2.1 Graph manifolds

We recall graph manifolds, which are defined and classified (with minor exceptions) by Waldhausen [Reference Waldhausen38]. In particular, we state several facts for graph manifolds with boundary. (In [Reference Waldhausen38], the same facts are proved for graph manifolds without boundary except for several cases.)

Let M be a $3$ -manifold, and let $\mathbb T:=T_1\cup \dots \cup T_n\subset M^\circ $ ( $n\geq 0$ ) be a disjoint union of tori. Let $U(\mathbb T)$ be a regular neighborhood of $\mathbb T$ in M. We call $\mathbb T$ a graph structure for M if every connected component of $M\setminus U(\mathbb T)^\circ $ is homeomorphic to an $S^1$ -bundle over $2$ -manifolds. A $3$ -manifold with a graph structure is called a graph manifold.

A graph structure $\mathbb T$ of a $3$ -manifold M is said to be reduced if it does not satisfy any of the ten conditions in [Reference Waldhausen38, Definition 6.2]. We omit to state the ten conditions, but that $M\setminus U(\mathbb T)^\circ $ has no component homeomorphic to $I\times S^1\times S^1$ . A $3$ -manifold with a fixed reduced graph structure is called a reduced graph manifold. For reduced graph manifolds with boundary, Waldhausen proved the following theorem.

For a reduced graph manifold M, Waldhausen defined a weighted graph $\Gamma _{\mathrm {W}}(M)$ , which is called W-graph in [Reference Neumann28], and proved Theorem 2.2 below. The W-graph $\Gamma _{\mathrm {W}}(M)$ consists of the following data:

• • there is a bijection from $\mathcal {V}(\Gamma _{\mathrm {W}}(M))$ to the set of connected components of $M\setminus U(\mathbb T)^\circ $ , and let $M_v$ be the component corresponding to the vertex v for each $v\in \mathcal {V}(\Gamma _{\mathrm {W}}(M))$ ;

• • if $M_v\not \cong S^1\times { \mathbb {D}^2}$ for $v\in \mathcal {V}(\Gamma _{\mathrm {W}}(M))$ , $M_v$ has the unique $S^1$ -bundle structure $p_v\colon M_v\to B_v$ up to isotopy by Theorem A.6, and v is assigned a triple of integers $(g_v,r_v,s_v),$ where $g_v$ is the genus of $B_v$ , $r_v$ is the number of boundary components of M contained in $M_v$ , and $s_v$ is the Euler number of $p_v\colon M_v\to B_v$ if $r_v=0$ , $s_v$ vanishes if $r_v>0$ ;

• • each $v\in \mathcal {V}(\Gamma _{\mathrm {W}}(M))$ with $M_v\cong S^1\times {\mathbb {D}^2}$ is left unweighted;

• • there is a bijection from $\mathcal {Y}(\Gamma _{\mathrm {W}}(M))$ to the set of components of $\partial U(\mathbb T)$ , write $y\mapsto T_y\subset \partial U(\mathbb T)$ for each $y\in \mathcal {Y}(\Gamma _{\mathrm {W}}(M))$ , satisfying

  • – $T_y\subset M_{t(y)}$ for each $y\in \mathcal {Y}(\Gamma _{\mathrm {W}}(M))$ ;

  • – for each $T\subset \mathbb T$ , there is $y\in \mathcal {Y}(\Gamma _{\mathrm {W}}(M))$ such that $\partial U(T)=T_y\cup T_{\overline {y}}$

  (note that M can be regarded as the manifold given by gluing $M_{t(y)}$ and $M_{t(\overline {y})}$ via certain homeomorphisms $T_y\to T_{\overline {y}}$ for $y\in \mathcal {Y}(\Gamma _{\mathrm {W}}(M))$ since $U(T)\cong I\times S^1\times S^1$ for each component $T\subset \mathbb T$ );

• • each $y\in \mathcal {Y}(\Gamma _{\mathrm {W}}(M))$ is assigned a pair $(\alpha _y,\beta _y)$ of integers satisfying

  $$ \begin{align*} &\gcd(\alpha_y,\beta_y)=1, \qquad 0\leq\beta_y<\alpha_y, \qquad \alpha_{\overline{y}}=\alpha_{y}, \qquad \beta_{\overline{y}}\beta_y\equiv 1 \pmod{\alpha_y} \end{align*} $$
  and $\beta _y>0$ if y is adjacent to an unweighted vertex, which is uniquely determined by the pasting of $M_{o(y)}$ and $M_{t(y)}$ at $T_y$ ;

• • each edge $y\in \mathcal {Y}(\Gamma _{\mathrm {W}}(M))$ is assigned a sign $\varepsilon _y\in \{\pm 1\}$ , which depends on orientations of fibers of $M_{o(y)}$ and $M_{t(y)}$ . Let $\Gamma _{\mathrm {W}}^\ast (M)$ be the full subgraph of $\Gamma _{\mathrm {W}}(M)$ given by removing $v\in \mathcal {V}(\Gamma _{\mathrm {W}}(M))$ with $g_v<0$ . Then $\Gamma _{\mathrm {W}}(M)$ is assigned the map $\varepsilon _M\colon H_1(\Gamma _{\mathrm {W}}^\ast (M))\to \mathbb Z/2\mathbb Z$ defined by $\varepsilon _M(C)$ equal to the number modulo $2$ of $y\in \mathcal {Y}(C)$ with $\varepsilon _y=-1$ for any cycle C of $\Gamma _{\mathrm {W}}^\ast (M)$ , which is uniquely determined.

Moreover, the equivalence of W-graphs is also defined in [Reference Waldhausen38]. Under our notation of graphs, for reduced graph manifolds $M_i$ with boundary, W-graphs $\Gamma _{\mathrm {W}}(M_1)$ and $\Gamma _{\mathrm {W}}(M_2)$ are equivalent if they satisfy one of the following conditions:

• • there is an isomorphism $\varphi \colon \Gamma _{\mathrm {W}}(M_1)\to \Gamma _{\mathrm {W}}(M_2)$ of graphs preserving weights $(g_v,r_v,s_v)$ of $v\in \mathcal {V}(\Gamma _{\mathrm {W}}(M_1))$ and $(\alpha _y,\beta _y)$ of $y\in \mathcal {Y}(\Gamma _{\mathrm {W}}(M_1))$ , and $\varphi $ is compatible with $\varepsilon _{M_i}\colon H_1(\Gamma _{\mathrm {W}}^\ast (M_i))\to \mathbb Z/2\mathbb Z$ , that is, $\varepsilon _{M_1}=\varepsilon _{M_2}\circ \varphi _\ast $ ;

• • the pair of $\Gamma _{\mathrm {W}}(M_1)$ and $\Gamma _{\mathrm {W}}(M_2)$ is that of the following graphs:

  (2.1)

  

We call the isomorphism $\varphi \colon \Gamma _{\mathrm {W}}(M_1)\to \Gamma _{\mathrm {W}}(M_2)$ above an equivalence map of W-graphs. The proof of [Reference Waldhausen38, Theorem 9.4] shows the following theorem.

2.2 Plumbing graphs

Neumann [Reference Neumann28] defined plumbed 3-manifolds corresponding to plumbing graphs based on Waldhausen’s work [Reference Waldhausen38]. In the appendix of [Reference Neumann28], he also defined decorated plumbing graphs to describe boundary framed graph manifolds, which are graph manifolds with fixed trivialization of each boundary component. In this note, we associate a decorated plumbing graph with the graph manifold constructed from the induced plumbing graph as mentioned in [Reference Neumann28, p. 339], which is called the plumbed manifold. We recall the (connected) decorated plumbing graph and such a graph manifold.

A decorated plumbing graph $\Gamma _{\mathrm {dpg}}=(\Gamma , \boldsymbol {g}, \boldsymbol {e}, \mathrm {sgn}; \overline {\mathcal {A}},\mathcal {A})$ is a finite connected graph $\Gamma $ together with a partition $\mathcal {V}(\Gamma )=\overline {\mathcal {A}}(\Gamma )\sqcup \mathcal {A}(\Gamma )$ such that $\boldsymbol {d}(v)\leq 1$ for each $v\in \mathcal {A}(\Gamma )$ , and maps

$$ \begin{align*} \boldsymbol{g} &\colon \overline{\mathcal{A}}(\Gamma) \longrightarrow \mathbb Z, & \boldsymbol{e} &\colon \overline{\mathcal{A}}(\Gamma) \longrightarrow \mathbb Z, & \mathrm{sgn} &\colon \mathcal{Y}(\Gamma) \longrightarrow \{\pm1\} \end{align*} $$

satisfying $\mathrm {sgn}(y)=\mathrm {sgn}(\overline {y})$ for any $y\in \mathcal {Y}(\Gamma )$ . A vertex in $\mathcal {A}(\Gamma )$ is called a boundary vertex, and drawn in the graph as an arrow head . An edge $y\in \mathcal {Y}(\Gamma )$ is called a $(+)$ -edge if $\mathrm {sgn}(y)=+1$ , and a $(-)$ -edge otherwise. In the case where $\mathcal {A}(\Gamma )=\emptyset $ , $\Gamma _{\mathrm {dpg}}$ is a plumbing graph, and we denote it by $\Gamma _{\mathrm {pg}}=(\Gamma ,\boldsymbol {g}, \boldsymbol {e}, \mathrm {sgn})$ .

Let $\Gamma _{\mathrm {dpg}}=(\Gamma ,\boldsymbol {g},\boldsymbol {e}, \mathrm {sgn}; \overline {\mathcal {A}}, \mathcal {A})$ be a decorated plumbing graph with $\overline {\mathcal {A}}(\Gamma )\ne \emptyset $ . The plumbed manifold $\mathscr {P}(\Gamma _{\mathrm {dpg}})$ for $\Gamma _{\mathrm {dpg}}$ is the graph manifold constructed as follows:

1. (I) For $v\in \overline {\mathcal {A}}(\Gamma )$ , let $B_v$ be a compact $2$ -manifold of genus $\boldsymbol {g}(v)$ with $\boldsymbol {d}(v)$ boundary components $\partial B_{y}$ , labeled by edges $y\in \mathcal {Y}(\Gamma )$ with $t(y)=v$ ; and let $p_v\colon M_v\to B_v$ be an $S^1$ -bundle, where, for each $y\in \mathcal {Y}(\Gamma )$ with $t(y)=v$ , the boundary component $T_{y}:=p_{v}^{-1}(\partial B_y)$ of $M_{v}$ has a fixed trivialization $ \tau _y\colon S^1\times S^1\to T_y $ such that the Euler number of $p_v$ is $\boldsymbol {e}(v)$ .

2. (II) The plumbed manifold $\mathscr {P}(\Gamma _{\mathrm {dpg}})$ is constructed by gluing $M_{t(y)}$ and $M_{t(\overline {y})}$ at $T_y\subset M_{t(y)}$ and $T_{\overline {y}}\subset M_{t(\overline {y})}$ by the homeomorphism

   (2.2) $$ \begin{align} T_{\overline{y}}\xrightarrow{\ \tau_{\overline{y}\ }^{-1}} S^1\times S^1 \xrightarrow{\mathrm{sgn}(y) J} S^1\times S^1\xrightarrow{\ \tau_y\ } T_y \end{align} $$
   for each $y\in \mathcal {Y}$ with $o(y),t(y)\in \overline {\mathcal {A}}(\Gamma )$ , where $J:=(\begin {smallmatrix} 0&1\\1&0 \end {smallmatrix})$ .

For $v\in \overline {\mathcal {A}}(\Gamma )$ , we call $p_v\colon M_v\to B_v$ (or $M_v$ ) the $S^1$ -bundle of v, $B_v$ the base 2-manifold of v, and a fiber of $p_v$ a fiber of v. An edge $y\in \mathcal {Y}(\Gamma )$ with $o(y)\in \mathcal {A}(\Gamma )$ corresponds to a boundary component $T_y$ of $\mathscr {P}(\Gamma _{\mathrm {dpg}})$ . Since the gluing maps reverse orientation, $\mathscr {P}(\Gamma _{\mathrm {dpg}})$ inherits compatible orientations from all pieces $M_v$ . The number of boundary components of $\mathscr {P}(\Gamma _{\mathrm {dpg}})$ is equal to $\#\mathcal {A}(\Gamma )$ . By abuse of notation, let $T_y$ denote the resulting torus in $\mathscr {P}(\Gamma _{\mathrm {dpg}})$ by gluing $T_y\subset M_{t(y)}$ and $T_{\overline {y}}\subset M_{t(\overline {y})}$ , hence $T_y=T_{\overline {y}}$ in $\mathscr {P}(\Gamma _{\mathrm {dpg}})$ .

In this note, we consider $\mathscr {P}(\Gamma _{\mathrm {dpg}})$ as a closed submanifold of a graph manifold M. An edge $y\in \mathcal {Y}(\Gamma )$ with $t(y)\in \mathcal {A}(\Gamma )$ corresponds to the boundary component $T_y$ of $M\setminus \mathscr {P}(\Gamma _{\mathrm {dpg}})^\circ $ which is identified with $T_{\overline {y}}\subset M_{o(t)}\subset \mathscr {P}(\Gamma )$ in M, and $t(y)\in \mathcal {A}(\Gamma )$ corresponds to an $S^1$ -bundle part $M_{t(y)}$ of $M\setminus \mathscr {P}(\Gamma )^\circ $ such that $T_y\subset M_{t(y)}$ .

Notation. We follow [Reference Neumann28, Appendix] to describe a decorated plumbing graph. Namely, a vertex $v\in \mathcal {V}(\Gamma )$ is written as follows:

We omit the weight of $\{y,\overline {y}\}$ if $\mathrm {sgn}(y)=+1$ , and we write $-$ if $\mathrm {sgn}(y)=-1$ . If $\boldsymbol {g}(v)=0$ , we may omit it. For example, the left decorated plumbing graph is “shorthand” for the right plumbing graph in the following graphs:

A chain of length k in $\Gamma _{\mathrm {dpg}}$ is any part $\mathfrak {c}$ of $\Gamma _{\mathrm {dpg}}$ of the following form with arbitrary edge signs (after relabeling vertices):

say $v_i$ the vertex on the above parts with $\boldsymbol {e}(v_i)=e_i$ . The chain $\mathfrak {c}$ above is said to be incident to a boundary vertex if either $v_1$ or $v_k$ is adjacent to a boundary vertex. An edge $y\in \mathcal {Y}(\Gamma )$ is regarded as a chain of length $0$ . A chain is maximal if it can be included in no larger chain.

In [Reference Neumann28, Proposition 2.1], nine operations (R0)–(R8) and their inverses to plumbing graphs are given, which do not change the diffeomorphism type of their plumbed $3$ -manifolds (hence also homeomorphism type). In [Reference Neumann28, Appendix], (R0)–(R7) are modified to F-calculus for applying decorated plumbing graphs ((R8) is removed). Moreover, the F-normal forms of decorated plumbing graphs are defined, which any decorated plumbing graph can be reduced to F-normal form using only F-calculus and their inverses.

Remark 2.5 In this article, we use the following five of F-calculus to obtain F-normal forms:

where $g\#(-1):=-2g-1$ if $g\geq 0$ , and $g\#(-1):=g-1$ otherwise. In [Reference Neumann28], the operations ( $\mathrm {R1}_0^+$ ), ( ${\mathrm {R1}}^+$ ), ( ${\mathrm {R1}}^-$ ), and ( ${\mathrm {R1}}^{++}$ ) are called blowing-downs and their reverse are called blowing-ups, and (R2) is called an ${\mathbb {R}}{\mathbb {R}}{\mathbb {P}}^2$ -absorption and its reverse is called an ${\mathbb {R}}{\mathbb {P}}^2$ -extrusion. Note that the blowing-down ( $\mathrm {R1}^{++}$ ) is different from the blowing-down of the dual graph for an algebraic curve on a complex surface.

Definition 2.7 Let $\Gamma _{\mathrm {dpg}}=(\Gamma ,\boldsymbol {g},\boldsymbol {e};\overline {\mathcal {A}}, \mathcal {A})$ be a decorated plumbing graph with $\mathrm {sgn}(y)=+1$ for each $y\in \mathcal {Y}(\Gamma )$ . The intersection form $S(\Gamma _{\mathrm {dpg}})$ of $\Gamma _{\mathrm {dpg}}$ can be defined as in [Reference Neumann28, Section 8]. Namely, if $\overline {\mathcal {A}}(\Gamma )=\{v_1,\dots ,v_r\}$ , then $S(\Gamma _{\mathrm {dpg}}):=(a_{ij})_{i,j=1,\dots ,r}$ with

$$ \begin{align*} a_{ij}:=\left\{\begin{array}{@{}ll} \boldsymbol{e}(v_i) + \#\{y\in\mathcal{Y}(\Gamma)\mid o(y)=t(y)=v_i\} & (i=j), \\[0.3em] \#\{ y\in\mathcal{Y}(\Gamma) \mid o(y)=v_i, t(y)=v_j \} & (i\ne j). \end{array}\right. \end{align*} $$

In Appendix B, we recall properties of plumbing graphs which are used in this note.

2.3 Modified plumbing graphs

Hironaka [Reference Hironaka20] defined modified plumbing graphs to describe normal surface–curve pairs $(X,\mathcal C)$ which are pairs of a complex normal surface X and an algebraic curve $\mathcal C\subset X$ . We will use modified plumbing graph to define a G-combinatorics for a plane curve. In this section, we recall the definition of the modified plumbing graph without boundary vertices.

Let $\mathrm {Fin}(S^1\times S^1)$ be the set of finite unbranched coverings from $S^1\times S^1$ to itself. A modified plumbing graph $\Gamma _{\mathrm {pg}}^{\mathrm {m}}=(\Gamma , \boldsymbol {g}, \boldsymbol {e}, \boldsymbol {m})$ is a plumbing graph $\Gamma _{\mathrm {pg}}=(\Gamma ,\boldsymbol {g},\boldsymbol {e})$ satisfying $\mathrm {sgn}(y)=+1$ for each $y\in \mathcal {Y}(\Gamma )$ together with a map $\boldsymbol {m}\colon \mathcal {Y}(\Gamma ) \to \mathrm {Fin}(S^1\times S^1)$ such that

1. (i) by choosing certain bases of tori $S^1\times S^1$ , the unbranched coverings $\boldsymbol {m}(y)\colon S^1\times S^1\to S^1\times S^1$ are represented by matrices in $M_2(\mathbb Z)$ of the following form:

   (2.3) $$ \begin{align} \boldsymbol{m}(y)=\begin{pmatrix} c(y) & 0 \\ b(y) & a(y) \end{pmatrix}\colon S^1\times S^1\longrightarrow S^1\times S^1, \end{align} $$
   where $0\leq b(y)<a(y)$ and $c(y)>0$ ;

2. (ii) the induced maps $\boldsymbol {m}(y)_\ast $ and $(J\,\boldsymbol {m}(\overline {y}))_\ast $ from $\pi _1(S^1\times S^1)$ to $\pi _1(S^1\times S^1)$ have the same image in $\pi _1(S^1\times S^1)\cong \mathbb Z\oplus \mathbb Z$ .

Given a modified plumbing graph $\Gamma _{\mathrm {pg}}^{\mathrm {m}}=(\Gamma ,\boldsymbol {g},\boldsymbol {e},\boldsymbol {m})$ , an associated graph manifold $\mathscr {P}(\Gamma _{\mathrm {pg}}^{\mathrm {m}})$ is defined as follows: For each $v\in \mathcal {V}(\Gamma )$ , let $B_v$ be a compact $2$ -manifold of genus $\boldsymbol {g}(v)$ with boundary components, labeled by $y\in \mathcal {Y}(\Gamma )$ with $t(y)=v$ , and let $p_v\colon M_v\to B_v$ be an $S^1$ -bundle, where $M_v$ is an oriented $3$ -manifold with trivializations at boundary components such that the Euler number of $p_v$ is $\boldsymbol {e}(v)$ . For each $y\in \mathcal {Y}(\Gamma )$ , let $T_y\subset M_{t(y)}$ be the boundary component associated with y with trivialization $\tau _y\colon S^1\times S^1\to T_y$ , and put $R_y:=\boldsymbol {m}(y)^{-1} J \boldsymbol {m}(\overline {y})$ . By condition (ii) of $\boldsymbol {m}$ , we have $R_y\in \mathrm {GL}(2,\mathbb Z)$ , $\det R_y=-1$ , and $R_y$ represent homeomorphisms $T_{\overline {y}}\to T_y$ :

The manifold $\mathscr {P}(\Gamma _{\mathrm {pg}}^{\mathrm {m}})$ is obtained by gluing $M_{t(y)}$ and $M_{t(\overline {y})}$ by $R_y$ .

Remark 2.9 For $y\in \mathcal {Y}(\Gamma )$ , $\boldsymbol {m}(y)\colon S^1\times S^1\to S^1\times S^1$ coincides with the map induced by the map

$$\begin{align*}\mathbb C^2\setminus\{u_1v_1\ne0\}\ni (u_1,v_1)\longmapsto(u_1^{c(y)},u_1^{b(y)}v_1^{a(y)})=(u_2,v_2)\in\mathbb C^2\setminus\{u_2v_2\ne0\}\end{align*}$$

by regarding $S^1\times S^1$ as $\{|u_i|=|v_i|=1\}$ . This map appears in [Reference Laufer24, Chapter 2] to construct a resolution of singularities of two-dimensional analytic spaces. Note that the dual graph of the exceptional divisor on the resolution is a chain, and $\mathscr {P}(\Gamma _{\mathrm {pg}}^{\mathrm {m}})$ can be constructed as the plumbed manifold $\mathscr {P}(\Gamma _{\mathrm {pg}})$ of $\Gamma _{\mathrm {pg}}$ , where $\Gamma _{\mathrm {pg}}$ is the plumbing graph constructed by replacing each edge of $\Gamma _{\mathrm {pg}}^{\mathrm {m}}$ with the chain of the exceptional divisor. By the resolution in [Reference Laufer24], we can check that the basis of $H_1(T_y,\mathbb Z)$ giving the matrix (2.3) corresponding to the basis given by $\Gamma _{\mathrm {pg}}$ .

3.1 Combinatorial type

Let $\mathcal C\subset {\mathbb {P}}^2$ be a plane curve of degree d, and let $\mathrm {Irr}(\mathcal C)$ denote the set of irreducible components of $\mathcal C$ . Let $\sigma \colon Y\to {\mathbb {P}}^2$ be the minimal good embedded resolution of the singularities $ \mathrm {Sing}(\mathcal C)$ of $\mathcal C$ , that is, the minimal sequence of blowing-ups which resolves $\mathop {\mathrm {Sing}}(\mathcal C)$ and such that $\sigma ^{-1}(\mathcal C)$ is a simple normal crossing divisor. Put

$$\begin{align*}{\mathcal C'}:=\sigma^{-1}(\mathcal C). \end{align*}$$

In [Reference Artal Bartolo, Bannai, Shirane and Tokunaga3], the combinatorial type (or the combinatorics for short) $ \mathrm { Cmb}(\mathcal C)$ of the plane curve $\mathcal C$ is redefined as the $3$ -tuple $ \mathrm {Cmb}(\mathcal C):=(\Gamma _{\mathcal C}, \mathrm {Str}_{\mathcal C}, \boldsymbol {e}_{\mathcal C})$ , where

• • $\Gamma _{\mathcal C}$ is the dual graph of $ \mathcal C'$ ,

• • $ \mathrm {Str}_{\mathcal C}\subset \mathcal {V}(\Gamma _{\mathcal C})$ is the set of vertices corresponding to the strict transforms of irreducible components of $\mathcal C$ , and

• • $\boldsymbol {e}_{\mathcal C}$ is the map $\boldsymbol {e}_{\mathcal C}\colon \mathcal {V}(\Gamma _{\mathcal C})\to \mathbb Z$ of self-intersection numbers.

Since ${\mathcal C'}$ is simple normal crossing, each edge of $\Gamma _{\mathcal C}$ is not a loop. We can identify the sets $\mathrm {Irr}({\mathcal C'})$ and $\mathrm {Irr}(\mathcal C)$ of irreducible components with $\mathcal {V}(\Gamma _{\mathcal C})$ and $ \mathrm {Str}_{\mathcal C}$ , respectively,

$$ \begin{align*} \mathrm{Irr}({\mathcal C'})&=\mathcal{V}(\Gamma_{\mathcal C}), & \mathrm{Irr}(\mathcal C)&= \mathrm{Str}_{\mathcal C}. \end{align*} $$

Each node $P\in \mathrm {Sing}({\mathcal C'})$ corresponds to a pair of edges $\{y, \overline {y}\}$ for some $y\in \mathcal {Y}(\Gamma _{\mathcal C})$ . For each $v\in \mathcal {V}(\Gamma _{\mathcal C})$ , let $ D^{\prime }_v$ denote the irreducible component of ${\mathcal C'}$ corresponding to v. By contracting exceptional components of $\sigma $ , we can compute the degrees of irreducible components of $\mathcal C$ and the map $\boldsymbol {g}_{\mathcal C}\colon \mathcal {V}(\Gamma _{\mathcal C})\to \mathbb Z$ of genera from $ \mathrm {Cmb}(\mathcal C)$ . Hence, the combinatorics $ \mathrm {Cmb}(\mathcal C)$ can be regarded as the plumbing graph

$$\begin{align*}\mathrm{Cmb}(\mathcal C)=(\Gamma_{\mathcal C}, \boldsymbol{g}_{\mathcal C},\boldsymbol{e}_{\mathcal C}; \mathrm{ Str}_{\mathcal C}) \end{align*}$$

together with the marking $ \mathrm {Str}_{\mathcal C}\subset \mathcal {V}(\Gamma _{\mathcal C})$ . Let $\mathscr {P}(\mathcal C)$ denote the plumbed manifold associated with $ \mathrm {Cmb}(\mathcal C)$ :

$$\begin{align*}\mathscr{P}(\mathcal C):=\mathscr{P}(\Gamma_{\mathcal C},\boldsymbol{g}_{\mathcal C},\boldsymbol{e}_{\mathcal C}). \end{align*}$$

For two plane curves $\mathcal C_1, \mathcal C_2\subset {\mathbb {P}}^2$ , an equivalence map $\varphi \colon \mathrm { Cmb}(\mathcal C_1)\to \mathrm {Cmb}(\mathcal C_2)$ is an isomorphism $\varphi \colon \Gamma _{\mathcal C_1}\to \Gamma _{\mathcal C_2}$ of graphs such that $\varphi ( \mathrm {Str}_{\mathcal C_1})= \mathrm { Str}_{\mathcal C_2}$ , $\boldsymbol {g}_{\mathcal C_1}=\boldsymbol {g}_{\mathcal C_2}\circ \varphi ,$ and $\boldsymbol {e}_{\mathcal C_1}=\boldsymbol {e}_{\mathcal C_2}\circ \varphi $ . We say that $\mathcal C_1$ and $\mathcal C_2$ have the same combinatorics if there is an equivalence map $\varphi \colon \mathrm {Cmb}(\mathcal C_1)\to \mathrm {Cmb}(\mathcal C_2)$ .

3.2 The boundary manifold $\mathscr {B}(\mathcal C)$

According to [Reference Durfee16], we define an algebraic neighborhood $\mathcal {T}(\mathcal C)$ of a plane curve $\mathcal C\subset {\mathbb {P}}^2$ as follows: Let $f\in \mathbb C[x_0,x_1,x_2]$ be a homogeneous polynomial of degree d defining $\mathcal C$ . Let $\eta _f\colon {\mathbb {P}}^2\to {\mathbb {R}}$ be the map defined by

$$\begin{align*}\eta_f(x_0,x_1,x_2):= \frac{|f(x_0,x_1,x_2)|^2}{(|x_0|^2+|x_1|^2+|x_2|^2)^d}\in{\mathbb{R}} \end{align*}$$

for $(x_0:x_1:x_2)\in {\mathbb {P}}^2$ . Let $\mathcal {T}(\mathcal C):=\{(x_0:x_1:x_2)\in {\mathbb {P}}^2\mid \eta _f(x_0,x_1,x_2)\leq \delta \}$ for sufficiently small $\delta>0$ , which is called an algebraic neighborhood of $\mathcal C$ . By [Reference Łojasiewicz25], $\mathcal {T}(\mathcal C)$ is a regular neighborhood of $\mathcal C$ . By [Reference Durfee16], $\mathcal {T}(\mathcal C)$ is unique up to isotopy as an algebraic neighborhood. However, the author is not sure whether $\mathcal {T}(\mathcal C)$ is unique as a regular neighborhood. Let $\mathscr {B}(\mathcal C)$ be the boundary of $\mathcal {T}(\mathcal C)$ , $ \mathscr {B}(\mathcal C):=\partial \mathcal {T}(\mathcal C)$ , called the boundary manifold of $\mathcal C$ .

Let $\sigma \colon Y\to {\mathbb {P}}^2$ be the minimal good embedded resolution of $\mathcal C$ , and put $\mathcal C':=\sigma ^{-1}(\mathcal C)$ . Since ${ \eta _f}\circ \sigma \colon Y\to {\mathbb {R}}$ is admissible in the sense of [Reference Mumford27], we can check that $\sigma ^{-1}(\mathcal {T}({\mathcal C}))$ is given by plumbing tubular neighborhoods $\mathcal {T}({ D'})$ of irreducible components ${ D'}$ of ${\mathcal C'}$ (cf. [Reference Hirzebruch, Neumann and Koh22, Section 8] and [Reference Mumford27, Section 1]). Note that, for each irreducible component ${ D'}\subset {\mathcal C'}$ , the Euler number of the $S^1$ -bundle $ \overline {p}_D\colon \partial \mathcal {T}(D')\to D'$ is equal to the self-intersection number of ${ D'}$ . From this point of view, $\mathscr {B}(\mathcal C)$ is constructed from $ \mathrm {Cmb}(\mathcal C)$ as follows. For $v\in \mathcal {V}(\Gamma _{\mathcal C})$ , put

$$ \begin{align*} B_v&:=D^{\prime}_v\setminus U( \mathrm{Sing}(\mathcal C'))^\circ, & M_v&:=\overline{p}_{D_v}^{-1}(B_v), & p_v&:=\overline{p}_{D_v}|_{M_v}\colon M_v\longrightarrow B_v. \end{align*} $$

For $y\in \mathcal {Y}(\Gamma _{\mathcal C})$ , let $\partial B_{y}$ be the connected component of $\partial B_{t(y)}$ around the node $P_y$ of $\mathcal C'$ corresponding to y, and put $T_y:=p_{t(y)}^{-1}(\partial B_{y})$ . We may assume that $\mathcal C'$ and $D^{\prime }_{t(y)}$ are locally defined at $P_y$ by $z_1z_2=0$ and $z_2=0$ , respectively, in $\mathbb {D}^2\times \mathbb {D}^2$ with coordinates $(z_1,z_2)$ , and that $T_y$ is a torus given by $|z_1|=|z_2|=\delta $ for small $\delta>0$ . Let $\tau _y\colon S^1\times S^1\to T_y$ be the trivialization given by $(z_1,z_2)\mapsto (\delta z_1,\delta z_2)$ . Then $\mathscr {B}(\mathcal C)$ is constructed by gluing $M_{t(y)}$ and $M_{t(\overline {y})}$ by (2.2) with $\mathrm {sgn}(y)=+1$ . Hence, $\mathscr {B}(\mathcal C)$ is a graph manifold with graph structure $\mathbb T_{\mathcal C}:=\bigcup _y T_y$ , where $T_{\overline {y}}$ is identified with $T_y$ in $\mathscr {B}(\mathcal C)$ for each $y\in \mathcal {Y}(\Gamma _{\mathcal C})$ . We call $\mathbb T_{\mathcal C}\subset \mathscr {B}(\mathcal C)$ the graph structure with respect to $ \mathrm { Cmb}(\mathcal C)$ . It follows from the definition of $\mathscr {P}(\mathcal C)$ that $\mathscr {B}(\mathcal C)$ is homeomorphic to $\mathscr {P}(\mathcal C)$ . Therefore, if two plane curves $\mathcal C_1,\mathcal C_2\subset {\mathbb {P}}^2$ have the same combinatorics, then there is a homeomorphism $h\colon \mathcal {T}(\mathcal C_1)\to \mathcal {T}(\mathcal C_2)$ with $h(\mathcal C_1)=\mathcal C_2$ .

3.3 Embedded topology in regular neighborhoods

Let $\mathcal C_1,\mathcal C_2\subset {\mathbb {P}}^2$ be two plane curves, and let $\sigma _i\colon Y_i\to {\mathbb {P}}^2$ be the minimal good embedded resolution of $\mathcal C_i$ . We put ${\mathcal C}^{\prime }_{i}:=\sigma _i^{-1}(\mathcal C_i)$ , and identify the combinatorics $ \mathrm {Cmb}(\mathcal C_i)$ with the corresponding plumbing graph, write

$$\begin{align*}\mathrm{Cmb}(\mathcal C_i)=(\Gamma_i,\boldsymbol{g}_i,\boldsymbol{e}_i; \mathrm{Str}_i). \end{align*}$$

For $v_i\in \mathrm {Str}_i$ (resp. $v_i\in \mathcal {V}(\Gamma _i)$ ), let $ D_{v_i}$ (resp. ${D}^{\prime }_{v_i}$ ) be the irreducible component of $\mathcal C_i$ (resp. ${\mathcal C}^{\prime }_i$ ) corresponding to $v_i$ . For $v_i\in \mathcal {V}(\Gamma _i)$ and $y_i\in \mathcal {Y}(\Gamma _i)$ , let $M_{v_i}$ , $B_{v_i}$ , $p_{v_i}$ , $T_{y_i}$ , and $\tau _{y_i}$ be as in Section 3.2. Let $ \mathbb T_i=\bigcup _{y_i} T_{y_i}\subset \mathscr {B}(\mathcal C_i)$ be the graph structure with respect to $ \mathrm { Cmb}(\mathcal C_i)$ .

Assume that there is a homeomorphism $h\colon U(\mathcal C_1)\to U(\mathcal C_2)$ with $h(\mathcal C_1)=\mathcal C_2$ for some regular neighborhoods $U(\mathcal C_i)$ of $\mathcal C_i$ . As in the proof of [Reference Artal Bartolo, Cogolludo-Agustín and Morales7, Theorem 2.2], h induces an isomorphism

$$\begin{align*}\Psi_h^\infty\colon \pi_1(\mathscr{B}(\mathcal C_1))\longrightarrow\pi_1(\mathscr{B}(\mathcal C_2)). \end{align*}$$

The isomorphism $\Psi _h^\infty $ can be constructed as follows: Let $\mathcal {T}(\mathcal C_2)\subset h(U(\mathcal C_1))$ be an algebraic neighborhood of $\mathcal C_2$ , and let $\mathcal {T}(\mathcal C_1)\subset h^{-1}(\mathcal {T}(\mathcal C_2))$ be an algebraic neighborhood of $\mathcal C_1$ . Then there is an algebraic neighborhood $\mathcal {T}'(\mathcal C_2)\subset h(\mathcal {T}(\mathcal C_1))$ . Since $\mathscr {B}(\mathcal C_i)$ is homotopic to $\mathcal {T}(\mathcal C_i)\setminus \mathcal C_i$ , we obtain the homomorphism

$$\begin{align*}\Psi_h^\infty\colon\pi_1(\mathscr{B}(\mathcal C_1))\cong\pi_1(\mathcal{T}(\mathcal C_1)\setminus\mathcal C_1)\overset{h_\ast}{\longrightarrow}\pi_1(\mathcal{T}(\mathcal C_2)\setminus\mathcal C_2)\cong\pi_1(\mathscr{B}(\mathcal C_2)). \end{align*}$$

Since we can similarly construct the inverse map of $\Psi _h^\infty $ , it is an isomorphism.

In [Reference Artal Bartolo, Cogolludo-Agustín and Morales7], it is also proved by using [Reference Waldhausen39, Corollary 6.5] that there exists a homeomorphism $\Psi _{\mathscr {B}}\colon \mathscr {B}(\mathcal C_1)\to \mathscr {B}(\mathcal C_2)$ with $\Psi _{\mathscr {B}\,\ast }=\Psi _h^\infty $ in several cases. Here, we construct such a homeomorphism $\Psi _{\mathscr {B}}\colon \mathscr {B}(\mathcal C_1)\to \mathscr {B}(\mathcal C_2)$ homotopic to $h|_{\mathscr {B}(\mathcal C_1)}$ in general case and prove that $\Psi _{\mathscr {B}}$ maps meridians of irreducible components of ${\mathcal C}^{\prime }_1$ to those of ${\mathcal C}^{\prime }_2$ .

Assume that $\mathcal C_i$ are singular. We locally construct a homeomorphism of boundary manifolds around each singular point homotopic to h. Let $P_1$ be a singular point of $\mathcal C_1$ , and put $P_2:=h(P_1)$ . Let $U(P_i)\subset {\mathbb {P}}^2$ be a regular neighborhood of $P_i$ for each $i=1,2$ . We may assume that

$$ \begin{align*} \mathscr{B}(\mathcal C_i,P_i):=\mathscr{B}(\mathcal C_i)\cap U(P_i) \subset \mathscr{B}(\mathcal C_i)\setminus\bigcup_{v_i\in \mathrm{Str}_i} { M_{v_i}^\circ} \end{align*} $$

is a connected component of the complement of the union of $\{{ M_{v_i}^\circ }\mid v_i\in \mathrm {Str}_i\}$ in $\mathscr {B}(\mathcal C_i)$ . We can regard $\mathscr {B}(\mathcal C_i,P_i)$ as a boundary framed graph manifold associated with the decorated plumbing graph

$$\begin{align*}\Gamma_{\mathrm{dpg}}(\mathcal C_i,P_i)=(\Gamma_{i,P_i}, \boldsymbol{g}_{i,P_i}, \boldsymbol{e}_{i,P_i};\overline{\mathcal{A}}, \mathcal{A}) \end{align*}$$

defined as follows: If $P_i\in \mathcal C_i$ is not a node at which two distinct components intersect, then we define $\Gamma _{\mathrm {dpg}}(\mathcal C_i,P_i)$ by the following steps:

1. (I) let $\overline {\mathcal {A}}(\Gamma _{i,P_i})$ be the subset of $\mathcal {V}(\Gamma _i)$ consisting of vertices $v_i$ with $\sigma _i({ D^{\prime }_{v_i}})=\{P_i\}$ ;

2. (II) $\mathcal {Y}(\Gamma _{i,P_i})$ is the subset of $\mathcal {Y}(\Gamma _i)$ consisting of edges $y_i$ with either $t(y_i)\in \overline {\mathcal {A}}(\Gamma _{i,P_i})$ or $o(y_i)\in \overline {\mathcal {A}}(\Gamma _{i,P_i})$ ;

3. (III) $\mathcal {A}(\Gamma _{i,P_i})$ is in bijection with the set of edges $y_i\in \mathcal {Y}(\Gamma _{i,P_i})$ with $t(y_i)\notin \overline {\mathcal {A}}(\Gamma _{i,P_i})$ as edges of $\Gamma _i$ ; let $v_{i,y_i}$ be the vertex of $\mathcal {A}(\Gamma _{i,P_i})$ corresponding to $y_i$ ;

4. (IV) for each $y_i\in \mathcal {Y}(\Gamma _{i,P_i})$ , let $t(y_i)$ be the same terminal vertex $t(y_i)$ of $y_i\in \mathcal {Y}(\Gamma _i)$ if $t(y_i)\in \overline {\mathcal {A}}(\Gamma _{i,P_i})$ as a vertex of $\Gamma _i$ ; and put $t(y_i):=v_{i,y_i}\in \mathcal {A}(\Gamma _{i,P_i})$ otherwise;

5. (V) for each $y_i\in \mathcal {Y}(\Gamma _{i,P_i})$ , put $o(y_i):=t(\overline {y}_i)$ as edges of $\Gamma _{i,P_i}$ ;

6. (VI) $\boldsymbol {g}_{i,P_i}, \boldsymbol {e}_{i,P_i}\colon \overline {\mathcal {A}}(\Gamma _{i,P_i})\to \mathbb Z$ are the restrictions of $\boldsymbol {g}_i$ and $\boldsymbol {e}_i$ , respectively.

Note that $\Gamma _{\mathrm {dpg}}(\mathcal C_i,P_i)$ corresponds to the dual graph of the minimal good embedded resolution of the singularity $P_i\in \mathcal C_i$ locally. Hence, $\Gamma _{i,P_i}$ has no cycle.

According to [Reference Eisenbud and Neumann17, Chapter V], if $P_i\in \mathcal C_i$ is a node at which two distinct components intersect, we take the blowing-up at $P_i$ ; replace $ \mathrm {Cmb}(\mathcal C_i)$ with the plumbing graph $\Gamma _{\mathrm {pg}}(\mathcal C_i)$ given by the blowing-up ( $\mathrm {R1}^+$ ) of $\mathop {\mathrm {Cmb}}(\mathcal C_i)$ at the edge corresponding to $P_i$ ; and define $\Gamma _{\mathrm {dpg}}(\mathcal C_i,P_i)$ as above. Namely, if $P_i$ is a node of $\mathcal C_i$ , then $\Gamma _{\mathrm {dpg}}(\mathcal C_i,P_i)$ is the following decorated plumbing graph:

We denote the F-normal form of $\Gamma _{\mathrm {dpg}}(\mathcal C_i,P_i)$ by $\Gamma _{\mathrm {dpg}}^{\mathrm {nml}}(\mathcal C_i,P_i)$ . For $v_i\in \overline {\mathcal {A}}(\Gamma _{i,P_i})$ and $y_i\in \mathcal {Y}(\Gamma _{i,P_i})$ , we may define $ p_{v_i}\colon M_{v_i}\to B_{v_i}$ , $ T_{y_i}$ and $ \tau _{y_i}\colon S^1\times S^1\to T_{y_i}$ as the same things for $\Gamma _{\mathrm {pg}}(\mathcal C_i)$ . Let $\mathbb T_{i.P_i}\subset \mathscr {B}(\mathcal C_i,P_i)$ be the graph structure with respect to $\Gamma _{\mathrm {dpg}}(\mathcal C_i,P_i)$ .

Note that $\mathcal {A}(\Gamma _{i,P_i})$ corresponds to the set of local branches of $\mathcal C_i$ at $P_i$ . Hence, h induces a bijection

$$\begin{align*}\xi_{h,P_1}\colon\mathcal{A}(\Gamma_{1,P_1})\longrightarrow\mathcal{A}(\Gamma_{2,P_2}) \end{align*}$$

since $h(\mathcal C_1)=\mathcal C_2$ . Let $h_{P_1}\colon \mathscr {B}(\mathcal C_1,P_1)\to \mathscr {B}(\mathcal C_2,P_2)$ be the restriction of $h'\colon \mathscr {B}(\mathcal C_1)\to \mathscr {B}(\mathcal C_2)$ constructed in Remark 3.2. We have the induced isomorphism

$$\begin{align*}h_{P_1\ast}\colon \pi_1(\mathscr{B}(\mathcal C_1,P_1))\longrightarrow\pi_1(\mathscr{B}(\mathcal C_2,P_2)) \end{align*}$$

of fundamental groups $\pi _1(\mathscr {B}(\mathcal C_i,P_i))$ such that $h_{P_1\ast }(i_{y_1\ast }(\pi _1({ T_{y_1}})))=i_{y_2\ast }(\pi _1({ T_{y_2}}))$ for each $y_i\in \mathcal {Y}(\Gamma _{i,P_i})$ with $t(y_1)\in \mathcal {A}(\Gamma _{1,P_1})$ and $t(y_2)=\xi _{h,P_1}(t(y_1))$ , where $i_{y_i}\colon { T_{y_i}}\to \mathscr {B}(\mathcal C_i,P_i)$ is the inclusion map. Note that it follows from [Reference Brieskorn and Knörrer13, III, Theorem 21] that there exists an isomorphism

(3.1) $$ \begin{align} \varphi_{P_1}\colon\Gamma_{\mathrm{dpg}}(\mathcal C_1,P_1)\longrightarrow \Gamma_{\mathrm{dpg}}(\mathcal C_2,P_2) \end{align} $$

satisfying $\varphi _{P_1}|_{\mathcal {A}(\Gamma _{1,P_1})}=\xi _{h,P_1}$ , and hence $\mathscr {B}(\mathcal C_i,P_i)$ are homeomorphic.

Remark 3.3 We can prove from $\varphi _{P_1}$ for each $P_1\in \mathrm {Sing}(\mathcal C_1)$ that $\mathcal C_1$ and $\mathcal C_2$ have the same combinatorics if they have the same embedded topology in regular neighborhoods. Indeed, if $v_i\in \mathrm {Str}_i$ satisfy $ h(D_{v_1})=D_{v_2}$ , then we have $\boldsymbol {g}_1(v_1)=\boldsymbol {g}_2(v_2)$ , and

$$\begin{align*}\boldsymbol{g}_i(v_i)=\frac{(d_i-1)(d_i-2)}{2}-{ \sum_{P_i\in \mathrm{Sing}(D_{v_i})}\delta_{D_{v_i},P_i}}, \end{align*}$$

where $d_i:=\deg { D_{v_i}}$ , and $\delta _{ D_{v_i},P_i}\in \mathbb Z$ is the delta invariant of $ D_{v_i}$ at $P_i\in \mathrm { Sing}({ D_{v_i}})$ . The existence of $\varphi _{P_1}$ implies $\sum \delta _{ D_{v_1},P_1}=\sum \delta _{ D_{v_2},P_2}$ . If $d_1\leq 2$ , then we obtain $d_1=d_2$ by Remark 3.2. If $d_1>2$ , then $d_1=d_2$ follows from $d_i>0$ . Hence, we have $\boldsymbol {e}_1(v_1)=\boldsymbol {e}_2(v_2)$ by $d_1=d_2$ and the existence of $\varphi _{P_1}$ . Therefore, we can construct an equivalence map $\varphi \colon \mathrm {Cmb}(\mathcal C_1)\to \mathrm { Cmb}(\mathcal C_2)$ . The arguments in Section 3.2 and this remark prove Fact 1.1.

We here construct a homeomorphism $\mathfrak {g}_{P_1}\colon \mathscr {B}(\mathcal C_1,P_1)\to \mathscr {B}(\mathcal C_2,P_2),$ which is homotopic to $h_{P_1}$ and satisfies certain properties. Put $E:=(\begin {smallmatrix} 1&0\\0&1 \end {smallmatrix})$ , the identity matrix.

Note that the homeomorphism $\mathfrak {g}_{P_1}$ in Lemma 3.4 is orientation-preserving by condition (iii). We consider the case where $P_1$ is not a node of $\mathcal C_1$ .

Lemma 3.5 (cf. [Reference Neumann28, Theorem 8.2])

If $P_i\in \mathcal C_i$ is a singular point which is not node, then the F-normal form $\Gamma _{\mathrm {dpg}}^{\mathrm {nml}}(\mathcal C_i,P_i)$ is obtained from $\Gamma _{\mathrm {dpg}}(\mathcal C_i,P_i)$ by applying the following operations to $\Gamma _{\mathrm {dpg}}(\mathcal C_i,P_i)$ wherever possible:

Furthermore, $\Gamma _{\mathrm {dpg}}^{\mathrm {nml}}(\mathcal C_i,P_i)$ satisfies the following:

1. (i) All edges of $\Gamma _{\mathrm {dpg}}^{\mathrm {nml}}(\mathcal C_i,P_i)$ are $(+)$ -edges.

2. (ii) Any vertex $v_i$ of $\Gamma _{\mathrm {dpg}}^{\mathrm {nml}}(\mathcal C_i,P_i)$ satisfies $\boldsymbol {g}(v_i)\geq -1$ .

3. (iii) Any vertex $v_i$ of $\Gamma _{\mathrm {dpg}}^{\mathrm {nml}}(\mathcal C_i,P_i)$ with $\boldsymbol {g}(v_i)=-1$ satisfies $\boldsymbol {d}(v_i)=1$ , $\boldsymbol {e}(v_i)\geq 0$ , and if $\boldsymbol {e}(v_i)=0$ , the maximal chain adjoining $v_i$ has length $\geq 1$ .

4. (iv) $\Gamma _{\mathrm {dpg}}(\mathcal C_i,P_i)$ is uniquely determined by $\Gamma _{\mathrm {dpg}}^{\mathrm {nml}}(\mathcal C_i,P_i)$ .

We prove that the homeomorphism $\mathfrak {g}^{\prime }_{P_1}$ is orientation-preserving.

Proof Assume that $\mathfrak {g}_{P_1}'$ is orientation-reversing. Let $y_{i1}\in \mathcal {Y}(\Gamma _{i,P_i})$ be edges with $t({ y_{i1}})\in \mathcal {A}(\Gamma _{i,P_i})$ and $\xi _{h,P_1}(t({ y_{11}}))={ y_{21}}$ . Since the isomorphism $\varphi _{P_1}$ satisfies $\varphi _{P_1}|_{\mathcal {A}(\mathcal C_{1,P_1})}=\xi _{h,P_1}$ , the maximal chains incident to $ y_{i1}$ are the same form:

(3.2)

where $-b_j\leq -2$ are the Euler numbers of corresponding vertices, and $y_{i2}\in \mathcal {Y}(\Gamma _{i,P_i})$ are edges such that $v_{i2}=t(y_{i2})$ and $\boldsymbol {d}(v_{i2})\geq 3$ for $i=1,2$ . Let $v_{i1}\in \mathrm {Str}_i$ be the vertices corresponding to $t(y_{i1})\in \mathcal {A}(\Gamma _{i,P_i})$ . Let $T_{ij}$ be the boundary component $ T_{y_{ij}}$ of $M_{v_{ij}}$ corresponding $y_{ij}$ for each $i,j=1,2$ . By [Reference Neumann28, Lemma 5.3], $T_{i1}$ and $T_{i2}$ are identified in $\mathscr {B}(\mathcal C_i)$ via homeomorphism given by the matrix

$$\begin{align*}B_k:=\begin{pmatrix} d_k & c_k \\ - d_{k-1} & -c_{k-1} \end{pmatrix}, \end{align*}$$

where $c_i, d_i\in \mathbb Z$ are coprime integers such that $c_i/d_i$ is equal to the continued fraction $[b_1,\dots ,b_i]$ for each $i=1,\dots ,k$ (see [Reference Neumann28, Lemma 5.2] for details). Note that $c_{-1}=0$ , $c_0=1$ , $d_{-1}=-1$ , $d_0=0$ , and $\det B_k=-1$ . By [Reference Artal Bartolo, Bannai, Shirane and Tokunaga3, Proposition 3.4] and [Reference Waldhausen38, Theorem 5.5], we may assume that $ \tau _{y_{2j}}\circ \mathfrak {g}^{\prime }_{P_1}\circ \tau _{y_{1j}}^{-1}$ is the homeomorphism of $S^1\times S^1$ given by the matrix

$$\begin{align*}H_j:=\begin{pmatrix} \alpha_j&0\\ \beta_j & -\alpha_j \end{pmatrix} \end{align*}$$

for each $j=1,2$ , where $\alpha _j\in \{\pm 1\}$ and $\beta _j\in \mathbb Z$ . Hence, we have the following commutative diagram:

(3.3)

The $(1,2)$ entry of $B_kH_1B_k^{-1}=H_2$ is $(2\alpha _1 d_k+\beta _1 c_k)c_k=0$ . Since $\gcd (c_k,d_k)=1$ , $c_k\geq k+1,$ and $d_k\geq k$ by [Reference Neumann28, Lemma 5.2(ii)], we obtain either $d_k=0$ or $c_k=2$ . If $d_k=0$ , then $k=0$ , $\beta _1=0$ , and $H_2=-H_1$ . If $c_k=2$ , then we have $k=1$ , $d_k=1$ , $b_1=2$ , $\beta _1=-\alpha _1,$ and $H_2=-H_1$ . Thus, we can construct $\Gamma _{\mathrm {dpg}}^{\mathrm {nml}}(\mathcal C_2,P_2)$ from $-\Gamma _{\mathrm {dpg}}^{\mathrm {nml}}(\mathcal C_1,P_1)$ by Theorem B.2.

By the above fact, we can apply the argument of [Reference Neumann28, Section 9] as follows. By Theorem B.2(ii) and Lemma 3.5(iii), $\Gamma _{\mathrm {dpg}}^{\mathrm {nml}}(\mathcal C_i,P_i)$ for $i=1,2$ have no vertex v with $\boldsymbol {g}(v)=-1$ . Hence, $\Gamma _{\mathrm {dpg}}(\mathcal C_i,P_i)=\Gamma _{\mathrm {dpg}}^{\mathrm {nml}}(\mathcal C_i,P_i)$ for $i=1,2$ by Lemma 3.5, and $\mathrm {sgn}(y_i)=+1$ for any $y_i\in \mathcal {Y}(\Gamma _{i,P_i})$ . Since $\Gamma _{i,P_i}$ is the dual graph of embedded resolution of the singularity $P_i\in \mathcal C_i$ , $\Gamma _{i,P_i}$ has no cycle for each $i=1,2$ . Moreover, there is $v_1\in \mathcal {V}(\Gamma _{1,P_i})$ with $\boldsymbol {d}(v_1)\geq 3$ since $P_1$ is not node. Let

$$\begin{align*}\Gamma_{\mathrm{dpg}}^{1,v_1}=(\Gamma_{1,P_1}^{v_1},\boldsymbol{g}_{1,P_1}^{v_1},\boldsymbol{e}_{1,P_1}^{v_1};\overline{\mathcal{A}},\mathcal{A})\end{align*}$$

be the full subgraph of $\Gamma _{\mathrm {dpg}}(\mathcal C_1,P_1)$ such that $\mathcal {V}(\Gamma _{1,P_1}^{v_1})$ consists of $v_1$ , all vertices on maximal chains adjoining $v_1$ and all boundary vertices adjacent to $v_1$ . The decorated plumbing graph $\Gamma _{\mathrm {dpg}}^{1,v_1}$ is a star-shaped decorated plumbing graph, and $\mathscr {P}(\Gamma _{\mathrm {dpg}}^{1,v_1})$ is a Seifelt manifold. Let $\widetilde {\Gamma }_{\mathrm {dpg}}^{1,v_1}$ be the F-normal form of $-\Gamma _{\mathrm {dpg}}^{1,v_1}$ . By Theorem B.2, $\widetilde {\Gamma }_{\mathrm {dpg}}^{1,v_1}$ is also a star-shaped decorated plumbing graph and is a subgraph of $\Gamma _{\mathrm {dpg}}(\mathcal C_2,P_2)$ . Since the intersection form $S(\Gamma _{\mathrm {dpg}}^{1,v_1})$ is negative definite, the Euler number of the Seifelt manifold $\mathscr {P}(\Gamma _{\mathrm {dpg}}^{1,v_1})$ is negative by Theorem B.4. Thus, the Euler number of $\mathscr {P}(\widetilde {\Gamma }_{\mathrm {dpg}}^{1,v_1})=\mathscr {P}(-\Gamma _{\mathrm {dpg}}^{1,v_1})$ is positive by the equality (B.2), which is contradiction to Theorem B.3 since $\widetilde {\Gamma }_{\mathrm {dpg}}^{1,v_1}$ is a subgraph of $\Gamma _{\mathrm {dpg}}(\mathcal C_2,P_2)$ , and the intersection form $S(\widetilde {\Gamma }_{\mathrm {dpg}}^{1,v_1})$ is negative definite. Therefore, $\mathfrak {g}^{\prime }_{P_1}$ is orientation-preserving.

Proof We construct the desired homeomorphism $\mathfrak {g}_{P_1}$ by deforming $\mathfrak {g}^{\prime }_{P_1}$ in Lemma 3.7. Let $\mathcal {S}_i$ be the subset of edges of $\Gamma _{\mathrm {dpg}}^{\mathrm {nml}}(\mathcal C_i,P_i)$ as in the proof of Lemma 3.7, and put $ \mathbb T_{\mathcal {S}_i}:=\bigcup _{y_i\in \mathcal {S}_i}T_{y_i}$ which is a reduce graph structure of $\mathscr {B}(\mathcal C_i,P_i)$ . By [Reference Waldhausen38, Theorem 8.1] and Lemma 3.8, there is an isotopy $\mathfrak {h}_{u}\colon \mathscr {B}(\mathcal C_1,P_1)\to \mathscr {B}(\mathcal C_2,P_2)$ ( ${0\leq u\leq 1}$ ), which is constant on $\partial \mathscr {B}(\mathcal C_i,P_i)$ and satisfies $\mathfrak {h}_{0}=\mathfrak {g}^{\prime }_{P_1}$ and $ \mathfrak {h}_1(\mathbb T_{\mathcal {S}_1})=\mathbb T_{\mathcal {S}_2}$ .

Let $y_{i1}\in \mathcal {Y}(\Gamma _{i,P_i})$ be edges satisfying $t(y_{i1})\in \mathcal {A}(\Gamma _{i,P_i})$ and $\varphi _{P_1}(y_{11})=y_{21}$ . For each $i=1,2$ , let $\mathfrak {c}_i$ be the maximal chain incident to $t(y_{i1})$ , and let $y_{i2}\in \mathcal {Y}(\Gamma _{i,P_i})$ be the edge on $\mathfrak {c}_i$ such that $\boldsymbol {d}(t(y_{i2}))>2$ as in (3.2). By the same argument in the proof of Lemma 3.8, we may assume that $ \tau _{y_{2j}}\circ \mathfrak {g}^{\prime }_{P_1}\circ \tau _{y_{1j}}^{-1}$ is given by

$$\begin{align*}H_j:=\begin{pmatrix} \alpha_j&0\\ \beta_j & \alpha_j \end{pmatrix} \end{align*}$$

for each $j=1,2$ , where $\alpha _j\in \{\pm 1\}$ and $\beta _j\in \mathbb Z$ . It follows from the diagram (3.3) that $\alpha _1=\alpha _2$ and $\beta _j=0$ for $j=1,2$ .

For $y_i\in \mathcal {S}_i$ ( $i=1,2$ ) with $ \mathfrak {h}_{1}(T_{y_1})=T_{y_2}$ , the maximal chains of $\Gamma _{\mathrm {dpg}}^{\mathrm {nml}}(\mathcal C_i,P_i)$ containing $y_i$ have the same number of edges. Since each maximal chain of $\Gamma _{\mathrm {dpg}}^{\mathrm {nml}}(\mathcal C_i,P_i)$ corresponds to a part homeomorphic to $I\times S^1\times S^1$ in $\mathscr {B}(\mathcal C_i,P_i)$ , there is an isotopy $\mathfrak {h}_{u}\colon \mathscr {B}(\mathcal C_1,P_1)\to \mathscr {B}(\mathcal C_2,P_2)$ ( $1\leq u\leq 2$ ), which is constant on $\partial \mathscr {B}(\mathcal C_i,P_i)$ and satisfies $\mathfrak {h}_{2}(\mathbb T_{1,P_1}^{\mathrm {nml}})=\mathbb T_{2,P_2}^{\mathrm {nml}}$ , where $\mathbb T_{i,P_i}^{\mathrm {nml}}$ is the graph structure corresponding to $\Gamma _{\mathrm {dpg}}^{\mathrm {nml}}(\mathcal C_i,P_i)$ . Tori in $\mathbb T_{i,P_i}$ exchanged by the operations in Lemma 3.5 are in the $S^1$ -bundle over the Möbius band studied in [Reference Waldhausen38, Section 3], and the tori give reduced graph structure of the $S^1$ -bundle. Hence, there is an isotopy $\mathfrak {h}_{u}\colon \mathscr {B}(\mathcal C_1,P_1)\to \mathscr {B}(\mathcal C_2,P_2)$ ( $2\leq u\leq 3$ ) by Theorem 2.2, which is constant on $\partial \mathscr {B}(\mathcal C_i,P_i)$ and satisfies $\mathfrak {h}_{3}(\mathbb T_{1,P_1})=\mathbb T_{2,P_2}$ .

If $v_{1}\in \overline {\mathcal {A}}(\Gamma _{1,P_1})$ is not on any chain of $\Gamma _{\mathrm {dpg}}(\mathcal C_1,P_1)$ , and $v_2:=\varphi _{P_1}(v_1)\in \overline {\mathcal {A}}(\Gamma _{2,P_2})$ , then, for a fiber $ F_{v_1}$ of $v_1$ , the image $ \mathfrak {h}_{3}(F_{v_1})$ is ambient isotopic in $ M_{v_2}$ to a fiber $ F_{v_2}$ of $v_2$ by Theorem A.6. If $v_1\in \overline {\mathcal {A}}(\Gamma _{1,P_1})$ is on a chain of $\Gamma _{\mathrm {dpg}}(\mathcal C_1,P_1)$ , then the argument of [Reference Neumann28, pp. 322–323] shows that, for a fiber $ F_{v_1}$ of $v_1$ , the image $ \mathfrak {h}_{3}(F_{v_1})$ is homotopic in $ M_{v_2}$ to a fiber $ F_{v_2}$ of $v_2$ (cf. [Reference Laufer24, Chapter II]).

For $y_1\in \mathcal {Y}(\Gamma _{1,P_1})$ with $t(y_1), t(\overline {y}_1)\notin \mathcal {A}(\Gamma _{1,P_1})$ , we can identify $ T_{\overline {y}_1}$ with $ T_{y_1}$ in $\mathscr {B}(\mathcal C_1,P_1)^\circ $ . Let $y_{i,1},\dots ,y_{i,r}\in \mathcal {Y}(\Gamma _{i,P_i})$ be edges such that

$$ \begin{align*} & t(y_{i,j})\in\overline{\mathcal{A}}(\Gamma_{i,P_i}), && y_{i,j'}\notin\{y_{i,j},\overline{y}_{i,j}\} \quad \text{if } j\ne j', \\ & \varphi_P(y_{1,j})=y_{2,j}, && \mathcal{Y}(\Gamma_{i,P_i})=\{y_{i,j}, \overline{y}_{i,j}\mid j=1,\dots,r\}. \end{align*} $$

By Lemma A.4, there is an isotopy $\mathfrak {h}_{u}$ ( $3\leq u\leq 4$ ) from $\mathfrak {h}_{3}$ , which is constant on $\partial \mathscr {B}(\mathcal C_i,P_i)$ and preserves graph structures $\mathbb T_{i,P_i}$ , such that $\mathfrak {h}_{4}$ maps every fiber of $t(\overline {y}_{1,j})$ in $ T_{\overline {y}_{1,j}}$ to a fiber of $t(\overline {y}_{2,j})$ in $ T_{\overline {y}_{2,j}}$ . Since each fiber $F_{i,j}\subset { T_{y_{i,j}}}$ of $t(y_{i,j})$ intersects at just one point with each fiber $ \overline {F}_{i,j}\subset T_{\overline {y}_{i,j}}=T_{y_{i,j}}$ of $t(\overline {y}_{i,j})$ , there is an isotopy $\mathfrak {h}_{u}$ ( $4\leq u\leq 5$ ) from $\mathfrak {h}_{4}$ such that

$$ \begin{align*} & \mathfrak{h}_{u}(\mathbb T_{1,P_1})=\mathbb T_{2,P_2}, && \mathfrak{h}_{u}(\overline{F}_{1,j})=\mathfrak{h}_{4}(\overline{F}_{1,j}) \end{align*} $$

for each fiber $\overline {F}_{1,j}\subset { T_{\overline {y}_{1,j}}}$ of $t(\overline {y}_{1,j})$ and any $4\leq u\leq 5$ , and for each fiber $ F_{1,y}\subset T_{y_1}$ of $ t(y_1)$ with $ y_1\in \mathcal {Y}(\Gamma _{1,P_1})$ , $ \mathfrak {h}_5(F_{1,y_1})$ is a fiber of $ t(\varphi _{P_1}(y_1))$ . By Lemmas A.5 and A.8, Theorem A.6, and Remark A.7, there is an isotopy $\mathfrak {h}_{u}$ ( $5\leq u\leq 6$ ) from $\mathfrak {h}_{5}$ such that

$$ \begin{align*} & \mathfrak{h}_{u}|_{\mathbb T_{1,P_1}}=\mathfrak{h}_{5}|_{\mathbb T_{1,P_1}}, && \mathfrak{h}_{u}|_{\partial\mathscr{B}(\mathcal C_1,P_1)}=\mathfrak{h}_{5}|_{\partial\mathscr{B}(\mathcal C_1,P_1)} \end{align*} $$

for any $5\leq u\leq 6$ , and $\mathfrak {h}_{6}$ maps every fiber of all $v_1\in \overline {\mathcal {A}}(\Gamma _{1,P_1})$ to that of $\varphi _{P_1}(v_1)$ . Put $\mathfrak {g}_{P_1}:=\mathfrak {h}_{6}$ , which satisfies (i)–(iii) and (v). Furthermore, it follows from [Reference Artal Bartolo, Bannai, Shirane and Tokunaga3, Remark 3.5] that $\mathfrak {g}_{P_1}$ satisfies (iv).

By identifying $ Y_i\setminus \mathcal C_i'$ with ${\mathbb {P}}^2\setminus \mathcal C_i$ , let $ \mathfrak {m}_{v_i}\in \pi _1({\mathbb {P}}^2\setminus \mathcal C_i)$ be a meridian of $ D^{\prime }_{v_i}\subset \mathcal C_i'$ for $v_i\in \mathcal {V}(\Gamma _i)$ . Theorem 1.2 follows from the next theorem.