Dtjytyk

I'm trying to do the following drawing in Tikz:

I almost finished the leftmost picture:

begin{tikzpicture}

fill[blue!20] (7.5,0) ellipse (1 and .75);

draw (7.5,0) ellipse (1 and .75);

begin{scope}

  clip (7.5,-.9) ellipse (1 and 1.25);

  draw(7.5,1.1) ellipse (1 and 1.25);

  clip (7.5,1.1) ellipse (1 and 1.25);

  draw (7.5,-1.1) ellipse (1 and 1.25);

  fill[white] (7.5,-1.1) ellipse (1 and 1.25);

end{scope}

draw (7.5,0) ellipse (0.8 and .47);

draw (7.5,.47) node[scale=0.8] {$<$} node[above] {$a$};

node (a) at (7.61,-.142894){};

node (b) at (8,-.649519){};

node (c) at ($(a)!0.5!(b)$) {};

begin{scope}[shift={(c)},x={(a)}, scale=0.7]

draw (1,0) arc (0:180:1 and 0.3);

draw[dashed] (-1,0) arc (180:360:1 and 0.3);

end{scope}

draw (7.942,-0.555) node[scale=0.8,rotate=-85] {$<$};

draw (7.67,-0.60) node {$b$};

end{tikzpicture}

However I failed to do the "bubble" in the first drawing and to the second one. I appreciate any help.

The closest question I found here was this one: Tikz: Once punctured torus?

However, the answers there don't solve most of my problems.

If you really intend to play with these tori, you may eventually want to switch to 3d coordinates, where it is possible to find out whether a coordinate is on the visible or hidden patch.

documentclass[tikz,border=3.14mm]{standalone}

usepackage{tikz-3dplot}

begin{document}

tdplotsetmaincoords{60}{0}

tikzset{declare function={torusx(u,v,R,r)=cos(u)*(R + r*cos(v)); 

torusy(u,v,R,r)=(R + r*cos(v))*sin(u);

torusz(u,v,R,r)=r*sin(v);

vcrit1(u,th)=atan(tan(th)*sin(u));% first critical v value

vcrit2(u,th)=180+atan(tan(th)*sin(u));% second critical v value

disc(th,R,r)=((pow(r,2)-pow(R,2))*pow(cot(th),2)+% 

pow(r,2)*(2+pow(tan(th),2)))/pow(R,2);% discriminant

umax(th,R,r)=ifthenelse(disc(th,R,r)>0,asin(sqrt(abs(disc(th,R,r)))),0);

}}



begin{tikzpicture}[tdplot_main_coords]

pgfmathsetmacro{R}{4}

pgfmathsetmacro{r}{1.5}

 draw[thick,fill=gray,even odd rule,fill opacity=0.2] plot[variable=x,domain=0:360,smooth,samples=71]

 ({torusx(x,vcrit1(x,tdplotmaintheta),R,r)},

 {torusy(x,vcrit1(x,tdplotmaintheta),R,r)},

 {torusz(x,vcrit1(x,tdplotmaintheta),R,r)}) 

 plot[variable=x,

 domain={-180+umax(tdplotmaintheta,R,r)}:{-umax(tdplotmaintheta,R,r)},smooth,samples=51]

 ({torusx(x,vcrit2(x,tdplotmaintheta),R,r)},

 {torusy(x,vcrit2(x,tdplotmaintheta),R,r)},

 {torusz(x,vcrit2(x,tdplotmaintheta),R,r)})

 plot[variable=x,

 domain={umax(tdplotmaintheta,R,r)}:{180-umax(tdplotmaintheta,R,r)},smooth,samples=51]

 ({torusx(x,vcrit2(x,tdplotmaintheta),R,r)},

 {torusy(x,vcrit2(x,tdplotmaintheta),R,r)},

 {torusz(x,vcrit2(x,tdplotmaintheta),R,r)});

 draw[thick] plot[variable=x,

 domain={-180+umax(tdplotmaintheta,R,r)/2}:{-umax(tdplotmaintheta,R,r)/2},smooth,samples=51]

 ({torusx(x,vcrit2(x,tdplotmaintheta),R,r)},

 {torusy(x,vcrit2(x,tdplotmaintheta),R,r)},

 {torusz(x,vcrit2(x,tdplotmaintheta),R,r)});

 foreach X  in {300}  

 {draw[thick,dashed] 

  plot[smooth,variable=x,domain={360+vcrit1(X,tdplotmaintheta)}:{vcrit2(X,tdplotmaintheta)},samples=71]   

 ({torusx(X,x,R,r)},{torusy(X,x,R,r)},{torusz(X,x,R,r)});

 draw[thick] 

  plot[smooth,variable=x,domain={vcrit2(X,tdplotmaintheta)}:{vcrit1(X,tdplotmaintheta)},samples=71]   

 ({torusx(X,x,R,r)},{torusy(X,x,R,r)},{torusz(X,x,R,r)});

 draw[thick,-latex] 

  plot[smooth,variable=x,domain={vcrit1(X,tdplotmaintheta)}:90,samples=71]   

 ({torusx(X,x,R,r)},{torusy(X,x,R,r)},{torusz(X,x,R,r)});

 }

 draw[thick,-latex] plot[smooth,variable=x,domain=00:360,samples=71]   

 ({torusx(x,90,R,r)},

 {torusy(x,90,R,r)},

 {torusz(x,90,R,r)}); 

 begin{scope}[declare function={myu(x)=sin(2*x)*sin(x);

 myv(x)=sin(2*x)*cos(x);}]

 draw[thick,fill=white] plot[smooth,variable=x,domain=00:90,samples=71]   

 ({torusx(-60+45*myu(x),90-45*myv(x),R,r)},

 {torusy(-60+45*myu(x),90-45*myv(x),R,r)},

 {torusz(-60+45*myu(x),90-45*myv(x),R,r)});

 end{scope}

end{tikzpicture}

end{document}

If you want a cartoon, consider e.g.

documentclass[tikz,border=3.14mm]{standalone}

usetikzlibrary{arrows.meta,bending,decorations.markings,intersections}

% https://tex.stackexchange.com/a/430239/121799

tikzset{% inspired by https://tex.stackexchange.com/a/316050/121799

    arc arrow/.style args={%

    to pos #1 with length #2}{

    decoration={

        markings,

         mark=at position 0 with {pgfextra{%

         pgfmathsetmacro{tmpArrowTime}{#2/(pgfdecoratedpathlength)}

         xdeftmpArrowTime{tmpArrowTime}}},

        mark=at position {#1-tmpArrowTime} with {coordinate(@1);},

        mark=at position {#1-2*tmpArrowTime/3} with {coordinate(@2);},

        mark=at position {#1-tmpArrowTime/3} with {coordinate(@3);},

        mark=at position {#1} with {coordinate(@4);

        draw[-{Stealth[length=#2,bend]}]       

        (@1) .. controls (@2) and (@3) .. (@4);},

        },

     postaction=decorate,

     },bent arrow/.style={arc arrow=to pos #1 with length 2mm},

}



begin{document}

begin{tikzpicture}[scale=4]

 begin{scope}[local bounding box=left]

  draw[fill=blue!20,even odd rule] (0,0) ellipse (1 and .75) 

   (-0.5,0) arc(120:60:1 and 1.25)  arc(-60:-120:1 and 1.25) coordinate[pos=0.25] (xt);

  draw (-0.5,0) arc(-120:-130:1 and 1.25) (0.5,0) arc(-60:-50:1 and 1.25);

  draw[bent arrow=0.2,thick,name path=b] (-65:1 and .75) to[out=40,in=10] 

   node[pos=0.2,right]{$b$} (xt);

  draw[dashed] (xt) to[out=-170,in=-140] (-65:1 and .75);

  draw[bent arrow=0.98,thick,name path=a] (0.8,0.05) arc(0:360:0.8 and .5)

   node[pos=0.2,below]{$ell$} node[pos=0.98,right]{$a$};

  draw[name intersections={of=a and b,by=i},fill=white] (i) 

  to[out=45,in=-45] ++ (0.2,0.4) to[out=135,in=45](i);

 end{scope}

 %

 begin{scope}[local bounding box=right,xshift=2.5cm]

  draw[fill=blue!20,even odd rule] 

  (-0.7,-1) to[out=90,in=-90] (-1,0) arc(180:0:1 and .75)

  to[out=-90,in=90] coordinate[pos=0.7] (ys) (0.7,-1) arc(0:180:0.7 and 0.12) coordinate[pos=0.5] (p)

   (-0.5,0) arc(120:60:1 and 1.25)  arc(-60:-120:1 and 1.25) coordinate[pos=0.5] (yt);

  draw (-0.5,0) arc(-120:-130:1 and 1.25) (0.5,0) arc(-60:-50:1 and 1.25);  

  draw (0.7,-1) arc(0:-180:0.7 and 0.12);

  draw[bent arrow=0.5,thick] (p) to[out=70,in=-120] (-20:0.8 and .5) 

  arc(-20:200:0.8 and .5) node[pos=0.5,below]{$a$}  to[out=-60,in=110] cycle;

  draw[bent arrow=0.5,thick] (p) to[out=80,in=180] node[pos=0.5,right]{$b$}  (yt);

  draw[dashed] (yt) to[out=0,in=70] (ys);

  draw[thick] (ys) to[out=-110,in=20] (p);

 end{scope}

 path (left) -- (right) node[midway,scale=2]{$simeq$};

end{tikzpicture}

end{document}

Unlike in the above picture, you cannot adjust the view angle.

Using the tqft package:

documentclass{article}

%url{https://tex.stackexchange.com/q/481212/86}

usepackage{tikz}

usetikzlibrary{

  tqft,

  decorations.markings,

  arrows.meta,

  hobby,

  calc

}



begin{document}



begin{tikzpicture}[use Hobby shortcut]

pic[

  scale=2,

  tqft,

  incoming boundary components = 0,

  outgoing boundary components = 2,

  cobordism edge/.style={draw},

  fill=gray!50,

  name=top

];

pic[

  scale=2,

  tqft,

  incoming boundary components = 2,

  outgoing boundary components = 0,

  cobordism edge/.style={draw},

  fill=gray!50,

  name=bottom,

  at=(top-outgoing boundary 1)

];

draw[

  decoration={

    markings,

    mark=at position .25 with {arrow{Latex}},

  },

  postaction={decorate}

]

(bottom-between first incoming and last incoming) to[out=45,in=-45]  node[pos=.25,right] {(b)} coordinate[pos=.5] (a) (bottom-between incoming 1 and 2);

draw[dashed] (bottom-between first incoming and last incoming) to[out=135,in=-135] (bottom-between incoming 1 and 2);

draw[

  decoration={

    markings,

    mark=at position .25 with {arrow{Latex}},

    mark=at position .25 with {node[right] {(a)};},

  },

  postaction={decorate}

] ([closed]$(bottom-between first incoming and last incoming)!.5!(bottom-between incoming 1 and 2)$) .. (bottom-incoming boundary 2.north) .. ($(top-between first and last outgoing)!.5!(top-between outgoing 1 and 2)$) .. (bottom-incoming boundary 1.north);

draw[fill=white] ([out angle=30]a) .. ++(1,.5) .. ++(.5,.6) .. ++(-.25,0) .. ([in angle=30]a);



pic[

  scale=2,

  tqft,

  incoming boundary components = 0,

  outgoing boundary components = 2,

  cobordism edge/.style={draw},

  fill=gray!50,

  name=secondtop,

  at={(7,0)}

];

pic[

  scale=2,

  tqft,

  incoming boundary components = 2,

  outgoing boundary components = 1,

  offset=.5,

  cobordism edge/.style={draw},

  every outgoing boundary component/.style={transform shape,draw},

  fill=gray!50,

  name=secondbottom,

  at=(secondtop-outgoing boundary 1)

];

draw[

  decoration={

    markings,

    mark=at position .5 with {arrow{Latex}},

    mark=at position .5 with {node[right] {(b)};},

  },

  postaction={decorate}

]

(secondbottom-outgoing boundary 1.north) to[out=90,in=-135] (secondbottom-between incoming 1 and 2);

draw (secondbottom-outgoing boundary 1.north) to[out=90,in=-135] (secondbottom-between last incoming and last outgoing);

draw[dashed] (secondbottom-between last incoming and last outgoing) to[out=45,in=0] (secondbottom-between incoming 1 and 2);

draw[

    decoration={

    markings,

    mark=at position .25 with {arrow{Latex}},

    mark=at position .25 with {node[right] {(a)};},

  },

    postaction={decorate}

]

([out angle=90]secondbottom-outgoing boundary 1.north) .. (secondbottom-incoming boundary 2.north) .. ($(secondtop-between first and last outgoing)!.5!(secondtop-between outgoing 1 and 2)$) .. (secondbottom-incoming boundary 1.north) .. ([in angle=90]secondbottom-outgoing boundary 1.north);



node at ($(top-outgoing boundary 2.east)!.5!(secondtop-outgoing boundary 1.west)$) {(simeq)};

end{tikzpicture}

end{document}