(-15:1) .. controls +(80:0.2) and (lı)(-140:0.2) .. controls +(40:0.2) and (eı)(-75:0.1);
[white] (lı) circle (0.1);
(-10:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(140:0.1) and (iı)(-75:0.1);

[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);
[white!40!gray] (lı) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(-110:0.1) and (iı)(105:0.1) .. controls +(-75:0.1) and (lı)(140:0.1);

[purple] (10:1.25) arc (10:260:1.25);

[yshift = -1.8cm, xshift = 0.5 cm]

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

(40:1.5) arc (40:350:1.5);
(50:1) arc (50:345:1);
(iı) .. controls +(105:0.1) and (hı)(-110:0.1) .. controls +(70:0.1) and (40:1.5)(-50:0.2);
[white] (hı) circle (0.1);
(eı) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(170:0.2) and (50:1)(-50:0.2);
(-15:1) .. controls +(80:0.2) and (lı)(-140:0.2) .. controls +(40:0.2) and (eı)(-75:0.1);
[white] (lı) circle (0.1);
(-10:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(140:0.1) and (iı)(-75:0.1);

[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);
[white!40!gray] (lı) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(-110:0.1) and (iı)(105:0.1) .. controls +(-75:0.1) and (lı)(140:0.1);

(0,0) rectangle (1.5,-1.5);
[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);

[purple] (10:1.25) arc (10:-100:1.25);
[green!80!purple] ($(10:1.25)+(0.5,-1.8)$) arc (10:260:1.25);

[yshift = -3.6cm, xshift = 1 cm]

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

(40:1.5) arc (40:350:1.5);
(50:1) arc (50:345:1);
(iı) .. controls +(105:0.1) and (hı)(-110:0.1) .. controls +(70:0.1) and (40:1.5)(-50:0.2);
[white] (hı) circle (0.1);
(eı) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(170:0.2) and (50:1)(-50:0.2);
(-15:1) .. controls +(80:0.2) and (lı)(-140:0.2) .. controls +(40:0.2) and (eı)(-75:0.1);
[white] (lı) circle (0.1);
(-10:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(140:0.1) and (iı)(-75:0.1);

[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);
[white!40!gray] (lı) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(-110:0.1) and (iı)(105:0.1) .. controls +(-75:0.1) and (lı)(140:0.1);

[yshift = -1.8cm, xshift = 0.5 cm]
(0,0) rectangle (1.5,-1.5);
[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);

[green!80!purple] ($(10:1.25)+(0.5,-1.8)$) arc (10:-100:1.25);
[orange] (1,-3.6) circle (1.25);

[->] (2.5,-1) – (3.4,-1);
(4,0) – (4,-2);
[purple] (4,0) circle (0.1cm);
[green!80!purple] (4,-1) circle (0.1cm);
[orange] (4,-2) circle (0.1cm);

[xshift = 8cm]

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

(40:1.5) arc (40:350:1.5);
(50:1) arc (50:345:1);
(iı) .. controls +(105:0.1) and (hı)(-110:0.1) .. controls +(70:0.1) and (40:1.5)(-50:0.2);
[white] (hı) circle (0.1);
(eı) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(170:0.2) and (50:1)(-50:0.2);
(-15:1) .. controls +(80:0.2) and (lı)(-140:0.2) .. controls +(40:0.2) and (eı)(-75:0.1);
[white] (lı) circle (0.1);
(-10:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(140:0.1) and (iı)(-75:0.1);

[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);
[white!40!gray] (lı) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(-110:0.1) and (iı)(105:0.1) .. controls +(-75:0.1) and (lı)(140:0.1);

[purple] (10:1.25) arc (10:260:1.25);

[yshift = -1.8cm, xshift = 0.5 cm]

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

(40:1.5) arc (40:350:1.5);
(50:1) arc (50:345:1);
(iı) .. controls +(105:0.1) and (hı)(-110:0.1) .. controls +(70:0.1) and (40:1.5)(-50:0.2);
[white] (hı) circle (0.1);
(eı) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(170:0.2) and (50:1)(-50:0.2);
(-15:1) .. controls +(80:0.2) and (lı)(-140:0.2) .. controls +(40:0.2) and (eı)(-75:0.1);
[white] (lı) circle (0.1);
(-10:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(140:0.1) and (iı)(-75:0.1);

[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);
[white!40!gray] (lı) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(-110:0.1) and (iı)(105:0.1) .. controls +(-75:0.1) and (lı)(140:0.1);

(0,0) rectangle (1.5,-1.5);
[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);

[purple] (10:1.25) arc (10:-25:1.25) .. controls +(-115:0.2) and ($(60:1.15)+(0.5,-1.8)$)(150:0.2);
[green!80!purple] ($(10:1.25)+(0.5,-1.8)$) arc (10:260:1.25);

[purple] ($(-90:1.15)+(0.5,-1.8)$) arc (-90:-270:1.15) .. controls +(0:0.3) and (-60:1.25)(30:0.3) arc (-60:-100:1.25);

[yshift = -3.6cm, xshift = 1 cm]

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

(40:1.5) arc (40:350:1.5);
(50:1) arc (50:345:1);
(iı) .. controls +(105:0.1) and (hı)(-110:0.1) .. controls +(70:0.1) and (40:1.5)(-50:0.2);
[white] (hı) circle (0.1);
(eı) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(170:0.2) and (50:1)(-50:0.2);
(-15:1) .. controls +(80:0.2) and (lı)(-140:0.2) .. controls +(40:0.2) and (eı)(-75:0.1);
[white] (lı) circle (0.1);
(-10:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(140:0.1) and (iı)(-75:0.1);

[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);
[white!40!gray] (lı) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(-110:0.1) and (iı)(105:0.1) .. controls +(-75:0.1) and (lı)(140:0.1);

[yshift = -1.8cm, xshift = 0.5 cm]
(0,0) rectangle (1.5,-1.5);
[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);
[purple] (-20:1.15) arc (-20:-100:1.15);

[yshift = -1.8cm, xshift = 0.5 cm]
(0,0) – (-30:1) arc (-30:-15:1) .. controls +(80:0.2) and (l2)(-140:0.2) – (h2) .. controls +(170:0.2) and (50:1)(-50:0.2) arc (50:90:1) – ++(90:0.5) – ++(0:1.5) – ++(0,-3) – ++(-1.5,0) – cycle;
[purple] (-20:1.15) arc (-20:60:1.15);

[yshift = -1.8cm, xshift = 0.5 cm]
(0,0) – (l2) .. controls +(40:0.2) and (e2)(-75:0.1) .. controls +(105:0.1) and (h1)(-10:0.2) – cycle;
[purple] (0:1.35) arc (0:25:1.35);

[green!80!purple] ($(10:1.25)+(0.5,-1.8)$) arc (10:-100:1.25);
[orange] (1,-3.6) circle (1.25);

[->] (2.5,-1) – (3.4,-1);
(4,-0.5) – +(-120:1) – +(-60:1) – cycle;
[purple] (4,-0.5) circle (0.1cm);
[green!80!purple] ($(4,-0.5)+(-120:1)$) circle (0.1cm);
[orange] ($(4,-0.5)+(-60:1)$) circle (0.1cm);

A Hopf arborescent plumbing with a cyclic plumbing graph.

§.§ Minors on surfaces, links, and plane trees

Since we focus our investigation on Hopf arborescent links, we define a stronger notion of minor that is well-tailored to these links. We say that a Hopf arborescent link $L_1$ is a link-minor of $L_2$ if there exists $T_1$ and $T_2$, two labelled plane trees such that $\Sigma(T_1)$ and $\Sigma(T_2)$ are canonical Seifert surfaces of $L_1$ and $L_2$ respectively and $T_1 \hookrightarrow T_2$. The main result of this section is the following one, establishing that if $L_1$ is a link-minor of $L_2$ then the Seifert surface of $L_1$ is a surface-minor of the Seifert surface of $L_2$.

Let $T_1$ and $T_2$ be two plane trees such that $T_1$ admits a homeomorphic embedding into $T_2$. Then the Hopf arborescent surface $\Sigma (T_1)$ is an incompressible subsurface of $\Sigma (T_2)$.

The proof relies on a series of lemmas that correspond to the different operations defining homeomorphic embeddings of trees.
The first one states that cutting a Hopf arborescent surface $\Sigma$ along an arc leaning on $\partial \Sigma$ yields an incompressible subsurface of $\Sigma$.

Let $\Sigma$ be surface and $\gamma$ be an arc that is not boundary-parallel in $\Sigma$ with both extremities in $\partial \Sigma$. Then cutting $\Sigma$ along $\gamma$ yields a surface $\Sigma' = \overline{\Sigma \smallsetminus \gamma}$ such that $\Sigma' \preccurlyeq \Sigma$.

By definition of $\Sigma'$, there is a natural map $h: \Sigma' \rightarrow \Sigma$ that is injective except on $h^{-1} (\gamma) = \gamma_1 \cup \gamma_2$. Let $T$ be a tubular neighbourhood of $\gamma$ in $\Sigma$. Its boundary can be decomposed into $t_1,t_2$, two arcs isotopic to $\gamma$ in $\Sigma$ and two open arcs of $\partial \Sigma$. Isotope $h$ within $\Sigma$ so that $h(\gamma_1) = t_1$, $h(\gamma_2) = t_2$, and $h(\Sigma') \cap T = t_1 \cup t_2$. It follows that $h(\Sigma')$ is a subsurface of $\Sigma$ such that $\Sigma \smallsetminus \Sigma'$ is not an open disc since $\gamma$ is not boundary-parallel.

Lemma <ref> essentially states that our surfaces behave well with respect to the surface-minor relation when cut along almost any arc.
An important point is that cutting along an arc that is the diagonal of a plumbing rectangle merges two bands into one new band with two extra crossings that are either negative or positive depending on the diagonal, see Figure <ref>.
So, cutting the plumbing of two positive Hopf bands along the diagonal that produces two negative crossings yields a positive Hopf band.
Symmetrically, one can merge two negative Hopf bands into one negative by cutting along the other diagonal.
Furthermore, when having a plumbing of two bands with opposite signs, one can merge them into a band with either sign depending on which cut one chooses.

[scale = 0.9]

(-1.6,1.6) rectangle (13.6,-3.6);

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (lı) circle (0.1);
[thick] (50:1) arc (50:345:1) .. controls +(80:0.2) and (lı)(-140:0.2) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(170:0.2) and (50:1)(-50:0.2);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (hı) circle (0.1);
[thick] (40:1.5) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(140:0.1) and (iı)(-75:0.1) .. controls +(105:0.1) and (hı)(-110:0.1) .. controls +(70:0.1) and (40:1.5)(-50:0.2);

[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);
[white!40!gray] (lı) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(-110:0.1) and (iı)(105:0.1) .. controls +(-75:0.1) and (lı)(140:0.1);

[yshift = -1.8cm, xshift = 0.5 cm]

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (lı) circle (0.1);
[thick] (50:1) arc (50:345:1) .. controls +(80:0.2) and (lı)(-140:0.2) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(170:0.2) and (50:1)(-50:0.2);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (hı) circle (0.1);
[thick] (40:1.5) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(140:0.1) and (iı)(-75:0.1) .. controls +(105:0.1) and (hı)(-110:0.1) .. controls +(70:0.1) and (40:1.5)(-50:0.2);

[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);
[white!40!gray] (lı) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(-110:0.1) and (iı)(105:0.1) .. controls +(-75:0.1) and (lı)(140:0.1);

(0,0) rectangle (1.5,-1.5);
[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);

[name path=ci1] (-90:1) arc (-90:0:1);
[name path=ce1] (-90:1.5) arc (-90:0:1.5);
[name path=ci2, yshift = -1.8cm, xshift = 0.5 cm] (90:1) arc (90:0:1);
[name path=ce2, yshift = -1.8cm, xshift = 0.5 cm] (90:1.5) arc (90:0:1.5);
[name intersections=of=ci1 and ce2,total=]
in 1,...,coordinate (h) at (intersection-);
[name intersections=of=ce1 and ci2,total=]
in 1,...,coordinate (l) at (intersection-);
[thick, brown] (h) .. controls +(-30:0.05) and (l)(30:0.05);
at (h) [above left] $\gamma_1$;

[->, thick] (2,-0.5) – (4.75,-0.5) node [above, midway] Cut along $\gamma_1$;

[xshift = 6.5cm]

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (lı) circle (0.1);
[thick] (50:1) arc (50:345:1) .. controls +(80:0.2) and (lı)(-140:0.2) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(170:0.2) and (50:1)(-50:0.2);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (hı) circle (0.1);
[thick] (40:1.5) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(140:0.1) and (iı)(-75:0.1) .. controls +(105:0.1) and (hı)(-110:0.1) .. controls +(70:0.1) and (40:1.5)(-50:0.2);

[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);
[white!40!gray] (lı) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(-110:0.1) and (iı)(105:0.1) .. controls +(-75:0.1) and (lı)(140:0.1);

[yshift = -1.8cm, xshift = 0.5 cm]

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (lı) circle (0.1);
[thick] (50:1) arc (50:345:1) .. controls +(80:0.2) and (lı)(-140:0.2) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(170:0.2) and (50:1)(-50:0.2);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (hı) circle (0.1);
[thick] (40:1.5) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(140:0.1) and (iı)(-75:0.1) .. controls +(105:0.1) and (hı)(-110:0.1) .. controls +(70:0.1) and (40:1.5)(-50:0.2);

[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);
[white!40!gray] (lı) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(-110:0.1) and (iı)(105:0.1) .. controls +(-75:0.1) and (lı)(140:0.1);

(0,0) rectangle (1.5,-1.5);
[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);

[white] (-18:1) .. controls +(-30:0.07) and ($(50:1)+(0.5,-1.8)$)(30:0.07) –  (-45:1.5) .. controls +(30:0.07) and ($(75:1.5)+(0.5,-1.8)$)(-30:0.07) – cycle;
(-18:1) .. controls +(-30:0.07) and ($(50:1)+(0.5,-1.8)$)(30:0.07);
($(75:1.5)+(0.5,-1.8)$) .. controls +(-30:0.07) and (-45:1.5)(30:0.07);

at (8.85,-0.5) $=$;

[xshift = 11cm]

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);
[yshift = -1.8cm, xshift = 0.5 cm]
(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

[white!40!gray] (l2) .. controls +(40:0.2) and (e2)(-75:0.1) .. controls +(105:0.1) and (h2)(-10:0.2) .. controls +(-110:0.1) and (i2)(105:0.1) .. controls +(-75:0.1) and (l2)(140:0.1);
[white!40!gray] (l1) .. controls +(40:0.2) and (e1)(-75:0.1) .. controls +(105:0.1) and (h1)(-10:0.2) .. controls +(-110:0.1) and (i1)(105:0.1) .. controls +(-75:0.1) and (l1)(140:0.1);
[blue!20!white] (h2) .. controls +(70:0.1) and ($(0.5,-1.8)+(40:1.5)$)(-50:0.2) .. controls +(130:0.4) and (-10:1.5)(-100:0.4) .. controls +(80:0.1) and (l1)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) .. controls +(-115:0.35) and ($(0.5,-1.8)+(50:1)$)(140:0.35) .. controls +(-50:0.2) and (h2)(-110:0.1);
[blue!20!white] (h1) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:180:1.5) .. controls +(-90:0.5) and (220:1.25)(170:0.3) .. controls +(50:0.3) and (180:1)(-90:0.5) arc (180:50:1) .. controls +(-40:1) and (50:1)(140:0.5) .. controls +(-50:0.2) and (h1)(170:0.2);
[blue!20!white] ($(160:1.25)+(0.5,-1.8)$) .. controls +(-170:0.3) and ($(0.5,-1.8)+(180:1.5)$)(90:0.1) arc (180:350:1.5) .. controls +(80:0.1) and (l2)(-40:0.2) .. controls +(-140:0.2) and ($(0.5,-1.8)+(-15:1)$)(80:0.2) arc (-15:-180:1) .. controls +(90:0.1) and ($(160:1.25)+(0.5,-1.8)$)(-30:0.3);
[white!40!gray] ($(160:1.25)+(0.5,-1.8)$) .. controls +(10:0.2) and (220:1.25)(-10:0.6) .. controls +(-140:0.6) and ($(160:1.25)+(0.5,-1.8)$)(150:0.25);

[even odd rule]

(-1.6,1.6) rectangle (2.1,-4) (220:1.25) circle (0.1) (h2) circle (0.1) (h1) circle (0.1);
(h2) .. controls +(70:0.1) and ($(0.5,-1.8)+(40:1.5)$)(-50:0.2) .. controls +(130:0.4) and (-10:1.5)(-100:0.4) .. controls +(80:0.1) and (l1)(-40:0.2) .. controls +(140:0.1) and (i1)(-75:0.1) .. controls +(105:0.1) and (h1)(-110:0.2) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:180:1.5) .. controls +(-90:0.5) and (220:1.25)(170:0.3) .. controls + (-10:0.6) and ($(160:1.25)+(0.5,-1.8)$)(10:0.2) .. controls +(-170:0.3) and ($(0.5,-1.8)+(180:1.5)$)(90:0.1) arc (180:350:1.5) .. controls +(80:0.1) and (l2)(-40:0.2) .. controls +(140:0.1) and (i2)(-75:0.1) .. controls +(105:0.1) and (h2)(-110:0.2);

[even odd rule]
(-1.6,1.6) rectangle (2.1,-4) ($(160:1.25)+(0.5,-1.8)$) circle (0.1) (l1) circle (0.1) (l2) circle (0.1);
(h2) .. controls +(-10:0.2) and (e2)(105:0.1) .. controls +(-75:0.1) and (l2)(40:0.2) .. controls +(-140:0.2) and ($(0.5,-1.8)+(-15:1)$)(80:0.2) arc (-15:-180:1) .. controls +(90:0.1) and ($(160:1.25)+(0.5,-1.8)$)(-30:0.3) .. controls +(150:0.25) and (220:1.25)(-140:0.6) .. controls +(50:0.3) and (180:1)(-90:0.5) arc (180:50:1) .. controls +(-50:0.2) and (h1)(170:0.2).. controls +(-10:0.2) and (e1)(105:0.1) .. controls +(-75:0.1) and (l1)(40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) .. controls +(-115:0.35) and ($(0.5,-1.8)+(50:1)$)(140:0.35) .. controls +(-50:0.2) and (h2)(-110:0.1);

[yshift = -8.5cm]
(-1.6,1.6) rectangle (13.6,-3.6);

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (lı) circle (0.1);
[thick] (50:1) arc (50:345:1) .. controls +(80:0.2) and (lı)(-140:0.2) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(170:0.2) and (50:1)(-50:0.2);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (hı) circle (0.1);
[thick] (40:1.5) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(140:0.1) and (iı)(-75:0.1) .. controls +(105:0.1) and (hı)(-110:0.1) .. controls +(70:0.1) and (40:1.5)(-50:0.2);

[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);
[white!40!gray] (lı) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(-110:0.1) and (iı)(105:0.1) .. controls +(-75:0.1) and (lı)(140:0.1);

[yshift = -1.8cm, xshift = 0.5 cm]

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (lı) circle (0.1);
[thick] (50:1) arc (50:345:1) .. controls +(80:0.2) and (lı)(-140:0.2) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(170:0.2) and (50:1)(-50:0.2);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (hı) circle (0.1);
[thick] (40:1.5) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(140:0.1) and (iı)(-75:0.1) .. controls +(105:0.1) and (hı)(-110:0.1) .. controls +(70:0.1) and (40:1.5)(-50:0.2);

[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);
[white!40!gray] (lı) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(-110:0.1) and (iı)(105:0.1) .. controls +(-75:0.1) and (lı)(140:0.1);

(0,0) rectangle (1.5,-1.5);
[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);

[yshift = -1.8cm, xshift = 0.5 cm]
(0,0) rectangle (1.5,-1.5);
[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);

[name path=ci1] (-90:1) arc (-90:0:1);
[name path=ce1] (-90:1.5) arc (-90:0:1.5);
[name path=ci2, yshift = -1.8cm, xshift = 0.5 cm] (90:1) arc (90:0:1);
[name path=ce2, yshift = -1.8cm, xshift = 0.5 cm] (90:1.5) arc (90:0:1.5);
[name intersections=of=ci1 and ci2,total=]
in 1,...,coordinate (h) at (intersection-);
[name intersections=of=ce1 and ce2,total=]
in 1,...,coordinate (l) at (intersection-);
[thick, brown] (h) – (l);
at (l) [right] $\gamma_2$;

[->, thick] (2,-0.5) – (4.75,-0.5) node [above, midway] Cut along $\gamma_2$;

[xshift = 6.5cm]

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (lı) circle (0.1);
[thick] (50:1) arc (50:345:1) .. controls +(80:0.2) and (lı)(-140:0.2) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(170:0.2) and (50:1)(-50:0.2);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (hı) circle (0.1);
[thick] (40:1.5) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(140:0.1) and (iı)(-75:0.1) .. controls +(105:0.1) and (hı)(-110:0.1) .. controls +(70:0.1) and (40:1.5)(-50:0.2);

[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);
[white!40!gray] (lı) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(-110:0.1) and (iı)(105:0.1) .. controls +(-75:0.1) and (lı)(140:0.1);

[yshift = -1.8cm, xshift = 0.5 cm]

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (lı) circle (0.1);
[thick] (50:1) arc (50:345:1) .. controls +(80:0.2) and (lı)(-140:0.2) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(170:0.2) and (50:1)(-50:0.2);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (hı) circle (0.1);
[thick] (40:1.5) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(140:0.1) and (iı)(-75:0.1) .. controls +(105:0.1) and (hı)(-110:0.1) .. controls +(70:0.1) and (40:1.5)(-50:0.2);

[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);
[white!40!gray] (lı) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(-110:0.1) and (iı)(105:0.1) .. controls +(-75:0.1) and (lı)(140:0.1);

(0.5,-0.3) rectangle (1.5,-1.05);
[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);

[yshift = -1.8cm, xshift = 0.5 cm]
(0,0) rectangle (1.5,-1.5);
[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);

[white] ($(90:1)+(0.5,-1.8)$) .. controls +(0:0.05) and (-18:1.5)(-118:0.1) – ($(48:1.5)+(0.5,-1.8)$) .. controls +(138:0.1) and (-80:1)(10:0.1) – cycle;
($(90:1)+(0.5,-1.8)$) .. controls +(0:0.05) and (-18:1.5)(-118:0.1);
($(48:1.5)+(0.5,-1.8)$) .. controls +(138:0.1) and (-80:1)(10:0.1);

at (8.85,-0.5) $=$;

[xshift = 11cm]

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);
[yshift = -1.8cm, xshift = 0.5 cm]
(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

[white!40!gray] (l2) .. controls +(40:0.2) and (e2)(-75:0.1) .. controls +(105:0.1) and (h2)(-10:0.2) .. controls +(-110:0.1) and (i2)(105:0.1) .. controls +(-75:0.1) and (l2)(140:0.1);
[white!40!gray] (l1) .. controls +(40:0.2) and (e1)(-75:0.1) .. controls +(105:0.1) and (h1)(-10:0.2) .. controls +(-110:0.1) and (i1)(105:0.1) .. controls +(-75:0.1) and (l1)(140:0.1);
[blue!20!white] (h2) .. controls +(70:0.1) and ($(0.5,-1.8)+(40:1.5)$)(-50:0.2) .. controls +(130:0.4) and (-10:1.5)(-100:0.4) .. controls +(80:0.1) and (l1)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) .. controls +(-115:0.35) and ($(0.5,-1.8)+(50:1)$)(140:0.35) .. controls +(-50:0.2) and (h2)(-110:0.1);
[blue!20!white] (h1) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:180:1.5) .. controls +(-90:0.5) and (220:1.25)(170:0.3) .. controls +(50:0.3) and (180:1)(-90:0.5) arc (180:50:1) .. controls +(-40:1) and (50:1)(140:0.5) .. controls +(-50:0.2) and (h1)(170:0.2);
[blue!20!white] ($(160:1.25)+(0.5,-1.8)$) .. controls +(-170:0.3) and ($(0.5,-1.8)+(180:1.5)$)(90:0.1) arc (180:350:1.5) .. controls +(80:0.1) and (l2)(-40:0.2) .. controls +(-140:0.2) and ($(0.5,-1.8)+(-15:1)$)(80:0.2) arc (-15:-180:1) .. controls +(90:0.1) and ($(160:1.25)+(0.5,-1.8)$)(-30:0.3);
[white!40!gray] ($(160:1.25)+(0.5,-1.8)$) .. controls +(10:0.2) and (220:1.25)(-10:0.6) .. controls +(-140:0.6) and ($(160:1.25)+(0.5,-1.8)$)(150:0.25);

[even odd rule]
(-1.6,1.6) rectangle (2.1,-4) ($(160:1.25)+(0.5,-1.8)$) circle (0.1) (h1) circle (0.1) (h2) circle (0.1);
(h2) .. controls +(70:0.1) and ($(0.5,-1.8)+(40:1.5)$)(-50:0.2) .. controls +(130:0.4) and (-10:1.5)(-100:0.4) .. controls +(80:0.1) and (l1)(-40:0.2) .. controls +(140:0.1) and (i1)(-75:0.1) .. controls +(105:0.1) and (h1)(-110:0.2) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:180:1.5) .. controls +(-90:0.5) and (220:1.25)(170:0.3) .. controls + (-10:0.6) and ($(160:1.25)+(0.5,-1.8)$)(10:0.2) .. controls +(-170:0.3) and ($(0.5,-1.8)+(180:1.5)$)(90:0.1) arc (180:350:1.5) .. controls +(80:0.1) and (l2)(-40:0.2) .. controls +(140:0.1) and (i2)(-75:0.1) .. controls +(105:0.1) and (h2)(-110:0.2);

[even odd rule]
(-1.6,1.6) rectangle (2.1,-4) (220:1.25) circle (0.1) (l1) circle (0.1) (l2) circle (0.1);
(h2) .. controls +(-10:0.2) and (e2)(105:0.1) .. controls +(-75:0.1) and (l2)(40:0.2) .. controls +(-140:0.2) and ($(0.5,-1.8)+(-15:1)$)(80:0.2) arc (-15:-180:1) .. controls +(90:0.1) and ($(160:1.25)+(0.5,-1.8)$)(-30:0.3) .. controls +(150:0.25) and (220:1.25)(-140:0.6) .. controls +(50:0.3) and (180:1)(-90:0.5) arc (180:50:1) .. controls +(-50:0.2) and (h1)(170:0.2).. controls +(-10:0.2) and (e1)(105:0.1) .. controls +(-75:0.1) and (l1)(40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) .. controls +(-115:0.35) and ($(0.5,-1.8)+(50:1)$)(140:0.35) .. controls +(-50:0.2) and (h2)(-110:0.1);

[xshift = 4cm, yshift = -6cm, scale = 1.2]

(blı) at (0,0);
(tlı) at ($(bl\i)+(0,\l)$);
(mnlı) at ($(bl\i)+0.5*(0,\l)+(-\esp,0)$);
(brı) at (Ł,0);
(trı) at ($(br\i)+(0,\l)$);
(mnrı) at ($(mnl\i)+(\L,0)$);
(mlı) at ($(bl\i)+0.5*(0,\l)+(\esp,0)$);
(qblı) at ($(bl\i)+0.25*(0,\l)+(\esp,0)$);
(qtlı) at ($(bl\i)+0.75*(0,\l)+(\esp,0)$);
(qbrı) at ($(qbl\i)+(\L,0)$);
(mrı) at ($(ml\i)+(\L,0)$);
(qtrı) at ($(qtl\i)+(\L,0)$);
(qlı) at ($(bl\i)+0.4*(0,\l)+0.5*(\L,0)+(\esp,0)$);
(qhı) at ($(bl\i)+0.6*(0,\l)+0.5*(\L,0)+(\esp,0)$);

[white!40!gray] (blı) .. controls +(0:) and (qblı)(-90:0.5*) .. controls +(90:0.5*) and (qlı)(-150:)̧ .. controls +(-30:)̧ and (qbrı)(90:0.5*)  .. controls +(-90:0.5*) and (brı)(0:) – cycle;
[white!40!gray] (tlı) .. controls +(0:) and (qtlı)(90:0.5*) .. controls +(-90:0.5*) and (qhı)(150:)̧ .. controls +(30:)̧ and (qtrı)(-90:0.5*)  .. controls +(90:0.5*) and (trı)(0:) – cycle;
[blue!20!white] (qlı) .. controls +(150:)̧ and (mlı)(-90:0.25*) .. controls +(90:0.25*) and (qhı)(-150:)̧ .. controls +(-30:)̧ and (mrı)(90:0.25*) .. controls +(-90:0.25*) and (qlı)(30:)̧;
[even odd rule]
(-3,-3) rectangle (3,3) (qlı) circle (0.75*çm);
(blı) .. controls +(0:) and (qblı)(-90:0.5*) .. controls +(90:0.5*) and (qlı)(-150:)̧ .. controls +(30:)̧ and (mrı)(-90:0.25*) .. controls +(90:0.25*) and (qhı)(-30:)̧ .. controls +(150:)̧ and (qtlı)(-90:0.5*) .. controls +(90:0.5*) and (tlı)(0:);

[even odd rule]
(-3,-3) rectangle (3,3) (qhı) circle (0.75*çm);
(brı) .. controls +(0:) and (qbrı)(-90:0.5*) .. controls +(90:0.5*) and (qlı)(-30:)̧ .. controls +(150:)̧ and (mlı)(-90:0.25*) .. controls +(90:0.25*) and (qhı)(-150:)̧ .. controls +(30:)̧ and (qtrı)(-90:0.5*) .. controls +(90:0.5*) and (trı)(0:);

[blue!20!white] (blı) .. controls +(180:) and (mnlı)(-90:2*) .. controls +(90:2*) and (tlı)(180:) – (trı) .. controls +(180:) and (mnrı)(90:2*) .. controls +(-90:2*) and (brı)(180:);
[thick] (blı) .. controls +(180:) and (mnlı)(-90:2*) node (0) [pos = 0.355, inner sep=0]  node (1) [pos = 0.565, inner sep=0]  .. controls +(90:2*) and (tlı)(180:) ;
[thick] (brı) .. controls +(180:) and (mnrı)(-90:2*) node (2) [pos = 0.36, inner sep=0]  node (3) [pos = 0.57, inner sep=0]  .. controls +(90:2*) and (trı)(180:);
ıin 0,1,2,3(nı) at (ı);;

[xshift = -1.25cm, yshift = 0.2cm, rotate = -90]

(blı) at (0,0);
(tlı) at ($(bl\i)+(0,\l)$);
(mnlı) at ($(bl\i)+0.5*(0,\l)+(-\esp,0)$);
(brı) at (Ł,0);
(trı) at ($(br\i)+(0,\l)$);
(mnrı) at ($(mnl\i)+(\L,0)$);
(mlı) at ($(bl\i)+0.5*(0,\l)+(\esp,0)$);
(qblı) at ($(bl\i)+0.25*(0,\l)+(\esp,0)$);
(qtlı) at ($(bl\i)+0.75*(0,\l)+(\esp,0)$);
(qbrı) at ($(qbl\i)+(\L,0)$);
(mrı) at ($(ml\i)+(\L,0)$);
(qtrı) at ($(qtl\i)+(\L,0)$);
(qlı) at ($(bl\i)+0.4*(0,\l)+0.5*(\L,0)+(\esp,0)$);
(qhı) at ($(bl\i)+0.6*(0,\l)+0.5*(\L,0)+(\esp,0)$);

[blue!20!white] (blı) .. controls +(180:) and (mnlı)(-90:2*) .. controls +(90:2*) and (tlı)(180:) – (trı) .. controls +(180:) and (mnrı)(90:2*) .. controls +(-90:2*) and (brı)(180:);
(blı) .. controls +(180:) and (mnlı)(-90:2*) .. controls +(90:2*) and (tlı)(180:);
(brı) .. controls +(180:) and (mnrı)(-90:2*) .. controls +(90:2*) and (trı)(180:);

[white!40!gray] (blı) .. controls +(0:) and (qblı)(-90:0.5*) .. controls +(90:0.5*) and (qlı)(-150:)̧ .. controls +(-30:)̧ and (qbrı)(90:0.5*)  .. controls +(-90:0.5*) and (brı)(0:) – cycle;
[white!40!gray] (tlı) .. controls +(0:) and (qtlı)(90:0.5*) .. controls +(-90:0.5*) and (qhı)(150:)̧ .. controls +(30:)̧ and (qtrı)(-90:0.5*)  .. controls +(90:0.5*) and (trı)(0:) – cycle;
[blue!20!white] (qlı) .. controls +(150:)̧ and (mlı)(-90:0.25*) .. controls +(90:0.25*) and (qhı)(-150:)̧ .. controls +(-30:)̧ and (mrı)(90:0.25*) .. controls +(-90:0.25*) and (qlı)(30:)̧;
[even odd rule]

(blı) .. controls +(0:) and (qblı)(-90:0.5*) .. controls +(90:0.5*) and (qlı)(-150:)̧;
(blı) .. controls +(0:) and (qblı)(-90:0.5*) .. controls +(90:0.5*) and (qlı)(-150:)̧ .. controls +(30:)̧ and (mrı)(-90:0.25*) .. controls +(90:0.25*) and (qhı)(-30:)̧ .. controls +(150:)̧ and (qtlı)(-90:0.5*) .. controls +(90:0.5*) and (tlı)(0:);

[even odd rule]

(brı) .. controls +(0:) and (qbrı)(-90:0.5*) .. controls +(90:0.5*) and (qlı)(-30:)̧;
(brı) .. controls +(0:) and (qbrı)(-90:0.5*) .. controls +(90:0.5*) and (qlı)(-30:)̧ .. controls +(150:)̧ and (mlı)(-90:0.25*) .. controls +(90:0.25*) and (qhı)(-150:)̧ .. controls +(30:)̧ and (qtrı)(-90:0.5*) .. controls +(90:0.5*) and (trı)(0:);

[blue!20!white] (blı) .. controls +(180:) and (mnlı)(-90:2*) .. controls +(90:2*) and (tlı)(180:) – (trı) .. controls +(180:) and (mnrı)(90:2*) .. controls +(-90:2*) and (brı)(180:);

[brown, thick] (n0) – (n3) (n1) – (n2);
at (n0) [below left] $\gamma_2$;
at (n1) [above left] $\gamma_1$;

(2.45,-5) – (0,-5);
[->] (0,-5) .. controls +(180:1) and (-0.25,-3.25)(-145:1);
[->] (0,-5) .. controls +(180:1) and (-0.5,-6.9)(145:0.4);
at (1.125,-4.5) [align = center] flat

[xshift = 6.5cm, yshift = -4.5cm]
(0,1) .. controls +(-70:1) and (1,0.5)(-90:0.35);
(0.2,1) .. controls +(-70:0.75) and (0.8,0.5)(-90:0.15);
[white, line width = 0.1cm] (0.8,0.5) .. controls +(90:0.15) and (0.2,0)(70:0.75);
(0.8,0.5) .. controls +(90:0.15) and (0.2,0)(70:0.75);
[white, line width = 0.1cm] (1,0.5) .. controls +(90:0.35) and (0,0)(70:1);
(1,0.5) .. controls +(90:0.35) and (0,0)(70:1);

[->] (1.2,0.5) – (1.8,0.5);
at (1.5,0.5) [above] R1;

[xshift = 2cm]
(0,1) .. controls +(-70:1) and (1,0.5)(-90:0.35);
(0.2,1) .. controls +(-70:0.5) and (0.5,0.5)(90:0.1);
[white, line width = 0.1cm] (0.5,0.5) .. controls +(-90:0.1) and (0.2,0)(70:0.5);
(0.5,0.5) .. controls +(-90:0.1) and (0.2,0)(70:0.5);
[white, line width = 0.1cm] (1,0.5) .. controls +(90:0.35) and (0,0)(70:1);
(1,0.5) .. controls +(90:0.35) and (0,0)(70:1);

[->] (1.2,0.5) – (1.8,0.5);
at (1.5,0.5) [above] R3;

[xshift = 4cm]
(0,1) .. controls +(-70:1) and (1,0.5)(-90:0.35);
[white, line width = 0.1cm] (0.2,1) .. controls +(-90:0.35) and (0,0.5)(90:0.1);
(0.2,1) .. controls +(-90:0.35) and (0,0.5)(90:0.1);
(0,0.5) .. controls +(-90:0.1) and (0.2,0)(90:0.35);
[white, line width = 0.1cm] (1,0.5) .. controls +(90:0.35) and (0,0)(70:1);
(1,0.5) .. controls +(90:0.35) and (0,0)(70:1);

[->] (1.2,0.5) – (1.8,0.5);
at (1.5,0.5) [above] R1;

[xshift = 6cm]
(0,1) .. controls +(-90:0.35) and (0.2,0.5)(90:0.1);
[white, line width = 0.1cm] (0.2,1) .. controls +(-90:0.35) and (0,0.5)(90:0.1);
(0.2,1) .. controls +(-90:0.35) and (0,0.5)(90:0.1);
(0,0.5) .. controls +(-90:0.1) and (0.2,0)(90:0.35);
[white, line width = 0.1cm] (0,0) .. controls +(90:0.35) and (0.2,0.5)(-90:0.1);
(0,0) .. controls +(90:0.35) and (0.2,0.5)(-90:0.1);

[xshift = 6.5cm, yshift = -6.7cm]
(1,0) .. controls +(130:0.6) and (0.2,0.6)(-90:0.2);
(0.8,0) .. controls +(130:0.6) and (0,0.6)(-90:0.35);
[white, line width = 0.1cm] (0.2,0) .. controls +(90:0.75) and (1,0.6)(-90:0.45).. controls +(90:0.5) and (0,0.6)(90:0.5);
(0.2,0) .. controls +(90:0.75) and (1,0.6)(-90:0.45).. controls +(90:0.5) and (0,0.6)(90:0.5);
[white, line width = 0.1cm] (0,0) .. controls +(90:0.75) and (0.8,0.6)(-90:0.2) .. controls +(90:0.3) and (0.2,0.6)(90:0.3);
(0,0) .. controls +(90:0.75) and (0.8,0.6)(-90:0.2) .. controls +(90:0.3) and (0.2,0.6)(90:0.3);

[->] (1.2,0.5) – (1.8,0.5);
at (1.5,0.5) [above] R1;

[xshift = 2cm]
(1,0) .. controls +(130:0.5) and (0.3,0.45)(0:0.05);
(0.8,0) .. controls +(130:0.6) and (0,0.6)(-90:0.35);
[white, line width = 0.1cm] (0.2,0) .. controls +(90:0.75) and (1,0.6)(-90:0.45).. controls +(90:0.5) and (0,0.6)(90:0.5);
(0.2,0) .. controls +(90:0.75) and (1,0.6)(-90:0.45).. controls +(90:0.5) and (0,0.6)(90:0.5);
[white, line width = 0.1cm] (0,0) .. controls +(90:0.4) and (0.3,0.45)(-180:0.1);
(0,0) .. controls +(90:0.4) and (0.3,0.45)(-180:0.1);

[->] (1.2,0.5) – (1.8,0.5);
at (1.5,0.5) [above] R3;

[xshift = 4cm]
(0.8,0) – +(0,0.6);
[white, line width = 0.1cm] (1,0) .. controls +(130:0.5) and (0.3,0.45)(0:0.05);
(1,0) .. controls +(130:0.5) and (0.3,0.45)(0:0.05);
[white, line width = 0.1cm] (0.2,0) .. controls +(90:0.75) and (1,0.6)(-90:0.45).. controls +(90:0.5) and (0.8,0.6)(90:0.5);
(0.2,0) .. controls +(90:0.75) and (1,0.6)(-90:0.45).. controls +(90:0.5) and (0.8,0.6)(90:0.5);
[white, line width = 0.1cm] (0,0) .. controls +(90:0.4) and (0.3,0.45)(-180:0.1);
(0,0) .. controls +(90:0.4) and (0.3,0.45)(-180:0.1);

[->] (1.2,0.5) – (1.8,0.5);
at (1.5,0.5) [above] R1;

[xshift = 6cm]
(0.8,0) – +(0,0.25);
[white, line width = 0.1cm] (1,0) .. controls +(130:0.5) and (0.3,0.45)(0:0.05);
(1,0) .. controls +(130:0.5) and (0.3,0.45)(0:0.05);
[white, line width = 0.1cm] (0.2,0) .. controls +(90:0.65) and (0.8,0.25)(90:0.25);
(0.2,0) .. controls +(90:0.65) and (0.8,0.25)(90:0.25);
[white, line width = 0.1cm] (0,0) .. controls +(90:0.4) and (0.3,0.45)(-180:0.1);
(0,0) .. controls +(90:0.4) and (0.3,0.45)(-180:0.1);

Cutting a Hopf plumbing of two positive Hopf bands along the diagonal $\gamma_2$ (resp. $\gamma_1$) induces $2$ positive (resp. negative) crossings. The Reidemeister moves are showed to help understand which crossings are obtained. The top right surface is a Hopf band while the bottom right one is a band with $6$ positive twists.

Assume that $T_1$ is obtained from $T_2$ by deleting a leaf.
Then $\Sigma (T_1)$ is an incompressible subsurface of $\Sigma (T_2)$.

Let $D$ be the plumbing rectangle of the Hopf band $H$ associated to the additional leaf $v$ of $T_2$ compared to $T_1$. By definition, $D$ has two sides $\gamma_1, \gamma_2$ in $\partial \Sigma (T_1)$. Thus $\gamma_1$ is also an arc of $\Sigma (T_2)$ with its extremities in $\partial \Sigma (T_2)$. By Lemma <ref>, $\Sigma (T_2)$ cut along $\gamma_1$ is an incompressible subsurface $\Sigma'$ of $\Sigma (T_2)$. Furthermore, the remaining of $H$ is a disc that can be isotoped into a neighbourhood of $\gamma_2$ so that $\Sigma (T_1) = \Sigma' \preccurlyeq \Sigma (T_2)$, see Figure <ref>.

(-0.75,-2) rectangle (2.75,2.5);

[blue!20!white] (-2,-1) .. controls +(70:0.5) and (0,0)(-166:0.5) – (1,0.25) .. controls +(14:1) and (3.5,-0.5)(140:0.5) – (4.5,-1.25) .. controls +(140:0.5) and (2,-0.5)(14:1) – (1,-0.75) .. controls +(-166:0.5) and (-1,-1.75)(70:0.5) – cycle;
(-2,-1) .. controls +(70:0.5) and (0,0)(-166:0.5) – (1,0.25) .. controls +(14:1) and (3.5,-0.5)(140:0.5);
[xshift = 1cm, yshift = -0.75cm] (-2,-1) .. controls +(70:0.5) and (0,0)(-166:0.5) – (1,0.25) .. controls +(14:1) and (3.5,-0.5)(140:0.5);
[blue!80!white] at (0,-0.5) $\Sigma (T_2)$;

(h) at (1,1.95);
(l) at (1, 1.55);
(i) at (0.5, 1.65);
(e) at (1.5, 1.9);

(h) .. controls +(180:0.2) and (0,2)(-36.75:0.2) .. controls +(143.25:0.1) and (-0.25,2.15)(0:0.1) .. controls +(180:0.35) and (0,0)(143.25:1) – (1,-0.75) .. controls +(-36.75:0.1) and (1.25,-0.9)(180:0.1) .. controls +(0:0.35) and (1,1.25)(-36.75:1) – cycle;
[blue!20!white] (1,2.25) .. controls +(143.25:0.1) and (0.75,2.4)(0:0.1) .. controls +(180:0.35) and (1,0.25)(143.25:1) – (2,-0.5) .. controls +(-36.75:0.1) and (2.25,-0.65)(180:0.1) – ++(0, -1) – ++(-3.5,0) – ++(0,4.5) – ++(2.5,0) – cycle;
[purple] (1,-0.75) – (2,-0.5);

[blue!20!white] (h) .. controls +(-150:0.1) and (i)(143.25:0.2) .. controls +(-36.75:0.1) and (l)(180:0.2) .. controls +(60:0.2) and (e)(-36.75:0.2) .. controls +(143.25:0.1) and (h)(0:0.2);
[white!40!gray] (-0.25,2.15) .. controls +(0:0.1) and (0,2)(143.25:0.1) .. controls +(-36.75:0.2) and (h)(180:0.2) .. controls +(30:0.1) and (1,2.25)(-36.75:0.2) .. controls +(143.25:0.1) and (0.75,2.4)(0:0.1)– cycle;
[white!40!gray] (1,1.25) .. controls +(143.25:0.1) and (l)(-130:0.2) .. controls +(180:0.2) and (2,1.5)(143.25:0.2) .. controls +(-36.75:1) and (2.25,-0.65)(0:0.35) – (1.25,-0.9) .. controls +(0:0.35) and (1,1.25)(-36.75:1);

(1,1.25) .. controls +(143.25:0.1) and (l)(-130:0.2) .. controls +(180:0.2) and (2,1.5)(143.25:0.2) .. controls +(-36.75:1) and (2.25,-0.65)(0:0.35) – (1.25,-0.9) .. controls +(0:0.35) and (1,1.25)(-36.75:1);
[purple, dotted] (1,-0.75) – (2,-0.5);

(-0.25,2.15) .. controls +(0:0.1) and (0,2)(143.25:0.1) .. controls +(-36.75:0.2) and (h)(180:0.2) .. controls +(30:0.1) and (1,2.25)(-36.75:0.2) .. controls +(143.25:0.1) and (0.75,2.4)(0:0.1)– cycle;
[dashed] (0.75,2.4) .. controls +(180:0.35) and (1,0.25)(143.25:1);

(1,1.25) .. controls +(143.25:0.1) and (l)(-130:0.2) .. controls +(180:0.2) and (2,1.5)(143.25:0.2) .. controls +(-36.75:1) and (2.25,-0.65)(0:0.35) – (1.25,-0.9) .. controls +(0:0.35) and (1,1.25)(-36.75:1);
[dashed] (4.5, -1.25) .. controls +(140:0.5) and (2,-0.5)(14:1) .. controls +(-36.75:0.1) and (2.25,-0.65)(180:0.1);

(h) .. controls +(180:0.2) and (0,2)(-36.75:0.2) .. controls +(143.25:0.1) and (-0.25,2.15)(0:0.1) .. controls +(180:0.35) and (0,0)(143.25:1) – (1,-0.75) .. controls +(-36.75:0.1) and (1.25,-0.9)(180:0.1) .. controls +(0:0.35) and (1,1.25)(-36.75:1) – cycle;
(0.75,2.4) .. controls +(180:0.35) and (1,0.25)(143.25:1);

(1,2.25) .. controls +(143.25:0.1) and (0.75,2.4)(0:0.1);
(2.25,-0.65) .. controls +(0:0.35) and (2,1.5)(-36.75:1);
(0,2) .. controls +(143.25:0.1) and (-0.25,2.15)(0:0.1) .. controls +(180:0.35) and (0,0)(143.25:1) (1,-0.75) .. controls +(-36.75:0.1) and (1.25,-0.9)(180:0.1) .. controls +(0:0.35) and (1,1.25)(-36.75:1) ;
[even odd rule]
(0, 2.25) rectangle (3, 1) (l) circle (0.1cm);
(1,1.25) .. controls +(143.25:0.1) and (l)(-130:0.2) .. controls +(60:0.2) and (e)(-36.75:0.2);

(e) .. controls +(143.25:0.1) and (h)(0:0.2) .. controls +(180:0.2) and (0,2)(-36.75:0.2);
[even odd rule]
(0,2) – (1,1.25) – (2,1.5) – (1,2.25) – (0,2) (h) circle (0.13cm);
(1,2.25) .. controls +(-36.75:0.2) and (h)(30:0.1) .. controls +(-150:0.1) and (i)(143.25:0.2);

(i) .. controls +(-36.75:0.1) and (l)(180:0.2) .. controls +(0:0.2) and (2,1.5)(143.25:0.2);
at (1.85,0.75) $H$;
[purple] (0,0) – (1,0.25);
at (0,0) [above right, purple] $\gamma_2$;
at (1,-0.75) [above, purple] $\gamma_1$;

at (3.75,0) [align = center] cut
along $\gamma_1$;
[->] (3,-0.5) – (4.5,-0.5);

[xshift = 5.5cm]
(-0.75,-2) rectangle (2.75,2.5);

[blue!20!white] (-2,-1) .. controls +(70:0.5) and (0,0)(-166:0.5) – (1,0.25) .. controls +(14:1) and (3.5,-0.5)(140:0.5) – (4.5,-1.25) .. controls +(140:0.5) and (2,-0.5)(14:1) – (1,-0.75) .. controls +(-166:0.5) and (-1,-1.75)(70:0.5) – cycle;
(-2,-1) .. controls +(70:0.5) and (0,0)(-166:0.5) – (1,0.25) .. controls +(14:1) and (3.5,-0.5)(140:0.5);
[xshift = 1cm, yshift = -0.75cm] (-2,-1) .. controls +(70:0.5) and (0,0)(-166:0.5) – (1,0.25) .. controls +(14:1) and (3.5,-0.5)(140:0.5);
[blue!80!white] at (0,-0.5) $\Sigma'$;

(h) at (1,1.95);
(l) at (1, 1.55);
(i) at (0.5, 1.65);
(e) at (1.5, 1.9);

(h) .. controls +(180:0.2) and (0,2)(-36.75:0.2) .. controls +(143.25:0.1) and (-0.25,2.15)(0:0.1) .. controls +(180:0.35) and (0,0)(143.25:1) – (1,-0.75) .. controls +(-36.75:0.1) and (1.25,-0.9)(180:0.1) .. controls +(0:0.35) and (1,1.25)(-36.75:1) – cycle;
[blue!20!white] (1,2.25) .. controls +(143.25:0.1) and (0.75,2.4)(0:0.1) .. controls +(180:0.35) and (1,0.25)(143.25:1) – (2,-0.5)– (1,-0.75) – ++(0, -1) – ++(-3.5,0) – ++(0,4.5) – ++(2.5,0) – cycle;
(1,-0.75) – (2,-0.5);

[blue!20!white] (h) .. controls +(-150:0.1) and (i)(143.25:0.2) .. controls +(-36.75:0.1) and (l)(180:0.2) .. controls +(60:0.2) and (e)(-36.75:0.2) .. controls +(143.25:0.1) and (h)(0:0.2);
[white!40!gray] (-0.25,2.15) .. controls +(0:0.1) and (0,2)(143.25:0.1) .. controls +(-36.75:0.2) and (h)(180:0.2) .. controls +(30:0.1) and (1,2.25)(-36.75:0.2) .. controls +(143.25:0.1) and (0.75,2.4)(0:0.1)– cycle;
[white!40!gray] (1,1.25) .. controls +(143.25:0.1) and (l)(-130:0.2) .. controls +(180:0.2) and (2,1.5)(143.25:0.2) .. controls +(-36.75:1) and (2.25,-0.65)(0:0.35) – (1.25,-0.9) .. controls +(0:0.35) and (1,1.25)(-36.75:1);

(1,1.25) .. controls +(143.25:0.1) and (l)(-130:0.2) .. controls +(180:0.2) and (2,1.5)(143.25:0.2) .. controls +(-36.75:1) and (2.25,-0.65)(0:0.35) – (1.25,-0.9) .. controls +(0:0.35) and (1,1.25)(-36.75:1);

(-0.25,2.15) .. controls +(0:0.1) and (0,2)(143.25:0.1) .. controls +(-36.75:0.2) and (h)(180:0.2) .. controls +(30:0.1) and (1,2.25)(-36.75:0.2) .. controls +(143.25:0.1) and (0.75,2.4)(0:0.1)– cycle;
[dashed] (0.75,2.4) .. controls +(180:0.35) and (1,0.25)(143.25:1);

(1,1.25) .. controls +(143.25:0.1) and (l)(-130:0.2) .. controls +(180:0.2) and (2,1.5)(143.25:0.2) .. controls +(-36.75:1) and (2.25,-0.65)(0:0.35) – (1.25,-0.9) .. controls +(0:0.35) and (1,1.25)(-36.75:1);
[dashed] (4.5, -1.25) .. controls +(140:0.5) and (2,-0.5)(14:1) – (1,-0.75);

(h) .. controls +(180:0.2) and (0,2)(-36.75:0.2) .. controls +(143.25:0.1) and (-0.25,2.15)(0:0.1) .. controls +(180:0.35) and (0,0)(143.25:1) – (1,-0.75) .. controls +(-36.75:0.1) and (1.25,-0.9)(180:0.1) .. controls +(0:0.35) and (1,1.25)(-36.75:1) – cycle;
(0.75,2.4) .. controls +(180:0.35) and (1,0.25)(143.25:1);

(1,2.25) .. controls +(143.25:0.1) and (0.75,2.4)(0:0.1);
(2.25,-0.65) .. controls +(0:0.35) and (2,1.5)(-36.75:1);
(0,2) .. controls +(143.25:0.1) and (-0.25,2.15)(0:0.1) .. controls +(180:0.35) and (0,0)(143.25:1) (1,-0.75) (1.25,-0.9) .. controls +(0:0.35) and (1,1.25)(-36.75:1) ;
[even odd rule]
(0, 2.25) rectangle (3, 1) (l) circle (0.1cm);
(1,1.25) .. controls +(143.25:0.1) and (l)(-130:0.2) .. controls +(60:0.2) and (e)(-36.75:0.2);

(e) .. controls +(143.25:0.1) and (h)(0:0.2) .. controls +(180:0.2) and (0,2)(-36.75:0.2);
[even odd rule]
(0,2) – (1,1.25) – (2,1.5) – (1,2.25) – (0,2) (h) circle (0.13cm);
(1,2.25) .. controls +(-36.75:0.2) and (h)(30:0.1) .. controls +(-150:0.1) and (i)(143.25:0.2);

(i) .. controls +(-36.75:0.1) and (l)(180:0.2) .. controls +(0:0.2) and (2,1.5)(143.25:0.2);
at (1.85,0.75) $H$;
[purple] (0,0) – (1,0.25);
(2.25,-0.65) – (1.25,-0.9);
at (0,0) [above right, purple] $\gamma_2$;

at (8.75,-0.5) $=$;

[xshift = 10cm]
(-0.75,-2) rectangle (2.75,2.5);

[blue!20!white] (-2,-1) .. controls +(70:0.5) and (0,0)(-166:0.5) – (1,0.25) .. controls +(14:1) and (3.5,-0.5)(140:0.5) – (4.5,-1.25) .. controls +(140:0.5) and (2,-0.5)(14:1) – (1,-0.75) .. controls +(-166:0.5) and (-1,-1.75)(70:0.5) – cycle;
(-2,-1) .. controls +(70:0.5) and (0,0)(-166:0.5) – (1,0.25) .. controls +(14:1) and (3.5,-0.5)(140:0.5);
[xshift = 1cm, yshift = -0.75cm] (-2,-1) .. controls +(70:0.5) and (0,0)(-166:0.5) – (1,0.25) .. controls +(14:1) and (3.5,-0.5)(140:0.5);
[blue!80!white] at (0,-0.5) $\Sigma(T_1)$;

Deleting a leaf yields a surface-minor.

A very similar proof yields the following lemma.

Assume that $T_1$ is a plane tree whose root has only one child, $T_2$ is the subtree rooted at that child. Then $\Sigma (T_1)$ is an incompressible subsurface of $\Sigma (T_2)$.

The proof is identical to the previous one: in that situation, $\Sigma (T_1)$ is obtained from $\Sigma (T_2)$ by plumbing a Hopf band, and cutting along one of the boundaries of the plumbing disc provides the needed incompressible subsurface, as in Figure <ref>.

Assume that $T_2$ is a plane tree in which $u\to v \to w$ are three consecutive vertices where $v$ has degree $2$, and that
$T_1$ is obtained from $T_2$ by contracting $u\to v\to w$ into a single edge $u\to w$, while preserving the labels of the endpoints.
Then $\Sigma (T_1)$ is an incompressible subsurface of $\Sigma (T_2)$.

By the construction of Hopf arborescent surfaces, the edge between $u$ and $v$ in $T_2$ corresponds to a plumbing rectangle $D$. It is important to recall here our orientation convention: if $u$ is labelled positively, the cores $\alpha(u)$ and $\alpha(v)$ are oriented so that one turns to the left when going from $u$ to $v$ at the rectangle $D$, while if $v$ is labelled negatively, one turns to the right, see Figure <ref>. Now, we consider two diagonal arcs $\gamma_1$ and $\gamma_2$ on the plumbing rectangle $D$ as pictured in Figure <ref>. When cutting along such a diagonal arc, we obtain a new surface in which the cores $\alpha(u)$ and $\alpha(v)$ merge into a single core. However, their orientations might mismatch, depending on whether we cut along $\gamma_1$ or $\gamma_2$. We take the convention that $\gamma_1$ is the arc the preserves the orientations, while $\gamma_2$ induces an orientation mismatch, see Figures <ref> and <ref>.

Now, let us first consider the case where the labels of $u$ and $v$ are the same. In this case, we consider the subsurface $\Sigma'$ of $\Sigma (T_2)$ obtained by cutting along $\gamma_1$. This has the effect of merging the core curves $\alpha(u)$ and $\alpha(v)$  in a way that respects their orientations. However, it might seem that since each curve $\alpha(u)$ and $\alpha(v)$ corresponds to a Hopf band, merging them like that yields a band that twists too much. But a key observation is that cutting along $\gamma_1$ adds a twist between these two bands, as pictured in Figure <ref>, and this twist is negative when the bands are positive, while it is positive when the bands are negative (indeed, this is the reason for our orientation convention). Therefore, the resulting surface $\Sigma'$ is exactly the same as the one corresponding to the tree $T_1$, and therefore $\Sigma (T_1)$ is an incompressible subsurface of $\Sigma (T_2)$ by Lemma <ref>. See the top and bottom pictures of Figure <ref> for an illustration.

Now, let us consider the case where the label of $u$ is $+$ while the label of $v$ is $-$. In that case, we consider the surface $\Sigma'$ of $\Sigma (T_2)$ obtained by cutting along $\gamma_2$. This has the effect of merging the core curves $\alpha(u)$ and $\alpha(v)$ but with an orientation mismatch. We take the convention that the resulting core curve $\alpha'$ is oriented by $\alpha(u)$, and therefore disagrees with the orientation of $\alpha(v)$ while it follows it. Since $u$ and $v$ are labelled with opposite signs the two twists on their Hopf bands cancel out, but cutting along $\gamma_2$ adds a new positive twist, therefore we can consider $\alpha'$ as being the core curve of a positive Hopf band. Now, let us consider the plumbing rectangle $D'$ corresponding to the edge between $v$ and $w$. Due to the orientation mismatch, arriving at this rectangle from $\alpha'$, we are oriented in the direction opposed to the one we would arrive with if we were arriving from $\alpha(v)$. But due to the orientation convention, when going from $\alpha(v)$ to $\alpha(w)$ in $\Sigma (T_2)$ we turn to the right since $v$ is negative, while when going from $\alpha'$ to $\alpha(w)$ in $\Sigma'$ we turn to the left since $\alpha'$ is a positive band. Therefore, this effect cancels out the orientation mismatch, and $\Sigma'$ coincides exactly with the surface $\Sigma (T_1)$ corresponding to the tree $T_1$. Therefore $\Sigma (T_1)$ is an incompressible subsurface of $\Sigma (T_2)$ by Lemma <ref>. See the third picture of Figure <ref> for an illustration.

The same cancellation effect happens when the label of $u$ is $-$ and the label of $v$ is $+$: when cutting along $\gamma_2$ we have an orientation mismatch which is cancelled out by the the fact that the new band is negative, and thus the orientation convention makes it turn in the opposite direction in the plumbing rectangle between $v$ and $w$. Therefore, in that case $\Sigma (T_1)$ is also an incompressible subsurface of $\Sigma (T_2)$ thanks to Lemma <ref>. This is illustrated in the second picture of Figure <ref>.

[scale = 1]

(-1.6,1.6) rectangle (13.6,-3.6);

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (lı) circle (0.1);
[thick] (50:1) arc (50:345:1) .. controls +(80:0.2) and (lı)(-140:0.2) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(170:0.2) and (50:1)(-50:0.2);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (hı) circle (0.1);
[thick] (40:1.5) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(140:0.1) and (iı)(-75:0.1) .. controls +(105:0.1) and (hı)(-110:0.1) .. controls +(70:0.1) and (40:1.5)(-50:0.2);

[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);
[white!40!gray] (lı) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(-110:0.1) and (iı)(105:0.1) .. controls +(-75:0.1) and (lı)(140:0.1);

[purple] (10:1.25) arc (10:260:1.25);

[yshift = -1.8cm, xshift = 0.5 cm]

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (lı) circle (0.1);
[thick] (50:1) arc (50:345:1) .. controls +(80:0.2) and (lı)(-140:0.2) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(170:0.2) and (50:1)(-50:0.2);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (hı) circle (0.1);
[thick] (40:1.5) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(140:0.1) and (iı)(-75:0.1) .. controls +(105:0.1) and (hı)(-110:0.1) .. controls +(70:0.1) and (40:1.5)(-50:0.2);

[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);
[white!40!gray] (lı) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(-110:0.1) and (iı)(105:0.1) .. controls +(-75:0.1) and (lı)(140:0.1);

(0,0) rectangle (1.5,-1.5);
[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);

[purple] (10:1.25) arc (10:-100:1.25);
at (-10:1.25) [purple, rotate = 80] $\blacktriangleright$;
[green!80!purple] ($(10:1.25)+(0.5,-1.8)$) arc (10:260:1.25);
at ($(80:1.25)+(0.5,-1.8)$) [green!80!purple, rotate = -10] $\blacktriangleleft$;

[yshift = -3.6cm, xshift = 1 cm]
(-1.6,1.6) rectangle (1.6,0.2);

(hı) at (-10:1.25);

[thick] (hı) .. controls +(30:0.1) and (20:1.5)(-70:0.5) arc (20:340:1.5) – (-15:1) arc (-15:-335:1) .. controls +(-65:0.2) and (hı)(130:0.2);

[blue!20!white] (hı) .. controls +(30:0.1) and (20:1.5)(-70:0.5) arc (20:340:1.5) – (-15:1) arc (-15:-335:1) .. controls +(-65:0.2) and (hı)(130:0.2);

[yshift = -1.8cm, xshift = 0.5 cm]
(0,0) rectangle (1.5,-1.5);
[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);

[green!80!purple] ($(10:1.25)+(0.5,-1.8)$) arc (10:-100:1.25);
[yshift = -3.6cm, xshift = 1 cm]
(-1.6,1.6) rectangle (1.6,0.2);
[orange] (0,0) circle (1.25);
at (80:1.25) [orange, rotate = -10] $\blacktriangleleft$;

[name path=ci1] (-90:1) arc (-90:0:1);
[name path=ce1] (-90:1.5) arc (-90:0:1.5);
[name path=ci2, yshift = -1.8cm, xshift = 0.5 cm] (90:1) arc (90:0:1);
[name path=ce2, yshift = -1.8cm, xshift = 0.5 cm] (90:1.5) arc (90:0:1.5);
[name intersections=of=ci1 and ce2,total=]
in 1,...,coordinate (h) at (intersection-);
[name intersections=of=ce1 and ci2,total=]
in 1,...,coordinate (l) at (intersection-);
[thick, dashed, brown] (h) .. controls +(-30:0.05) and (l)(30:0.05);

(g1) [purple, fill, circle, inner sep = 0.1cm] at (2.5,0) ;
at (2.6, 0) [right, black] $+$;
(g2) [green!80!purple, fill, circle, inner sep = 0.1cm] at (2.5,-1) ;
at (2.6, -1) [right, black] $+$;
(g3) [orange, fill, circle, inner sep = 0.1cm] at (2.5,-2) ;
at (2.6, -2) [right, black] $x$;
[->] (g1) – (g2);
[->] (g2) – (g3);

[->, thick] (3.25,-0.5) – (4,-0.5);

[xshift = 6.5cm]

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (lı) circle (0.1);
[thick] (50:1) arc (50:345:1) .. controls +(80:0.2) and (lı)(-140:0.2) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(170:0.2) and (50:1)(-50:0.2);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (hı) circle (0.1);
[thick] (40:1.5) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(140:0.1) and (iı)(-75:0.1) .. controls +(105:0.1) and (hı)(-110:0.1) .. controls +(70:0.1) and (40:1.5)(-50:0.2);

[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);
[white!40!gray] (lı) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(-110:0.1) and (iı)(105:0.1) .. controls +(-75:0.1) and (lı)(140:0.1);

[red!35!blue] (15:1.25) arc (15:260:1.25);

[yshift = -1.8cm, xshift = 0.5 cm]

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (lı) circle (0.1);
[thick] (50:1) arc (50:345:1) .. controls +(80:0.2) and (lı)(-140:0.2) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(170:0.2) and (50:1)(-50:0.2);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (hı) circle (0.1);
[thick] (40:1.5) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(140:0.1) and (iı)(-75:0.1) .. controls +(105:0.1) and (hı)(-110:0.1) .. controls +(70:0.1) and (40:1.5)(-50:0.2);

[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);
[white!40!gray] (lı) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(-110:0.1) and (iı)(105:0.1) .. controls +(-75:0.1) and (lı)(140:0.1);

(0,0) rectangle (1.5,-1.5);
[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);

[red!35!blue] (15:1.25) arc (15:-15:1.25) .. controls +(-115:0.25) and ($(0.5,-1.8)+(50:1.25)$)(140:0.25) arc (50:10:1.25) ($(-100:1.25)+(0.5,-1.8)$) arc (-100:-280:1.25) .. controls +(-10:0.35) and (-60:1.25)(30:0.35) arc (-60:-100:1.25);
at ($(90:1.25)+(0.5,-1.80)$) [red!35!blue] $\blacktriangleleft$;

[yshift = -3.6cm, xshift = 1 cm]
(-1.6,1.6) rectangle (1.6,0.2);

(hı) at (-10:1.25);

[thick] (hı) .. controls +(30:0.1) and (20:1.5)(-70:0.5) arc (20:340:1.5) – (-15:1) arc (-15:-335:1) .. controls +(-65:0.2) and (hı)(130:0.2);

[blue!20!white] (hı) .. controls +(30:0.1) and (20:1.5)(-70:0.5) arc (20:340:1.5) – (-15:1) arc (-15:-335:1) .. controls +(-65:0.2) and (hı)(130:0.2);

[yshift = -1.8cm, xshift = 0.5 cm]
(0,0) rectangle (1.5,-1.5);
[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);

[yshift = -3.6cm, xshift = 1 cm]
(-1.6,1.6) rectangle (1.6,0.2);
[orange] (0,0) circle (1.25);
at (80:1.25) [orange, rotate = -10] $\blacktriangleleft$;

[red!35!blue] ($(10:1.25)+(0.5,-1.8)$) arc (10:-100:1.25);

[white] (-18:1) .. controls +(-30:0.07) and ($(50:1)+(0.5,-1.8)$)(30:0.07) –  (-45:1.5) .. controls +(30:0.07) and ($(75:1.5)+(0.5,-1.8)$)(-30:0.07) – cycle;
(-18:1) .. controls +(-30:0.07) and ($(50:1)+(0.5,-1.8)$)(30:0.07);
($(75:1.5)+(0.5,-1.8)$) .. controls +(-30:0.07) and (-45:1.5)(30:0.07);

[xshift = -4.75cm]
(g1) [red!35!blue, fill, circle, inner sep = 0.1cm] at (2.5,0) ;
at (2.6, 0) [right, black] $+$;
(g3) [orange, fill, circle, inner sep = 0.1cm] at (2.5,-2) ;
at (2.6, -2) [right, black] $x$;
[->] (g1) – (g3);

at (8.85,-0.5) $=$;

[xshift = 11cm]

[yshift = -1.8cm, xshift = 0.5 cm]

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

[white!40!gray] (l2) .. controls +(40:0.2) and (e2)(-75:0.1) .. controls +(105:0.1) and (h2)(-10:0.2) .. controls +(-110:0.1) and (i2)(105:0.1) .. controls +(-75:0.1) and (l2)(140:0.1);
[blue!20!white] (h2) .. controls +(70:0.1) and ($(0.5,-1.8)+(40:1.5)$)(-50:0.2) .. controls +(130:0.75) and (40:1.5)(-50:1) arc (40:180:1.5) .. controls +(-90:1) and ($(0.5,-1.8)+(180:1.5)$)(90:1) arc (180:350:1.5) .. controls +(80:0.1) and (l2)(-40:0.2) .. controls +(-140:0.2) and ($(0.5,-1.8)+(-15:1)$)(80:0.2) arc (-15:-180:1) .. controls +(90:1) and (180:1)(-90:1) arc (180:50:1) .. controls +(-40:1) and ($(0.5,-1.8)+(50:1)$)(140:0.5) .. controls +(-50:0.2) and (h2)(170:0.2);

[even odd rule]
(-1.6,1.6) rectangle (2.1,-4) (l2) circle (0.1);
(l2) .. controls +(-140:0.2) and ($(0.5,-1.8)+(-15:1)$)(80:0.2) arc (-15:-180:1) .. controls +(90:1) and (180:1)(-90:1) arc (180:50:1)  .. controls +(-40:1) and ($(0.5,-1.8)+(50:1)$)(140:0.5) .. controls +(-50:0.2) and (h2)(170:0.2) .. controls +(-10:0.1) and (e2)(105:0.1) .. controls +(-75:0.1) and (l2)(40:0.1);

[even odd rule]
(-1.6,1.6) rectangle (2.1,-4) (h2) circle (0.1);
(h2) .. controls +(70:0.1) and ($(0.5,-1.8)+(40:1.5)$)(-50:0.2) .. controls +(130:0.75) and (40:1.5)(-50:1) arc (40:180:1.5) .. controls +(-90:1) and ($(0.5,-1.8)+(180:1.5)$)(90:1) arc (180:350:1.5) .. controls +(80:0.1) and (l2)(-40:0.2) .. controls +(140:0.1) and (i2)(-75:0.1) .. controls +(105:0.1) and (h2)(-110:0.1);

[red!35!blue] (180:1.25) arc (180:-15:1.25) .. controls +(-115:0.25) and ($(0.5,-1.8)+(40:1.25)$)(130:0.25) arc (40:10:1.25) ($(-100:1.25)+(0.5,-1.8)$) arc (-100:-180:1.25) .. controls +(90:1) and (180:1.25)(-90:1);

[yshift = -3.6cm, xshift = 1 cm]
(-1.6,1.6) rectangle (1.6,0.2);

(hı) at (-10:1.25);

[thick] (hı) .. controls +(30:0.1) and (20:1.5)(-70:0.5) arc (20:340:1.5) – (-15:1) arc (-15:-335:1) .. controls +(-65:0.2) and (hı)(130:0.2);

[blue!20!white] (hı) .. controls +(30:0.1) and (20:1.5)(-70:0.5) arc (20:340:1.5) – (-15:1) arc (-15:-335:1) .. controls +(-65:0.2) and (hı)(130:0.2);

[yshift = -1.8cm, xshift = 0.5 cm]
(0.5,-0.3) rectangle (1.5,-1.05);
[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);

[yshift = -3.6cm, xshift = 1 cm]
(-1.6,1.6) rectangle (1.6,0.2);
[orange] (0,0) circle (1.25);
at (80:1.25) [orange, rotate = -10] $\blacktriangleleft$;

[red!35!blue] ($(10:1.25)+(0.5,-1.8)$) arc (10:-100:1.25);
at ($(-10:1.25)+(0.5,-1.8)$) [red!35!blue, rotate = 80] $\blacktriangleright$;

[yshift =-5.5cm]
(-1.6,1.6) rectangle (13.6,-3.6);

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (hı) circle (0.1);
[thick] (50:1) arc (50:345:1) .. controls +(80:0.2) and (lı)(-140:0.2) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(170:0.2) and (50:1)(-50:0.2);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (lı) circle (0.1);
[thick] (40:1.5) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(140:0.1) and (iı)(-75:0.1) .. controls +(105:0.1) and (hı)(-110:0.1) .. controls +(70:0.1) and (40:1.5)(-50:0.2);

[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);
[white!40!gray] (lı) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(-110:0.1) and (iı)(105:0.1) .. controls +(-75:0.1) and (lı)(140:0.1);

[purple] (10:1.25) arc (10:260:1.25);

[yshift = -1.8cm, xshift = 0.5 cm]

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (lı) circle (0.1);
[thick] (50:1) arc (50:345:1) .. controls +(80:0.2) and (lı)(-140:0.2) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(170:0.2) and (50:1)(-50:0.2);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (hı) circle (0.1);
[thick] (40:1.5) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(140:0.1) and (iı)(-75:0.1) .. controls +(105:0.1) and (hı)(-110:0.1) .. controls +(70:0.1) and (40:1.5)(-50:0.2);

[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);
[white!40!gray] (lı) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(-110:0.1) and (iı)(105:0.1) .. controls +(-75:0.1) and (lı)(140:0.1);

(0,0) rectangle (1.5,-1.5);
[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);

[purple] (10:1.25) arc (10:-100:1.25);
at (-10:1.25) [purple, rotate = 80] $\blacktriangleright$;
[green!80!purple] ($(10:1.25)+(0.5,-1.8)$) arc (10:260:1.25);
at ($(40:1.25)+(0.5,-1.8)$) [green!80!purple, rotate = -50] $\blacktriangleright$;

[yshift = -3.6cm, xshift = 1 cm]
(-1.6,1.6) rectangle (1.6,0.2);

(hı) at (-10:1.25);

[thick] (hı) .. controls +(30:0.1) and (20:1.5)(-70:0.5) arc (20:340:1.5) – (-15:1) arc (-15:-335:1) .. controls +(-65:0.2) and (hı)(130:0.2);

[blue!20!white] (hı) .. controls +(30:0.1) and (20:1.5)(-70:0.5) arc (20:340:1.5) – (-15:1) arc (-15:-335:1) .. controls +(-65:0.2) and (hı)(130:0.2);

[yshift = -1.8cm, xshift = 0.5 cm]
(0,0) rectangle (1.5,-1.5);
[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);

[green!80!purple] ($(10:1.25)+(0.5,-1.8)$) arc (10:-100:1.25);
[yshift = -3.6cm, xshift = 1 cm]
(-1.6,1.6) rectangle (1.6,0.2);
[orange] (0,0) circle (1.25);
at (40:1.25) [orange, rotate = -50] $\blacktriangleright$;

[name path=ci1] (-90:1) arc (-90:0:1);
[name path=ce1] (-90:1.5) arc (-90:0:1.5);
[name path=ci2, yshift = -1.8cm, xshift = 0.5 cm] (90:1) arc (90:0:1);
[name path=ce2, yshift = -1.8cm, xshift = 0.5 cm] (90:1.5) arc (90:0:1.5);
[name intersections=of=ci1 and ce2,total=]
in 1,...,coordinate (h) at (intersection-);
[name intersections=of=ce1 and ci2,total=]
in 1,...,coordinate (l) at (intersection-);
[thick, dashed, brown] (h) .. controls +(-30:0.05) and (l)(30:0.05);

(g1) [purple, fill, circle, inner sep = 0.1cm] at (2.5,0) ;
at (2.6, 0) [right, black] $-$;
(g2) [green!80!purple, fill, circle, inner sep = 0.1cm] at (2.5,-1) ;
at (2.6, -1) [right, black] $+$;
(g3) [orange, fill, circle, inner sep = 0.1cm] at (2.5,-2) ;
at (2.6, -2) [right, black] $x$;
[->] (g1) – (g2);
[->] (g2) – (g3);

[->, thick] (3.25,-0.5) – (4,-0.5);

[xshift = 6.5cm]

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (hı) circle (0.1);
[thick] (50:1) arc (50:345:1) .. controls +(80:0.2) and (lı)(-140:0.2) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(170:0.2) and (50:1)(-50:0.2);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (lı) circle (0.1);
[thick] (40:1.5) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(140:0.1) and (iı)(-75:0.1) .. controls +(105:0.1) and (hı)(-110:0.1) .. controls +(70:0.1) and (40:1.5)(-50:0.2);

[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);
[white!40!gray] (lı) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(-110:0.1) and (iı)(105:0.1) .. controls +(-75:0.1) and (lı)(140:0.1);

[red!35!blue] (15:1.25) arc (15:260:1.25);

[yshift = -1.8cm, xshift = 0.5 cm]

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (lı) circle (0.1);
[thick] (50:1) arc (50:345:1) .. controls +(80:0.2) and (lı)(-140:0.2) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(170:0.2) and (50:1)(-50:0.2);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (hı) circle (0.1);
[thick] (40:1.5) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(140:0.1) and (iı)(-75:0.1) .. controls +(105:0.1) and (hı)(-110:0.1) .. controls +(70:0.1) and (40:1.5)(-50:0.2);

[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);
[white!40!gray] (lı) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(-110:0.1) and (iı)(105:0.1) .. controls +(-75:0.1) and (lı)(140:0.1);

(0,0) rectangle (1.5,-1.5);
[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);

[red!35!blue] (15:1.25) arc (15:-15:1.25) .. controls +(-115:0.25) and ($(0.5,-1.8)+(50:1.25)$)(140:0.25) arc (50:10:1.25) ($(-100:1.25)+(0.5,-1.8)$) arc (-100:-280:1.25) .. controls +(-10:0.35) and (-60:1.25)(30:0.35) arc (-60:-100:1.25);

[yshift = -3.6cm, xshift = 1 cm]
(-1.6,1.6) rectangle (1.6,0.2);

(hı) at (-10:1.25);

[thick] (hı) .. controls +(30:0.1) and (20:1.5)(-70:0.5) arc (20:340:1.5) – (-15:1) arc (-15:-335:1) .. controls +(-65:0.2) and (hı)(130:0.2);

[blue!20!white] (hı) .. controls +(30:0.1) and (20:1.5)(-70:0.5) arc (20:340:1.5) – (-15:1) arc (-15:-335:1) .. controls +(-65:0.2) and (hı)(130:0.2);

[yshift = -1.8cm, xshift = 0.5 cm]
(0,0) rectangle (1.5,-1.5);
[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);

[yshift = -3.6cm, xshift = 1 cm]
(-1.6,1.6) rectangle (1.6,0.2);
[orange] (0,0) circle (1.25);
at (40:1.25) [orange, rotate = -50] $\blacktriangleright$;

[red!35!blue] ($(10:1.25)+(0.5,-1.8)$) arc (10:-100:1.25);
at ($(-10:1.25)+(0.5,-1.8)$) [red!35!blue, rotate = 70] $\blacktriangleright$;

[white] (-18:1) .. controls +(-30:0.07) and ($(50:1)+(0.5,-1.8)$)(30:0.07) –  (-45:1.5) .. controls +(30:0.07) and ($(75:1.5)+(0.5,-1.8)$)(-30:0.07) – cycle;
(-18:1) .. controls +(-30:0.07) and ($(50:1)+(0.5,-1.8)$)(30:0.07);
($(75:1.5)+(0.5,-1.8)$) .. controls +(-30:0.07) and (-45:1.5)(30:0.07);

[xshift = -4.75cm]
(g1) [red!35!blue, fill, circle, inner sep = 0.1cm] at (2.5,0) ;
at (2.6, 0) [right, black] $-$;
(g3) [orange, fill, circle, inner sep = 0.1cm] at (2.5,-2) ;
at (2.6, -2) [right, black] $x$;
[->] (g1) – (g3);

at (8.85,-0.5) $=$;

[xshift = 11cm]

[yshift = -1.8cm, xshift = 0.5 cm]

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

[white!40!gray] (l2) .. controls +(40:0.2) and (e2)(-75:0.1) .. controls +(105:0.1) and (h2)(-10:0.2) .. controls +(-110:0.1) and (i2)(105:0.1) .. controls +(-75:0.1) and (l2)(140:0.1);
[blue!20!white] (h2) .. controls +(70:0.1) and ($(0.5,-1.8)+(40:1.5)$)(-50:0.2) .. controls +(130:0.75) and (40:1.5)(-50:1) arc (40:180:1.5) .. controls +(-90:1) and ($(0.5,-1.8)+(180:1.5)$)(90:1) arc (180:350:1.5) .. controls +(80:0.1) and (l2)(-40:0.2) .. controls +(-140:0.2) and ($(0.5,-1.8)+(-15:1)$)(80:0.2) arc (-15:-180:1) .. controls +(90:1) and (180:1)(-90:1) arc (180:50:1) .. controls +(-40:1) and ($(0.5,-1.8)+(50:1)$)(140:0.5) .. controls +(-50:0.2) and (h2)(170:0.2);

[even odd rule]
(-1.6,1.6) rectangle (2.1,-4) (h2) circle (0.1);
(l2) .. controls +(-140:0.2) and ($(0.5,-1.8)+(-15:1)$)(80:0.2) arc (-15:-180:1) .. controls +(90:1) and (180:1)(-90:1) arc (180:50:1)  .. controls +(-40:1) and ($(0.5,-1.8)+(50:1)$)(140:0.5) .. controls +(-50:0.2) and (h2)(170:0.2) .. controls +(-10:0.1) and (e2)(105:0.1) .. controls +(-75:0.1) and (l2)(40:0.1);

[even odd rule]
(-1.6,1.6) rectangle (2.1,-4) (l2) circle (0.1);
(h2) .. controls +(70:0.1) and ($(0.5,-1.8)+(40:1.5)$)(-50:0.2) .. controls +(130:0.75) and (40:1.5)(-50:1) arc (40:180:1.5) .. controls +(-90:1) and ($(0.5,-1.8)+(180:1.5)$)(90:1) arc (180:350:1.5) .. controls +(80:0.1) and (l2)(-40:0.2) .. controls +(140:0.1) and (i2)(-75:0.1) .. controls +(105:0.1) and (h2)(-110:0.1);

[red!35!blue] (180:1.25) arc (180:-15:1.25) .. controls +(-115:0.25) and ($(0.5,-1.8)+(40:1.25)$)(130:0.25) arc (40:10:1.25) ($(-100:1.25)+(0.5,-1.8)$) arc (-100:-180:1.25) .. controls +(90:1) and (180:1.25)(-90:1);

[yshift = -3.6cm, xshift = 1 cm]
(-1.6,1.6) rectangle (1.6,0.2);

(hı) at (-10:1.25);

[thick] (hı) .. controls +(30:0.1) and (20:1.5)(-70:0.5) arc (20:340:1.5) – (-15:1) arc (-15:-335:1) .. controls +(-65:0.2) and (hı)(130:0.2);

[blue!20!white] (hı) .. controls +(30:0.1) and (20:1.5)(-70:0.5) arc (20:340:1.5) – (-15:1) arc (-15:-335:1) .. controls +(-65:0.2) and (hı)(130:0.2);

[yshift = -1.8cm, xshift = 0.5 cm]
(0.5,-0.3) rectangle (1.5,-1.05);
[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);

[yshift = -3.6cm, xshift = 1 cm]
(-1.6,1.6) rectangle (1.6,0.2);
[orange] (0,0) circle (1.25);
at (40:1.25) [orange, rotate = -50] $\blacktriangleright$;

[red!35!blue] ($(10:1.25)+(0.5,-1.8)$) arc (10:-100:1.25);
at ($(-10:1.25)+(0.5,-1.8)$) [red!35!blue, rotate = 80] $\blacktriangleright$;

[yshift = -11cm]
(-1.6,1.6) rectangle (13.6,-3.6);

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (lı) circle (0.1);
[thick] (50:1) arc (50:345:1) .. controls +(80:0.2) and (lı)(-140:0.2) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(170:0.2) and (50:1)(-50:0.2);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (hı) circle (0.1);
[thick] (40:1.5) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(140:0.1) and (iı)(-75:0.1) .. controls +(105:0.1) and (hı)(-110:0.1) .. controls +(70:0.1) and (40:1.5)(-50:0.2);

[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);
[white!40!gray] (lı) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(-110:0.1) and (iı)(105:0.1) .. controls +(-75:0.1) and (lı)(140:0.1);

[purple] (10:1.25) arc (10:260:1.25);

[yshift = -1.8cm, xshift = 0.5 cm]

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (hı) circle (0.1);
[thick] (50:1) arc (50:345:1) .. controls +(80:0.2) and (lı)(-140:0.2) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(170:0.2) and (50:1)(-50:0.2);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (lı) circle (0.1);
[thick] (40:1.5) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(140:0.1) and (iı)(-75:0.1) .. controls +(105:0.1) and (hı)(-110:0.1) .. controls +(70:0.1) and (40:1.5)(-50:0.2);

[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);
[white!40!gray] (lı) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(-110:0.1) and (iı)(105:0.1) .. controls +(-75:0.1) and (lı)(140:0.1);

(0,0) rectangle (1.5,-1.5);
[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);

[purple] (10:1.25) arc (10:-100:1.25);
at (-10:1.25) [purple, rotate = 80] $\blacktriangleright$;
[green!80!purple] ($(10:1.25)+(0.5,-1.8)$) arc (10:260:1.25);
at ($(90:1.25)+(0.5,-1.8)$) [green!80!purple] $\blacktriangleleft$;

[yshift = -3.6cm, xshift = 1 cm]
(-1.6,1.6) rectangle (1.6,0.2);

(hı) at (-10:1.25);

[thick] (hı) .. controls +(30:0.1) and (20:1.5)(-70:0.5) arc (20:340:1.5) – (-15:1) arc (-15:-335:1) .. controls +(-65:0.2) and (hı)(130:0.2);

[blue!20!white] (hı) .. controls +(30:0.1) and (20:1.5)(-70:0.5) arc (20:340:1.5) – (-15:1) arc (-15:-335:1) .. controls +(-65:0.2) and (hı)(130:0.2);

[yshift = -1.8cm, xshift = 0.5 cm]
(0,0) rectangle (1.5,-1.5);
[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);

[green!80!purple] ($(10:1.25)+(0.5,-1.8)$) arc (10:-100:1.25);
[yshift = -3.6cm, xshift = 1 cm]
(-1.6,1.6) rectangle (1.6,0.2);
[orange] (0,0) circle (1.25);
at (40:1.25) [orange, rotate = -50] $\blacktriangleright$;

[name path=ci1] (-90:1) arc (-90:0:1);
[name path=ce1] (-90:1.5) arc (-90:0:1.5);
[name path=ci2, yshift = -1.8cm, xshift = 0.5 cm] (90:1) arc (90:0:1);
[name path=ce2, yshift = -1.8cm, xshift = 0.5 cm] (90:1.5) arc (90:0:1.5);
[name intersections=of=ci1 and ci2,total=]
in 1,...,coordinate (h) at (intersection-);
[name intersections=of=ce1 and ce2,total=]
in 1,...,coordinate (l) at (intersection-);
[thick, dashed, brown] (h) – (l);

(g1) [purple, fill, circle, inner sep = 0.1cm] at (2.5,0) ;
at (2.6, 0) [right, black] $+$;
(g2) [green!80!purple, fill, circle, inner sep = 0.1cm] at (2.5,-1) ;
at (2.6, -1) [right, black] $-$;
(g3) [orange, fill, circle, inner sep = 0.1cm] at (2.5,-2) ;
at (2.6, -2) [right, black] $x$;
[->] (g1) – (g2);
[->] (g2) – (g3);

[->, thick] (3.25,-0.5) – (4,-0.5);

[xshift = 6.5cm]

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (lı) circle (0.1);
[thick] (50:1) arc (50:345:1) .. controls +(80:0.2) and (lı)(-140:0.2) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(170:0.2) and (50:1)(-50:0.2);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (hı) circle (0.1);
[thick] (40:1.5) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(140:0.1) and (iı)(-75:0.1) .. controls +(105:0.1) and (hı)(-110:0.1) .. controls +(70:0.1) and (40:1.5)(-50:0.2);

[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);
[white!40!gray] (lı) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(-110:0.1) and (iı)(105:0.1) .. controls +(-75:0.1) and (lı)(140:0.1);

[red!35!blue] (15:1.25) arc (15:260:1.25);

[yshift = -1.8cm, xshift = 0.5 cm]

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (hı) circle (0.1);
[thick] (50:1) arc (50:345:1) .. controls +(80:0.2) and (lı)(-140:0.2) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(170:0.2) and (50:1)(-50:0.2);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (lı) circle (0.1);
[thick] (40:1.5) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(140:0.1) and (iı)(-75:0.1) .. controls +(105:0.1) and (hı)(-110:0.1) .. controls +(70:0.1) and (40:1.5)(-50:0.2);

[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);
[white!40!gray] (lı) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(-110:0.1) and (iı)(105:0.1) .. controls +(-75:0.1) and (lı)(140:0.1);

(0.5,-0.3) rectangle (1.5,-1.05);
[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);

[red!35!blue] ($(10:1.25)+(0.5,-1.8)$) arc (10:40:1.25) .. controls +(130:0.25) and (-70:1.25)(20:0.25) arc (-70:-100:1.25);
[red!35!blue] ($(-100:1.25)+(0.5,-1.8)$) arc (-100:-280:1.25) .. controls +(-10:0.15) and (-20:1.25)(-110:0.15) arc (-20:15:1.25);

[yshift = -3.6cm, xshift = 1 cm]
(-1.6,1.6) rectangle (1.6,0.2);

(hı) at (-10:1.25);

[thick] (hı) .. controls +(30:0.1) and (20:1.5)(-70:0.5) arc (20:340:1.5) – (-15:1) arc (-15:-335:1) .. controls +(-65:0.2) and (hı)(130:0.2);

[blue!20!white] (hı) .. controls +(30:0.1) and (20:1.5)(-70:0.5) arc (20:340:1.5) – (-15:1) arc (-15:-335:1) .. controls +(-65:0.2) and (hı)(130:0.2);

[yshift = -1.8cm, xshift = 0.5 cm]
(0,0) rectangle (1.5,-1.5);
[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);

[yshift = -3.6cm, xshift = 1 cm]
(-1.6,1.6) rectangle (1.6,0.2);
[orange] (0,0) circle (1.25);
at (40:1.25) [orange, rotate = -50] $\blacktriangleright$;

at ($(-60:1.25)+(0.5,-1.8)$) [red!35!blue, rotate = 30] $\blacktriangleleft$;
[red!35!blue] ($(10:1.25)+(0.5,-1.8)$) arc (10:-100:1.25);

[white] ($(90:1)+(0.5,-1.8)$) .. controls +(0:0.05) and (-18:1.5)(-118:0.1) – ($(48:1.5)+(0.5,-1.8)$) .. controls +(138:0.1) and (-80:1)(10:0.1) – cycle;
($(90:1)+(0.5,-1.8)$) .. controls +(0:0.05) and (-18:1.5)(-118:0.1);
($(48:1.5)+(0.5,-1.8)$) .. controls +(138:0.1) and (-80:1)(10:0.1);

[xshift = -4.75cm]
(g1) [red!35!blue, fill, circle, inner sep = 0.1cm] at (2.5,0) ;
at (2.6, 0) [right, black] $+$;
(g3) [orange, fill, circle, inner sep = 0.1cm] at (2.5,-2) ;
at (2.6, -2) [right, black] $x$;
[->] (g1) – (g3);

at (8.85,-0.5) $=$;

[xshift = 11cm]

[yshift = -1.8cm, xshift = 0.5 cm]

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

[white!40!gray] (l2) .. controls +(40:0.2) and (e2)(-75:0.1) .. controls +(105:0.1) and (h2)(-10:0.2) .. controls +(-110:0.1) and (i2)(105:0.1) .. controls +(-75:0.1) and (l2)(140:0.1);
[blue!20!white] (h2) .. controls +(70:0.1) and ($(0.5,-1.8)+(40:1.5)$)(-50:0.2) .. controls +(130:0.75) and (40:1.5)(-50:1) arc (40:180:1.5) .. controls +(-90:1) and ($(0.5,-1.8)+(180:1.5)$)(90:1) arc (180:350:1.5) .. controls +(80:0.1) and (l2)(-40:0.2) .. controls +(-140:0.2) and ($(0.5,-1.8)+(-15:1)$)(80:0.2) arc (-15:-180:1) .. controls +(90:1) and (180:1)(-90:1) arc (180:50:1) .. controls +(-40:1) and ($(0.5,-1.8)+(50:1)$)(140:0.5) .. controls +(-50:0.2) and (h2)(170:0.2);

[even odd rule]
(-1.6,1.6) rectangle (2.1,-4) (l2) circle (0.1);
(l2) .. controls +(-140:0.2) and ($(0.5,-1.8)+(-15:1)$)(80:0.2) arc (-15:-180:1) .. controls +(90:1) and (180:1)(-90:1) arc (180:50:1)  .. controls +(-40:1) and ($(0.5,-1.8)+(50:1)$)(140:0.5) .. controls +(-50:0.2) and (h2)(170:0.2) .. controls +(-10:0.1) and (e2)(105:0.1) .. controls +(-75:0.1) and (l2)(40:0.1);

[even odd rule]
(-1.6,1.6) rectangle (2.1,-4) (h2) circle (0.1);
(h2) .. controls +(70:0.1) and ($(0.5,-1.8)+(40:1.5)$)(-50:0.2) .. controls +(130:0.75) and (40:1.5)(-50:1) arc (40:180:1.5) .. controls +(-90:1) and ($(0.5,-1.8)+(180:1.5)$)(90:1) arc (180:350:1.5) .. controls +(80:0.1) and (l2)(-40:0.2) .. controls +(140:0.1) and (i2)(-75:0.1) .. controls +(105:0.1) and (h2)(-110:0.1);

[red!35!blue] (180:1.25) arc (180:-15:1.25) .. controls +(-115:0.25) and ($(0.5,-1.8)+(40:1.25)$)(130:0.25) arc (40:10:1.25) ($(-100:1.25)+(0.5,-1.8)$) arc (-100:-180:1.25) .. controls +(90:1) and (180:1.25)(-90:1);

[yshift = -3.6cm, xshift = 1 cm]
(-1.6,1.6) rectangle (1.6,0.2);

(hı) at (-10:1.25);

[thick] (hı) .. controls +(30:0.1) and (20:1.5)(-70:0.5) arc (20:340:1.5) – (-15:1) arc (-15:-335:1) .. controls +(-65:0.2) and (hı)(130:0.2);

[blue!20!white] (hı) .. controls +(30:0.1) and (20:1.5)(-70:0.5) arc (20:340:1.5) – (-15:1) arc (-15:-335:1) .. controls +(-65:0.2) and (hı)(130:0.2);

[yshift = -1.8cm, xshift = 0.5 cm]
(0.5,-0.3) rectangle (1.5,-1.05);
[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);

[yshift = -3.6cm, xshift = 1 cm]
(-1.6,1.6) rectangle (1.6,0.2);
[orange] (0,0) circle (1.25);
at (40:1.25) [orange, rotate = -50] $\blacktriangleright$;

[red!35!blue] ($(10:1.25)+(0.5,-1.8)$) arc (10:-100:1.25);
at ($(-60:1.25)+(0.5,-1.8)$) [red!35!blue, rotate = 30] $\blacktriangleleft$;

[yshift = -16.5cm]
(-1.6,1.6) rectangle (13.6,-3.6);

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (hı) circle (0.1);
[thick] (50:1) arc (50:345:1) .. controls +(80:0.2) and (lı)(-140:0.2) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(170:0.2) and (50:1)(-50:0.2);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (lı) circle (0.1);
[thick] (40:1.5) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(140:0.1) and (iı)(-75:0.1) .. controls +(105:0.1) and (hı)(-110:0.1) .. controls +(70:0.1) and (40:1.5)(-50:0.2);

[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);
[white!40!gray] (lı) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(-110:0.1) and (iı)(105:0.1) .. controls +(-75:0.1) and (lı)(140:0.1);

[purple] (10:1.25) arc (10:260:1.25);

[yshift = -1.8cm, xshift = 0.5 cm]

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (hı) circle (0.1);
[thick] (50:1) arc (50:345:1) .. controls +(80:0.2) and (lı)(-140:0.2) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(170:0.2) and (50:1)(-50:0.2);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (lı) circle (0.1);
[thick] (40:1.5) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(140:0.1) and (iı)(-75:0.1) .. controls +(105:0.1) and (hı)(-110:0.1) .. controls +(70:0.1) and (40:1.5)(-50:0.2);

[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);
[white!40!gray] (lı) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(-110:0.1) and (iı)(105:0.1) .. controls +(-75:0.1) and (lı)(140:0.1);

(0,0) rectangle (1.5,-1.5);
[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);

[purple] (10:1.25) arc (10:-100:1.25);
at (-10:1.25) [purple, rotate = 80] $\blacktriangleright$;
[green!80!purple] ($(10:1.25)+(0.5,-1.8)$) arc (10:260:1.25);
at ($(40:1.25)+(0.5,-1.8)$) [green!80!purple, rotate = -50] $\blacktriangleright$;

[yshift = -3.6cm, xshift = 1 cm]
(-1.6,1.6) rectangle (1.6,0.2);

(hı) at (-10:1.25);

[thick] (hı) .. controls +(30:0.1) and (20:1.5)(-70:0.5) arc (20:340:1.5) – (-15:1) arc (-15:-335:1) .. controls +(-65:0.2) and (hı)(130:0.2);

[blue!20!white] (hı) .. controls +(30:0.1) and (20:1.5)(-70:0.5) arc (20:340:1.5) – (-15:1) arc (-15:-335:1) .. controls +(-65:0.2) and (hı)(130:0.2);

[yshift = -1.8cm, xshift = 0.5 cm]
(0,0) rectangle (1.5,-1.5);
[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);

[green!80!purple] ($(10:1.25)+(0.5,-1.8)$) arc (10:-100:1.25);
[yshift = -3.6cm, xshift = 1 cm]
(-1.6,1.6) rectangle (1.6,0.2);
[orange] (0,0) circle (1.25);
at (80:1.25) [orange, rotate = -10] $\blacktriangleleft$;

[name path=ci1] (-90:1) arc (-90:0:1);
[name path=ce1] (-90:1.5) arc (-90:0:1.5);
[name path=ci2, yshift = -1.8cm, xshift = 0.5 cm] (90:1) arc (90:0:1);
[name path=ce2, yshift = -1.8cm, xshift = 0.5 cm] (90:1.5) arc (90:0:1.5);
[name intersections=of=ci1 and ci2,total=]
in 1,...,coordinate (h) at (intersection-);
[name intersections=of=ce1 and ce2,total=]
in 1,...,coordinate (l) at (intersection-);
[thick, dashed, brown] (h) – (l);

(g1) [purple, fill, circle, inner sep = 0.1cm] at (2.5,0) ;
at (2.6, 0) [right, black] $-$;
(g2) [green!80!purple, fill, circle, inner sep = 0.1cm] at (2.5,-1) ;
at (2.6, -1) [right, black] $-$;
(g3) [orange, fill, circle, inner sep = 0.1cm] at (2.5,-2) ;
at (2.6, -2) [right, black] $x$;
[->] (g1) – (g2);
[->] (g2) – (g3);

[->, thick] (3.25,-0.5) – (4,-0.5);

[xshift = 6.5cm]

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (hı) circle (0.1);
[thick] (50:1) arc (50:345:1) .. controls +(80:0.2) and (lı)(-140:0.2) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(170:0.2) and (50:1)(-50:0.2);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (lı) circle (0.1);
[thick] (40:1.5) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(140:0.1) and (iı)(-75:0.1) .. controls +(105:0.1) and (hı)(-110:0.1) .. controls +(70:0.1) and (40:1.5)(-50:0.2);

[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);
[white!40!gray] (lı) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(-110:0.1) and (iı)(105:0.1) .. controls +(-75:0.1) and (lı)(140:0.1);

[red!35!blue] (15:1.25) arc (15:260:1.25);

[yshift = -1.8cm, xshift = 0.5 cm]

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (hı) circle (0.1);
[thick] (50:1) arc (50:345:1) .. controls +(80:0.2) and (lı)(-140:0.2) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(170:0.2) and (50:1)(-50:0.2);

[even odd rule]
(-1.6,1.6) rectangle (1.6,-1.6) (lı) circle (0.1);
[thick] (40:1.5) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(140:0.1) and (iı)(-75:0.1) .. controls +(105:0.1) and (hı)(-110:0.1) .. controls +(70:0.1) and (40:1.5)(-50:0.2);

[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);
[white!40!gray] (lı) .. controls +(40:0.2) and (eı)(-75:0.1) .. controls +(105:0.1) and (hı)(-10:0.2) .. controls +(-110:0.1) and (iı)(105:0.1) .. controls +(-75:0.1) and (lı)(140:0.1);

(0.5,-0.3) rectangle (1.5,-1.05);
[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);

[red!35!blue] ($(10:1.25)+(0.5,-1.8)$) arc (10:40:1.25) .. controls +(130:0.25) and (-70:1.25)(20:0.25) arc (-70:-100:1.25);
[red!35!blue] ($(-100:1.25)+(0.5,-1.8)$) arc (-100:-280:1.25) .. controls +(-10:0.15) and (-20:1.25)(-110:0.15) arc (-20:15:1.25);

[yshift = -3.6cm, xshift = 1 cm]
(-1.6,1.6) rectangle (1.6,0.2);

(hı) at (-10:1.25);

[thick] (hı) .. controls +(30:0.1) and (20:1.5)(-70:0.5) arc (20:340:1.5) – (-15:1) arc (-15:-335:1) .. controls +(-65:0.2) and (hı)(130:0.2);

[blue!20!white] (hı) .. controls +(30:0.1) and (20:1.5)(-70:0.5) arc (20:340:1.5) – (-15:1) arc (-15:-335:1) .. controls +(-65:0.2) and (hı)(130:0.2);

[yshift = -1.8cm, xshift = 0.5 cm]
(0,0) rectangle (1.5,-1.5);
[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);

[yshift = -3.6cm, xshift = 1 cm]
(-1.6,1.6) rectangle (1.6,0.2);
[orange] (0,0) circle (1.25);
at (80:1.25) [orange, rotate = -10] $\blacktriangleleft$;

at ($(-60:1.25)+(0.5,-1.8)$) [red!35!blue, rotate = 30] $\blacktriangleleft$;
[red!35!blue] ($(10:1.25)+(0.5,-1.8)$) arc (10:-100:1.25);

[white] ($(90:1)+(0.5,-1.8)$) .. controls +(0:0.05) and (-18:1.5)(-118:0.1) – ($(48:1.5)+(0.5,-1.8)$) .. controls +(138:0.1) and (-80:1)(10:0.1) – cycle;
($(90:1)+(0.5,-1.8)$) .. controls +(0:0.05) and (-18:1.5)(-118:0.1);
($(48:1.5)+(0.5,-1.8)$) .. controls +(138:0.1) and (-80:1)(10:0.1);

[xshift = -4.75cm]
(g1) [red!35!blue, fill, circle, inner sep = 0.1cm] at (2.5,0) ;
at (2.6, 0) [right, black] $-$;
(g3) [orange, fill, circle, inner sep = 0.1cm] at (2.5,-2) ;
at (2.6, -2) [right, black] $x$;
[->] (g1) – (g3);

at (8.85,-0.5) $=$;

[xshift = 11cm]

[yshift = -1.8cm, xshift = 0.5 cm]

(eı) at (15:1.5);
(iı) at (15:1);
(lı) at (2.5:1.25);
(hı) at (27.5:1.25);

[white!40!gray] (l2) .. controls +(40:0.2) and (e2)(-75:0.1) .. controls +(105:0.1) and (h2)(-10:0.2) .. controls +(-110:0.1) and (i2)(105:0.1) .. controls +(-75:0.1) and (l2)(140:0.1);
[blue!20!white] (h2) .. controls +(70:0.1) and ($(0.5,-1.8)+(40:1.5)$)(-50:0.2) .. controls +(130:0.75) and (40:1.5)(-50:1) arc (40:180:1.5) .. controls +(-90:1) and ($(0.5,-1.8)+(180:1.5)$)(90:1) arc (180:350:1.5) .. controls +(80:0.1) and (l2)(-40:0.2) .. controls +(-140:0.2) and ($(0.5,-1.8)+(-15:1)$)(80:0.2) arc (-15:-180:1) .. controls +(90:1) and (180:1)(-90:1) arc (180:50:1) .. controls +(-40:1) and ($(0.5,-1.8)+(50:1)$)(140:0.5) .. controls +(-50:0.2) and (h2)(170:0.2);

[even odd rule]
(-1.6,1.6) rectangle (2.1,-4) (h2) circle (0.1);
(l2) .. controls +(-140:0.2) and ($(0.5,-1.8)+(-15:1)$)(80:0.2) arc (-15:-180:1) .. controls +(90:1) and (180:1)(-90:1) arc (180:50:1)  .. controls +(-40:1) and ($(0.5,-1.8)+(50:1)$)(140:0.5) .. controls +(-50:0.2) and (h2)(170:0.2) .. controls +(-10:0.1) and (e2)(105:0.1) .. controls +(-75:0.1) and (l2)(40:0.1);

[even odd rule]
(-1.6,1.6) rectangle (2.1,-4) (l2) circle (0.1);
(h2) .. controls +(70:0.1) and ($(0.5,-1.8)+(40:1.5)$)(-50:0.2) .. controls +(130:0.75) and (40:1.5)(-50:1) arc (40:180:1.5) .. controls +(-90:1) and ($(0.5,-1.8)+(180:1.5)$)(90:1) arc (180:350:1.5) .. controls +(80:0.1) and (l2)(-40:0.2) .. controls +(140:0.1) and (i2)(-75:0.1) .. controls +(105:0.1) and (h2)(-110:0.1);

[red!35!blue] (180:1.25) arc (180:-15:1.25) .. controls +(-115:0.25) and ($(0.5,-1.8)+(40:1.25)$)(130:0.25) arc (40:10:1.25) ($(-100:1.25)+(0.5,-1.8)$) arc (-100:-180:1.25) .. controls +(90:1) and (180:1.25)(-90:1);

[yshift = -3.6cm, xshift = 1 cm]
(-1.6,1.6) rectangle (1.6,0.2);

(hı) at (-10:1.25);

[thick] (hı) .. controls +(30:0.1) and (20:1.5)(-70:0.5) arc (20:340:1.5) – (-15:1) arc (-15:-335:1) .. controls +(-65:0.2) and (hı)(130:0.2);

[blue!20!white] (hı) .. controls +(30:0.1) and (20:1.5)(-70:0.5) arc (20:340:1.5) – (-15:1) arc (-15:-335:1) .. controls +(-65:0.2) and (hı)(130:0.2);

[yshift = -1.8cm, xshift = 0.5 cm]
(0.5,-0.3) rectangle (1.5,-1.05);
[blue!20!white] (hı) .. controls +(70:0.1) and (40:1.5)(-50:0.2) arc (40:350:1.5) .. controls +(80:0.1) and (lı)(-40:0.2) .. controls +(-140:0.2) and (-15:1)(80:0.2) arc (-15:-310:1) .. controls +(-50:0.2) and (hı)(170:0.2);

[yshift = -3.6cm, xshift = 1 cm]
(-1.6,1.6) rectangle (1.6,0.2);
[orange] (0,0) circle (1.25);
at (80:1.25) [orange, rotate = -10] $\blacktriangleleft$;

[red!35!blue] ($(10:1.25)+(0.5,-1.8)$) arc (10:-100:1.25);
at ($(-60:1.25)+(0.5,-1.8)$) [red!35!blue, rotate = 30] $\blacktriangleleft$;

All cases of contraction of a $3$-path to an edge preserving the labels of the endpoints.

As a corollary, contracting any path of $T_1$ into an edge whose labels match with the labels of the extremities of the path produces a tree $T_2$ such that $\Sigma (T_1) \preccurlyeq \Sigma (T_2)$.

By definition, if $T_1$ admits a homeomorphic embedding into $T_2$, it can be obtained iteratively from $T_2$ by (i) removing a child leaf, (ii) removing a parent leaf, (iii) reducing a label or (iv) contracting a path while preserving the labels of the endpoints. Since the well-quasi-order that we chose on the alphabet $A=\{+,-\}$ is that no two elements are comparable, no label reduction can happen. Then the cases (i), (ii) and (iv) are handled respectively by Lemma <ref>, Lemma <ref> and Lemma <ref>.

On the other hand, the Kruskal Tree Theorem directly yields the following proposition.

Hopf arborescent links are well-quasi-ordered under the link-minor relation.

Take an infinite sequence $(L_n)_{n \in \mathbb{N}}$ of Hopf arborescent links, let $(T_n)_{n \in \mathbb{N}}$ a sequence of plane trees such that for all $n \in \N,$ $\Sigma (T_n)$ is a Seifert surface of $L_n$. Then, by Theorem <ref>, there exists $i<j$ such that $T_i$ admits a homeomorphic embedding into $T_2$. Hence $L_i$ is a link-minor of $L_j$.

We can deduce Theorem <ref> as a direct corollary of Proposition <ref> and Proposition <ref>.

Take an infinite sequence $(\Sigma_n)_{n \in \mathbb{N}}$ of canonical Seifert surfaces of Hopf arborescent links. Then by Proposition <ref>, $\partial \Sigma_i \hookrightarrow \partial \Sigma_j$ for some $i<j$. By Proposition <ref> we have $\Sigma_i \preccurlyeq \Sigma_j$, i.e. the surface-minor order is a well-quasi-order on Hopf arborescent surfaces.

The proof of Proposition <ref> highlights that the minor relation on the set of Hopf arborescent surfaces is more subtle and fragile than one might expect.
Indeed, the cuts involved when taking an incompressible subsurface in the proof of Lemma <ref> inevitably merge Hopf bands and thus one needs to be careful in order to control the number of resulting twists.
In particular, the proof does not seem to generalise to the more general classes of surfaces obtained by plumbing bands with a bounded number of twists (even though everything works well at the level of trees).

§ DECIDABILITY OF THE GENUS DEFECT FOR HOPF ARBORESCENT LINKS

§.§ Monotonicity of the genus defect

Now that we proved that link-minor is a well-quasi-order on the set of Hopf arborescent links, we want to highlight a property that is stable for this minor relation.
Recall that the genus defect $\Delta_g(L)$ of an oriented link $L$ is the difference $g(L)-g_4(L)$ between its classical genus and its 4-dimensional genus.
The latter can be either in the topological locally flat or in the smooth category.
All statements in this section (and in particular Theorem <ref>) hold in both categories.
We reprove Lemma 6 of [5] in the form of Proposition <ref> using the fact that link-minor implies that the associated Seifert are surface-minors.

The genus defect $\Delta_g$ is monotone on the family of Hopf arborescent links with respect to the link-minor relation, i.e., if $L_1$ is a link-minor of $L_2$, then $\Delta g(L_1) \leq \Delta g (L_2)$.

We rely on the following lemma that highlights how the $4$-genus behaves with respect to surface-minors. It uses a cut-and-paste construction and an Euler characteristic argument.

Let $\Sigma$ be an oriented surface of $\Sp^3$ and $\Sigma'$ be a surface-minor of $\Sigma$. If we write  $L = \partial \Sigma$ and $L' = \partial \Sigma'$, then we have $g(\Sigma) - g_4(L) \geq g( \Sigma' ) - g_4(L')$.

Seeing $\Sp^3$ as the boundary of the $4$-ball $\B^4$, consider a surface $S'$ in $\B^4$ such that $\partial S' = L'$ and $S' \cap \Sp^3 = L'$.
Gluing the remaining pieces of $\Sigma \smallsetminus \Sigma'$ to $S'$ along $L'$ yields a surface $S$ in $B^4$ such that $\partial S = L$, see Figure <ref>.

[opacity = 0.2, purple] (0,0) circle (1 and 2);
[black, thick] (0,0) circle (1 and 2);
at (-1.2,-1) $L$;
[purple] at (0,1.5) $\Sigma$;

[xshift = 3cm]
[opacity = 0.2, purple] (0,0) circle (1 and 2);
[black, thick] (0,0) circle (1 and 2);
at (-1.2,-1) $L$;
[purple, fill opacity = 0.4] (0,0) circle (0.75 and 1.5);
[purple] at (0,0) $\Sigma'$;
[purple] at (0,-1.75) $L'$;

[xshift = 6cm]

[black, thick] (0,0) circle (1 and 2);
at (-1.2,-1) $L$;
[purple] at (0,-1.75) $L'$;
[opacity = 0.3, blue] (0,1.5) arc (90:270:0.75 and 1.5) .. controls +(2,0) and (0,1.5)(2,0);
[purple, thick] (0,1.5) arc (90:270:0.75 and 1.5);
[purple, thick, dotted] (0,-1.5) arc (-90:90:0.75 and 1.5);
[blue] (0,-1.5) .. controls +(2,0) and (0,1.5)(2,0);
at (1.8,0) ${\color{blue} S'}$;

[xshift = 9.5cm]
[opacity = 0.3, blue] (0,0) circle (1 and 2);
[white] (0,0) circle (0.75 and 1.5);
[black, thick] (0,0) circle (1 and 2);
at (-1.2,-1) $L$;
[opacity = 0.3, blue] (0,1.5) arc (90:270:0.75 and 1.5) .. controls +(2,0) and (0,1.5)(2,0);
[blue] (0,1.5) arc (90:270:0.75 and 1.5);
[blue, dotted] (0,-1.5) arc (-90:90:0.75 and 1.5);
[blue] (0,-1.5) .. controls +(2,0) and (0,1.5)(2,0);
[blue] at (1.8,0) $S$;

Illustration of the construction of $\Sigma$, $\Sigma'$, $L'$, $S,$ and $S'$. Purple surfaces are in $\Sp^3$ while blue one are considered to be in $B^4$.

By definition of $g_4(L)$, we have $g_4(L) \leq g(S)$, and thus $g( \Sigma ) - g_4( L ) \geq g( \Sigma ) - g( S )$.
Furthermore, the genus of $S$ is given by $g(S) = g(\Sigma) - g(\Sigma') + g(S')$. Thus one has $g( \Sigma ) - g_4( L ) \geq g( \Sigma' ) - g( S' )$.
Now if we assume $S'$ to minimise the 4-genus over surfaces bounded by $L'$, we conclude: $g(\Sigma) - g_4(L) \geq g( \Sigma' ) -g_4(L')$.

As Hopf arborescent surfaces are of minimal genus for Hopf arborescent links, Lemma <ref> can be used to prove Proposition <ref>.

Let $L$ and $L'$ be two Hopf arborescent links such that $L'$ is a link-minor of $L$.
Consider $\Sigma$ and $\Sigma'$ the corresponding canonical Seifert surfaces.
Since Hopf arborescent links are fibred, $\Sigma$ is a Seifert surface of $L$ of minimal genus (see Theorem <ref>), i.e. $g(L) = g(\Sigma)$ and similarly, $g(\Sigma') = g(L')$.
By Proposition <ref> one has $\Sigma \preccurlyeq \Sigma'$.
Hence, by Lemma <ref>, one gets $\Delta_g(L) \geq \Delta_g(L')$.

§.§ Proof of Theorem <ref>

We first show that the link-minor relation can be decided using the decidability of link equivalence. For knots, equivalence can be tested as a combination of an algorithm that allows to decide whether two $3$-manifolds with boundary are homeomorphic [23, 31] and the Gordon-Luecke Theorem [17] that states that two knots are equivalent if their complements, which are $3$-manifolds with boundaries, are equivalent. In the case of links, we additionally need to keep track of a longitude of each torus boundary component in the complement of the link. We refer to the survey of Lackenby <cit.> for a summary of the techniques that allow to prove the following theorem:

Given two links $L_1$ and $L_2$, the problem of testing whether $L_1$ is isotopic to $L_2$ is decidable.

Given a Hopf arborescent link $L$, denote by $\mathcal{T}(L)$ the set of plane trees $T$ whose associated Hopf arborescent surface $\Sigma(T)$ has $L$ as oriented boundary.
As a corollary, we obtain:

Given a Hopf arborescent link $L$, the set $\mathcal{T}(L)$ is computable.

For increasing $k\in \N$, we enumerate all plane tree $T_i$ with $k$ vertices labelled by $\ens{-,+}$ and store the trees $T_i$ such that $\partial \Sigma (T_i)$ is isotopic to $L$, where we test the isotopy using Theorem <ref>.
If we find a $k$ for which such a tree exists, we finish the enumeration for this value and return the stored trees.
Indeed, plumbing $n$ Hopf bands produces a surface with Betti number $n$.
By Theorem <ref>, all the trees $T_i$ such that $\partial \Sigma (T_i) = L$ produce surfaces $\Sigma(T_i)$ with the same genus, and hence have the same number of vertices.
As the entry is a Hopf arborescent link, there exists a tree $T$ such that $\partial \Sigma (T) = L$. Thus the algorithm terminates.

Alternatively, and if one wants some control on the complexity of that algorithm, one can avoid blindly testing for increasing $k$ by first computing the genus of the link [19, 31], or just computing an upper bound to it using, e.g., Seifert's algorithm, and then enumerate only the trees that produce surfaces up to that genus. From Lemma <ref>, we obtain:

Given two Hopf arborescent links $L_1$ and $L_2$, testing if $L_1$ is a link-minor of $L_2$ is decidable.

Using Lemma <ref>, we compute $\mathcal{T}(L_1)$ and $\mathcal{T}(L_2)$.
The trees in $\mathcal{T}(L_1)$ (resp. $\mathcal{T}(L_2)$) all have the same number $k_1$ (resp. $k_2$) of vertices.
Then we brute force every possible path contraction to an edge and iterated leaf deletion on trees of $\mathcal{T}(L_2)$ such that the result is a tree with $k_1$ vertices, and test whether it is equal to a tree of $\mathcal{T}(L_1)$. If such a test succeeds, we output yes, otherwise we return no.
There is a finite number of trees in both $\mathcal{T}(L_1)$ and $\mathcal{T}(L_2)$ and a finite number of a trees with $k_1$ vertices that homeomorphically embed into a tree of $\mathcal{T}(L_2)$.
Hence that algorithm eventually terminates. Its correctness follows directly from the definition of link-minor.

Finally we prove Theorem <ref> by using the stability of the genus-defect by link-minor, the previous algorithms, and the well-quasi-order properties.

By Proposition <ref>, the order defined by link-minors is a well-quasi-order on the set of Hopf arborescent links. Hence, the set of Hopf arborescent links $\mathcal{H}_{k}$ that have defect at most $k$ is characterized by a finite family $\mathcal{F}_k$ of forbidden minors. It follows, by Proposition <ref> that $\Delta_g (L) \leq k$ if and only if for all $f$ in $\mathcal{F}_k$,  $f$ is not a link-minor of $L$. Using Lemma <ref> we test for each $f \in \mathcal{F}_{k}$ if $f$ is a link-minor of $L$. If such a test succeeds, output no, otherwise the input link has $\Delta_g (L) \leq k$.

As said in the introduction, our proof in not constructive as it relies at its core on the existence of a set of forbidden minors for having defect at most $k$. This set of forbidden minors is not explicit and hard-coded in the algorithm. Furthermore, the sets of excluded minors will be different for the two different notions of defect (smooth and locally flat). It is likely that computing them is a topological challenge requiring arguments of different nature.

Theorem <ref> also provides the existence of a set of forbidden minors for having defect at most $k$ but for a different and stronger definition of minors on links that relies only on the surface-minor relation on the Seifert surface and not the trees. However deciding this relation, even by a brute force argument, seems challenging: in addition to the fact that no algorithm seems to be known for testing isotopy of surfaces, one would also need to control the complexity of the cutting arcs. Even in the case of positive Hopf arborescent links, this seems delicate, see for example [32].

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