $1$ -coherent and v-solenoidal. First, $1$ -coherency is a condition of the underlying unoriented graph and since $\Gamma $ admits an undirected Euler path, $\Gamma $ is $1$ -coherent. Second, it follows that $\Gamma $ is v-solenoidal for any vertex since the indegree and outdegree are infinite for any vertex.

Proof of Theorem 1.4.

We first show that $\Gamma _{\textrm {OT}}$ is strongly connected. Let $(M_{1},\xi _{1})$ and $(M_{2},\xi _{2})$ be two overtwisted contact manifolds. By [Reference Etnyre and Honda15], there exist a directed path p from $(M_{1},\xi _{1})$ to $(M_{2},\xi _{2})$ in $\Gamma $ . We argue that any vertex in p corresponds to an overtwisted contact manifold. Let us assume the contrary. Since $(M_{2},\xi _{2})$ is overtwisted, there exists an overtwisted contact manifold $(M_{\textrm {OT}},\xi _{\textrm {OT}})$ that can be obtained by contact $(-1)$ -surgery from a tight contact manifold $(M_{\textrm {tight}},\xi _{\textrm {tight}})$ contradicting Wand’s result which says that contact $(-1)$ -surgery preserves tightness [Reference Wand29].

Next, we consider $\Gamma _{*}$ with $*=\textrm {tight}$ , $\textrm {Stein}$ , $\textrm {strong}$ , $\textrm {weak}$ or $\textrm {c}\neq 0$ . To show that $\Gamma _{*}$ is connected, we show that there exists an undirected path in $\Gamma _{*}$ from $(S^{3},\xi _{\mathrm {st}})$ to any other contact manifold $(M,\xi )$ with property $*$ . We consider the contact $(+1)$ -surgery along the Legendrian unknot with Thurston–Bennequin invariant $\operatorname {\mathrm {tb}}=-2$ and rotation number $\operatorname {\mathrm {rot}}=1$ in $(S^{3},\xi _{\mathrm {st}})$ . It is well known that the resulting contact manifold is the overtwisted contact structure $\xi _{1}$ on $S^{3}$ with normalised $\mathrm{d}_{3}$ -invariant equal to $1$ [Reference Ding, Geiges and Stipsicz6]. Then by [Reference Etnyre and Honda15], there exists a directed path in $\Gamma $ of contact $(-1)$ -surgeries from $(S^{3},\xi _{1})$ to $(M,\xi )$ . However, this path runs through at least one overtwisted contact manifold and hence it is not in $\Gamma _{*}$ . In total, we get a surgery link L in $(S^{3},\xi _{\mathrm {st}})$ describing $(M,\xi )$ with only a single contact surgery coefficient $(+1)$ . Now we change the order of the surgeries and first perform all contact $(-1)$ -surgeries and at the end, we perform the single contact $(+1)$ -surgery. Since contact $(-1)$ -surgery is known to preserve any of the properties $*$ by [Reference Eliashberg10, Reference Ozsváth and Szabó27, Reference Wand29, Reference Weinstein30], we get a path in $\Gamma _{*}$ from $(S^{3},\xi _{\mathrm {st}})$ to $(M,\xi )$ .

The contact surgery subgraphs $\Gamma _{\textrm {tight}}$ , $\Gamma _{\textrm {strong}}$ , $\Gamma _{\textrm {weak}}$ and $\Gamma _{\textrm {c}\neq 0}$ are not strongly connected since there exists in each of these graphs a contact manifold $(M,\xi )$ which is not Stein fillable [Reference Eliashberg12, Reference Ghiggini18] and then there cannot be a directed path from $(S^{3},\xi _{\mathrm {st}})$ to $(M,\xi )$ .

Finally, we show that $\Gamma _{\textrm {Stein}}$ is not strongly connected. We show that there exists no directed path of contact $(-1)$ -surgeries from $(S^{3},\xi _{\mathrm {st}})$ to $(S^{1}\times S^{2},\xi _{\mathrm {st}})$ . Let us assume the contrary. Then there must be a simply connected Stein cobordism from $(S^{3},\xi _{\mathrm {st}})$ to ${(S^{1}\times S^{2},\xi _{\mathrm {st}})}$ . We glue this Stein cobordism to the standard $4$ -ball filling of $(S^{3},\xi _{\mathrm {st}})$ to get a simply connected Stein filling $(W,\omega _{\mathrm {st}})$ of $(S^{1}\times S^{2},\xi _{\mathrm {st}})$ . However, by [Reference Eliashberg10], any Stein filling of $(S^{1}\times S^{2},\xi _{\mathrm {st}})$ is diffeomorphic to $S^{1}\times D^{3}$ which is not simply connected.

Question 3.2. Is $\Gamma _{\mathrm{OT}}$ the only nontrivial strongly connected component of $\Gamma $ ?

Proof of Lemma 3.4.

Let L be a Legendrian knot in $(M,\xi ^{M}_{\textrm {stab}})$ such that $L(-1)=(N,\xi ^{N})$ . Let $L^{*}$ be the dual surgery knot of L in $(N,\xi ^{N})$ . By the cancellation lemma, $L^{*}(+1)$ is again contactomorphic to $(M,\xi ^{M}_{\textrm {stab}})$ . Now we choose a loose Legendrian realisation $L^{\prime }$ of $L^{*}$ such that if we stabilise $L^{\prime }$ , once positive and once negative, we get a Legendrian knot which is formally isotopic to $L^{*}$ . (This is possible since we can destabilise any loose Legendrian knot.)

We claim that $L^{\prime }(-1)$ yields $(M,\xi ^{M})$ . Since $L^{\prime }$ is topologically isotopic to $L^{*}$ and its contact framing and the contact framing of $L^{*}$ differ by 2, the contact $(-1)$ -surgery along $L^{\prime }$ yields topologically the same manifold as the contact $(+1)$ -surgery along $L^{*}$ , which is in fact M. Note that $L^{\prime }(-1)$ is overtwisted since $L^{\prime }$ is a loose knot. Then a straightforward computation in a local model, as in [Reference Ding, Geiges and Stipsicz7], shows that the homotopical invariants of the contact structures agree. Thus by Eliashberg’s classification of contact structures [Reference Eliashberg9], it follows that the contact structures are contactomorphic.

Proof of Theorem 1.5.

First we remark that by following the proof of Theorem 1.1, we conclude that each of the subgraphs $\Gamma _{\textrm {OT}}$ , $\Gamma _{\textrm {tight}}$ , $\Gamma _{\textrm {Stein}}$ , $\Gamma _{\textrm {strong}}$ , $\Gamma _{\textrm {weak}}$ and $\Gamma _{\textrm {c}\neq 0}$ is k-connected and k-edge-connected for any natural number k. Thus, each of these subgraphs admits Eulerian and Hamiltonian paths and also Hamiltonian walks. For the directed paths, we conclude as in the proof of Theorem 1.3 that $\Gamma _{*}$ does not admit a Hamiltonian diwalk, a Hamiltonian dipath and a Eulerian dipath for $*=\textrm {tight}$ , $\textrm {Stein}$ , $\textrm {strong}$ , $\textrm {weak}$ or $\textrm {c}\neq 0$ .

However, we show that an Eulerian dipath exists in $\Gamma _{\textrm {OT}}$ . As in the proof of Corollary 2.2, it follows that $\Gamma _{\textrm {OT}}$ is $1$ -coherent and v-solenoidal. By the main result of [Reference Nash-Williams25], it is therefore enough to show that $\Gamma _{\textrm {OT}}$ is unbiased. Assume there exists a subset X of the vertices of $\Gamma _{\textrm {OT}}$ such that there are infinitely many edges oriented from X to its complement $X^{c}$ , but only finitely many edges oriented from $X^{c}$ to X. As a first step, we show that there exists, for any $i\in \mathbb {N}$ , a Legendrian knot $K_{i}$ in an overtwisted contact manifolds $(M_{i},\xi _{i})$ in X such that all $M_{i}$ are pairwise nondiffeomorphic and the contact $(-1)$ -surgery $K_{i}(-1)$ along $K_{i}$ lies in $X^{c}$ . (This step will not use the overtwistedness.)

By assumption, we know that there are infinitely many edges pointing out of X. For any $i\in \mathbb {N}$ , we choose some Legendrian knot $K^{\prime }_{i}$ in an overtwisted contact manifold $(M,\xi )$ in X such that $K^{\prime }_{i}(-1)$ lies in $X^{c}$ . We show that we can obtain infinitely many of the $K^{\prime }_{i}(-1)$ by a contact $(-1)$ -surgery from infinitely many different manifolds in X. For that, we first observe that we can get an overtwisted contact structure $\xi _{\textrm {OT}}$ on $M\# L(i+1,1)$ by a single contact $(+1)$ -surgery along a Legendrian unknot $U_{i}$ in a Darboux ball in $M\!\setminus \!K_{i}$ . Since only finitely many edges point into X, we conclude that an infinite subset of the $(M\# L(i+1,1),\xi _{\textrm {OT}})$ are elements of X again. Performing a contact $(-1)$ -surgery along $K_{i}$ in $(M\# L(i+1,1),\xi _{\textrm {OT}})$ yields $K_{i}(-1)\#L(i+1,1)$ with an overtwisted contact structure. Then cancelling the contact $(+1)$ -surgery along $U_{i}$ by a contact $(-1)$ -surgery along a push-off of $U_{i}$ yields $K_{i}(-1)$ . Thus we can conclude that there exists an infinite family of Legendrian knots $K_{i}$ in different overtwisted contact manifolds $(M_{i},\xi _{i})$ in X such that $K_{i}(-1)$ are elements of $X^{c}$ .

In the next step, which only works for overtwisted contact manifolds, we show that we can get back from the $K_{i}(-1)$ to X by contact $(-1)$ -surgeries. We write $(M_{i},\xi _{i})$ as $(M,\xi _{i}^{\prime })\#(S^{3},\xi _{1})$ . By Lemma 3.4, there exists a contact $(-1)$ -surgery along a Legendrian knot in $K_{i}(-1)$ yielding $(M_{i},\xi _{i}^{\prime })$ , and from $(M_{i},\xi _{i}^{\prime })$ , we can get back to $(M_{i},\xi _{i})$ by another contact $(-1)$ -surgery. Thus we have constructed an infinite family of edges pointing from $X^{c}$ into X contradicting the assumption and finishing the proof of Theorem 1.5.

Proof. (1) Let p be a minimal (undirected) path between $(M_{1},\xi _{1})$ and $(M_{2},\xi _{2})$ in $\Gamma $ . If every vertex in p corresponds to an overtwisted contact manifold, the distances in $\Gamma $ and $\Gamma _{\textrm {OT}}$ agree. In general, however, the path p might run through tight contact manifolds. To prevent this, we choose an ordered surgery link L in $(M_{1},\xi _{1})$ such that contact $(\pm 1)$ -surgery in that given order along L corresponds to the path p. We denote by $\xi _{0}$ the overtwisted contact structure on $S^{3}$ with vanishing normalised $\operatorname {\mathrm {d}}_{3}$ -invariant. A $2$ -component surgery diagram of $(S^{3},\xi _{0})$ is shown in Figure 1 (centre). We add this surgery diagram in a Darboux ball in the exterior of L to the surgery link, where we first perform the surgeries along the two new Legendrian knots and afterwards the surgeries corresponding to p. Then any contact manifold in the path p is replaced by a connected sum with $(S^{3},\xi _{0})$ . By Eliashberg’s classification of overtwisted contact structures [Reference Eliashberg9], it follows that we have constructed a new path $p^{\prime }$ from $(M_{1},\xi _{1})$ to $(M_{2},\xi _{2})$ in $\Gamma _{\textrm {OT}}$ of length $2$ larger than the length of p.

Figure 1 Contact surgery diagrams of contact structures on $S^{3}$ : left, $(S^{3},\xi _{-1})$ ; centre, $(S^{3},\xi _{0})$ ; right, $(S^{3},\xi _{1})$ [Reference Ding, Geiges and Stipsicz6, Reference Etnyre, Kegel and Onaran16].

(2) By Wand’s theorem [Reference Wand29], any directed path in $\Gamma $ between two overtwisted contact manifolds cannot run through a tight contact manifold (see the proof of Theorem 1.4).

Proof. Recall that, by the work of Eliashberg, $\xi _{\mathrm {st}}$ is the unique tight contact structure on $S^{3}$ [Reference Eliashberg11] and the overtwisted contact structures are in one-to-one correspondence to the homotopy classes of tangential $2$ -plane fields [Reference Eliashberg9], which are on homology spheres in bijection with the integers via their normalised $\mathrm{d}_{3}$ -invariants [Reference Gompf19]. We denote the unique overtwisted contact structure on $S^{3}$ with normalised $\mathrm{d}_{3}$ -invariant equal to n by $\xi _{n}$ .

Contact surgery diagrams for all contact structures on $S^{3}$ were explicitly described in [Reference Ding, Geiges and Stipsicz6]. A contact surgery diagram of $\xi _{1}$ is given by the contact $(+1)$ -surgery along the Legendrian unknot with Thurston–Bennequin invariant $\mathrm{tb}=-2$ and rotation number $\mathrm{rot}=1$ , shown on the right of Figure 1. The contact $(\pm 1)$ -surgery diagram along the $2$ -component link, shown on the left of Figure 1, represents $\xi _{-1}$ . The disjoint union of two surgery diagrams describes a connected sum of the underlying contact manifolds. Since the $\mathrm{d}_{3}$ -invariant behaves additively under the connected sum, we get contact surgery diagrams of all contact structures on $S^{3}$ by taking appropriate disjoint unions of the contact surgery diagrams of $\xi _{1}$ and $\xi _{-1}$ .

It follows that we can get $(S^{3},\xi _{1})$ by a single contact $(+1)$ -surgery from $(S^{3},\xi _{\mathrm {st}})$ and that there exists a contact $(+1)$ -surgery on $(S^{3},\xi _{k})$ yielding $(S^{3},\xi _{k+1})$ and thus conversely a contact $(-1)$ -surgery from $(S^{3},\xi _{k})$ to $(S^{3},\xi _{k-1})$ .

Proof of Theorem 1.6.

First we discuss the case that M is a homology sphere. Then we can get any overtwisted contact structure on M from a fixed contact structure on M by connected summing with overtwisted contact structures of $S^{3}$ [Reference Ding, Geiges and Stipsicz6]. It follows that any two overtwisted contact structures on M can be connected in $\Gamma _{M}$ . Now let $\xi _{\textrm {tight}}$ be some tight contact structure on M. Then there exists a single contact $(+1)$ -surgery along a Legendrian unknot in $(M,\xi _{\textrm {tight}})$ yielding $(M,\xi _{\textrm {tight}})\# (S^{3},\xi _{1})$ which is overtwisted, and thus it follows that $\Gamma _{M}$ is connected.

If the underlying manifold is not a homology sphere, it gets slightly more complicated since the classification of tangential $2$ -plane fields is more involved. As a further invariant, we have the $\mathit {spin}^{c}$ structure of a contact structure. However, it is known that for a given contact structure with $\mathit {spin}^{c}$ structure $\mathfrak s$ , we can get any other overtwisted contact structure with the same $\mathit {spin}^{c}$ structure $\mathfrak s$ by connected summing with the overtwisted contact structures on $S^{3}$ [Reference Ding, Geiges and Stipsicz6]. Thus, we can apply the same argument as in the homology sphere case to deduce that $\Gamma _{(M,\mathfrak s)}$ is connected.

It remains to show that there is no edge connecting two different $\mathit {spin}^{c}$ structures on M. For that, we use Gompf’s $\Gamma $ -invariant which classifies $\mathit {spin}^{c}$ structures [Reference Gompf19]. Let $\mathfrak s$ be a $\mathit {spin}^{c}$ structure on M and $\xi $ be a contact structure inducing $\mathfrak s$ . Let $L_{0}$ in $(M,\xi )$ be a Legendrian knot such that contact $(+1)$ - or contact $(-1)$ -surgery along $L_{0}$ yields another contact structure $\xi ^{\prime }$ on M. We want to show that $\xi ^{\prime }$ induces the same $\mathit {spin}^{c}$ structure $\mathfrak s$ . For that, we choose a spin structure $\mathfrak t$ , describe $(M,\xi )$ by a contact surgery diagram along a Legendrian link $L=L_{1}\cup \cdots \cup L_{n}$ in $(S^{3},\xi _{\mathrm {st}})$ and present $L_{0}$ as a knot in the exterior of L. To show that $\xi $ and $\xi ^{\prime }$ induce the same $\mathit {spin}^{c}$ structures, it is enough to compute that Gompf’s $\Gamma $ invariants of $\xi $ and $\xi ^{\prime }$ with respect to $\mathfrak t$ agree. We present $\mathfrak t$ via a characteristic sublink $(L_{j})_{j\in J}$ , $J\subset \{1,\ldots ,n\}$ , of L. Since the homologies of M and $L_{0}(\pm 1)$ agree, we deduce that $\mu _{L_{0}}$ is nullhomologous in $L_{0}(\pm 1)$ and that J is also a characteristic sublink of the surgery diagram $L_{0}\cup L$ of $L_{0}(\pm 1)$ . Then we can use the formula for computing Gompf’s $\Gamma $ -invariant from [Reference Etnyre, Kegel and Onaran16] to compute

$$ \begin{align*} \Gamma(\xi^{\prime},\mathfrak t)-\Gamma(\xi,\mathfrak t)=\frac{1}{2} \bigg(\sum_{i=0}^{n} r_{i}\mu_{i}+\sum_{j\in J}(Q^{\prime}\mu)_{j}\bigg)-\frac{1}{2} \bigg(\sum_{i=1}^{n} r_{i}\mu_{i}+\sum_{j\in J}(Q\mu)_{j}\bigg)=0, \end{align*} $$

where $r_{i}$ denotes the rotation number and $\mu _{i}$ the meridian of $L_{i}$ , $Q^{\prime }$ and Q denote the linking matrices of $L_{0}\cup L$ and L, and all noncancelling terms are multiples of $\mu _{0}=0$ .

Proof. We relate the two contact surgery diagrams in Figure 2 by first performing two handle slides [Reference Avdek1, Reference Casals, Etnyre and Kegel2, Reference Ding and Geiges5] followed by a lantern destabilisation [Reference Lisca and Stipsicz24] (see also [Reference Etnyre, Kegel and Onaran16]). We first slide the red knot L over $U_{+}$ , as in Figure 3(i). Then, we slide $U_{+}$ over the red one as in Figure 3(ii), where the handle slides are indicated via the arrows. After isotoping Figure 3(iii) to get Figure 3(iv), we apply the lantern destabilisation to get Figure 3(v).

Figure 3 Proof of Lemma 4.2 via two handle slides and a lantern destabilisation.

Proof of Theorem 1.7.

Let $(N,\xi _{N})$ be a contact manifold in the link of $(M,\xi )$ . We construct a path of length one or two in $\operatorname {\mathrm {lk}}(M,\xi )$ from $(N,\xi _{N})$ to $(M,\xi )\#(S^{3},\xi _{1})$ . (We recall that $(M,\xi )\#(S^{3},\xi _{1})$ can be obtained from $(M,\xi )$ by a single contact $(+1)$ -surgery along a Legendrian unknot with $\operatorname {\mathrm {tb}}=-2$ in a standard Darboux ball in $(M,\xi )$ .)

First, we consider the case in which we can obtain $(N,\xi _{N})$ by a contact $(+1)$ -surgery along a Legendrian knot K in $(M,\xi )$ . In [Reference Avdek1] it is shown that performing a contact $(+1)$ -surgery K followed by a contact $(+1)$ -surgery along a Legendrian meridian U of K with $\operatorname {\mathrm {tb}}=-1$ corresponds to a negative stabilisation of $(M,\xi )$ and thus yields $(M,\xi )\#(S^{3},\xi _{1})$ .

The case in which $(N,\xi _{N})$ arises as contact $(-1)$ -surgery from $(M,\xi )$ can be reduced to the first case by applying Lemma 4.2 once.