Pensieve header: Drawing a inflated tori, as in classes/0506/Topology.

<< KnotTheory`

TubePlot[TorusKnot[8,3]]

TC[r1_, t1_,r2_,t2_ ] := {
(r1 +r2 Cos[2π t2])Cos[2π t1],
(r1 +r2 Cos[2π t2])Sin[2π t1],
r2 Sin[2π t2]
};
InflatedTorus[p_,q_,b_] :=ParametricPlot3D[
TC[2,p t-q s,
1+b((p^2)+(q^2))s(1-((p^2)+(q^2))s),q t+p s],
{t, 0, 1}, {s, 0,1/((p^2)+(q^2))},
PlotPoints -> {6((p^2)+(q^2))+1,7},
Boxed->False, Axes->False
]

InflatedTorus[3,8,1]

InflatedTorus[3,8,-1]

s=Sqrt[2]/2;
ParametricPlot3D[{{Cos[t],Sin[t],0},{-2/3+Cos[t],s Sin[t],s Sin[t]},{2/3+Cos[t],s Sin[t],-s Sin[t]}},{t,0,2Pi}, Axes->False, Boxed->False]/.Line[pts_]:>Tube[pts,0.1]

ParametricPlot3D[{{Cos[t],Sin[t],0},{-4/3+Cos[t],s Sin[t],s Sin[t]},{4/3+Cos[t],s Sin[t],-s Sin[t]}},{t,0,2Pi}, Axes->False, Boxed->False]/.Line[pts_]:>Tube[pts,0.1]