Mrthdrb

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

I almost finished the leftmost picture:

begintikzpicture
fill[blue!20] (7.5,0) ellipse (1 and .75);
draw (7.5,0) ellipse (1 and .75);
beginscope
 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);
endscope
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 ($(a)!0.5!(b)$) ;
beginscope[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);
endscope
draw (7.942,-0.555) node[scale=0.8,rotate=-85] $<$;
draw (7.67,-0.60) node $b$;
endtikzpicture

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
usepackagetikz-3dplot
begindocument
tdplotsetmaincoords600
tikzsetdeclare 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);


begintikzpicture[tdplot_main_coords]
pgfmathsetmacroR4
pgfmathsetmacror1.5
 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)); 
 beginscope[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));
 endscope
endtikzpicture
enddocument

If you want a cartoon, consider e.g.

documentclass[tikz,border=3.14mm]standalone
usetikzlibraryarrows.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%
 pgfmathsetmacrotmpArrowTime#2/(pgfdecoratedpathlength)
 xdeftmpArrowTimetmpArrowTime,
 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,


begindocument
begintikzpicture[scale=4]
 beginscope[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);
 endscope
 %
 beginscope[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);
 endscope
 path (left) -- (right) node[midway,scale=2]$simeq$;
endtikzpicture
enddocument

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

Using the tqft package:

documentclassarticle
%urlhttps://tex.stackexchange.com/q/481212/86
usepackagetikz
usetikzlibrary
 tqft,
 decorations.markings,
 arrows.meta,
 hobby,
 calc


begindocument

begintikzpicture[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 arrowLatex,
 ,
 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 arrowLatex,
 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 arrowLatex,
 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 arrowLatex,
 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);
endtikzpicture
enddocument