Yhyjyk

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.