{z,w}={Exp[I α]Sin[γ],Exp[I β]Cos[γ]};

SterProj[{z_,w_}]:=Module[{x=Re[z],y=Im[z],u=Re[w],v=Im[w]},{(x/(1-v)),(y/(1-v)),(u/(1-v))}//ComplexExpand]
σ[{z_,w_}]:={w,z}


 g[Α_]=SterProj@{z,w}/.α->Α;
f[Γ_]=SterProj/@{{z,w},σ@{z,w}}/.γ->Γ;

Show@Table[ParametricPlot3D[g[i],{γ,0,π/4},{β,0,2π},PlotRange->{{-π,π},{-π,π},{-π,π}}],{i,0,2π,π/8}]

Show@Table[ParametricPlot3D[f[i],{α,0,2π},{β,0,2π},PlotRange->{{-π,π},{-π,π},{-π,π}}],{i,0,π/4,π/4^2}]

Clear[f]

test[r_,θ_,n_]:=((Abs[r])^4)f2[θ]+2π n/q
αf[γ_,β_,n_]:=((Abs[4/π γ])^4)f2[β]+2π n/q

Plot[test[1,θ,0],{θ,0,2π}]
Plot[test2[π/4,β,0],{β,0,2π}]
Plot[InsideWasher[π/4,β,0],{β,0,2π}]

InsideWasher[γ_,β_,n_]:= Module[
{α=αf[γ,β,n]},
{z,w}={Exp[I β]Sin[γ],Exp[I α]Cos[γ]};
{z,w}]

knot2[θ_,k_]:=Module[
{α=θ},
{z,w}={Exp[I α]Cos[((q θ)/p)+2π k/p],Exp[I α]Sin[((q θ)/p)+2π k/p]};
{z,w}]

ParametricPlot3D[{SterProj@knot2[θ,0],SterProj@knot2[θ,1]},{θ,0,2π}]

Show[ParametricPlot3D[Table[SterProj@InsideWasher[γ,β,n],{n,0,3}],{γ,0,π/4},{β,0,2π}],ParametricPlot3D[Table[SterProj@σ@InsideWasher[γ,β,n],{n,0,3}],{γ,0,π/4},{β,0,2π}]]

ParametricPlot3D[Table[SterProj@σ@InsideWasher[γ,β,n],{n,0,3}],{γ,0,π/4},{β,0,2π}]