ReIm:={Re@#,Im@#}&
SterProj[{{z1_,z2_},{w1_,w2_}}]:={(z1/(1-w2)),(z2/(1-w2)),(w1/(1-w2))};

T2={α Exp[I θ]/Sqrt[2],Exp[I ϕ]/Sqrt[2]};

ReIm@(ComplexExpand/@T2)

T=ReIm/@{Sin[α]Exp[I θ], Cos[α] Exp[I ϕ]}

SterProj[T]

Manipulate[ParametricPlot3D[SterProj[T]/.α->Α,{θ,0,2Pi},{ϕ,0,2Pi}],{Α,0,Pi/2}]

ParametricPlot3D[SterProj[T]/.α->π/8,{θ,0,2Pi},{ϕ,0,2Pi}]

ParametricPlot3D[knot,{θ,0,2π},PlotStyle->{Thick,Blue}]

ParametricPlot3D[{r Sin[θ],r Cos[θ],1/3 ((Abs[Cos[θ]+I Sin[θ]])^3) Arg[-(((Cos[θ]+I Sin[θ]))^2)]},{θ,0,2Pi},{r,0,1},PlotPoints->50]

((Abs[x+I y])^3)Arg[(x+I y)^3]/2

Plot3D[(1/2) ((Abs[x+I y])^3) Arg[(((x+I y))^3)],{x,-20.`,20.`},{y,-20.`,20.`}]

surface =Table[{r Sin[θ],r Cos[θ],π n+(1/3) (E^(-3 Im[θ])) ((Abs[r])^3) Arg[(E^(2 I θ)) (r^2)]},{n,0,2}]
knot={{Sin[((3 θ)/2)],Cos[((3 θ)/2)],((2 n π)/3)+θ},{-Sin[((3 θ)/2)],-Cos[((3 θ)/2)],((2 n π)/3)+θ}}/.n->0

((Abs[x+I y])^3)Arg[(x+I y)^2]/3/.(x+I y)->r Exp[I θ]

((Abs[x+I y])^3)Arg[(x+I y)^2]/3/.(x+I y)->r Exp[I θ]//Simplify

Plot3D[((Abs[x+I y])^3)Arg[(x+I y)^2]/3,{x,-1,1},{y,-1,1}, Exclusions->None, RegionFunction->((((#1)^2)+((#2)^2))<1&)]
ParametricPlot3D[{r Sin[θ],r Cos[θ],((Abs[x+I y])^3)Arg[(x+I y)^2]/3/.(x+I y)->r Exp[I θ]},{θ,0,2Pi},{r,0,1}]

ParametricPlot3D[surface,{θ,0,2Pi},{r,0,1}]

outsidesurface=Table[{1/r Sin[θ],1/r Cos[θ],(π n+(1/3) ((Abs[x+I y])^3) Arg[Exp[n I π](((x+I y))^2)])/.{x->Cos[θ],y->Sin[θ]} },{n,0,2}]

Show[ParametricPlot3D[{((Sin[θ+π/2])/r),((Cos[θ+π/2])/r),2π/3+(1/3) ((Abs[Exp[I θ]])^3) Arg[-((Exp[I θ])^2)]},{θ,0,Pi/2},{r,0,1},PlotRange->{{-3,3},{-3,3},{0,7}},PlotPoints->20],
ParametricPlot3D[{((Sin[θ+3π/2])/r),((Cos[θ+3π/2])/r),2π/3+(1/3) ((Abs[Exp[I θ]])^3) Arg[-((Exp[I θ])^2)]},{θ,0,Pi/2},{r,0,1},PlotRange->{{-3,3},{-3,3},{0,7}},PlotPoints->20],
ParametricPlot3D[{((Sin[θ+π])/r),((Cos[θ+π])/r),5π/3+(1/3) ((Abs[Exp[I θ]])^3) Arg[-((Exp[I θ])^2)]},{θ,0,Pi/2},{r,0,1},PlotRange->{{-3,3},{-3,3},{0,7}},PlotPoints->20],
ParametricPlot3D[{((Sin[θ+2π])/r),((Cos[θ+2π])/r),5π/3+(1/3) ((Abs[Exp[I θ]])^3) Arg[-((Exp[I θ])^2)]},{θ,0,Pi/2},{r,0,1},PlotRange->{{-3,3},{-3,3},{0,7}},PlotPoints->20],
ParametricPlot3D[{{r Sin[θ],r Cos[θ],(1/3) (E^(-3 Im[θ])) ((Abs[r])^3) Arg[(E^(2 I θ)) (r^2)]},{r Sin[θ+π/2],r Cos[θ+π/2],π+(1/3) (E^(-3 Im[θ])) ((Abs[r])^3) Arg[(E^(2 I θ)) (r^2)]},{r Sin[θ],r Cos[θ],2 π+(1/3) (E^(-3 Im[θ])) ((Abs[r])^3) Arg[(E^(2 I θ)) (r^2)]}},{θ,0,2Pi},{r,0,1},Exclusions->None],
ParametricPlot3D[knot,{θ,0,2π},PlotStyle->{Thick,Blue}]
]

Show[ParametricPlot3D[{((Sin[θ+π/2])/r),((Cos[θ+π/2])/r),2π/3+(1/3) ((Abs[Exp[I θ]])^3) Arg[-((Exp[I θ])^2)]},{θ,0,Pi/2},{r,0,1},PlotRange->{{-3,3},{-3,3},{0,7}},PlotPoints->20],
ParametricPlot3D[{((Sin[θ+3π/2])/r),((Cos[θ+3π/2])/r),2π/3+(1/3) ((Abs[Exp[I θ]])^3) Arg[-((Exp[I θ])^2)]},{θ,0,Pi/2},{r,0,1},PlotRange->{{-3,3},{-3,3},{0,7}},PlotPoints->20],
ParametricPlot3D[{((Sin[θ+π])/r),((Cos[θ+π])/r),5π/3+(1/3) ((Abs[Exp[I θ]])^3) Arg[-((Exp[I θ])^2)]},{θ,0,Pi/2},{r,0,1},PlotRange->{{-3,3},{-3,3},{0,7}},PlotPoints->20],
ParametricPlot3D[{((Sin[θ+2π])/r),((Cos[θ+2π])/r),5π/3+(1/3) ((Abs[Exp[I θ]])^3) Arg[-((Exp[I θ])^2)]},{θ,0,Pi/2},{r,0,1},PlotRange->{{-3,3},{-3,3},{0,7}},PlotPoints->20],
ParametricPlot3D[{{r Sin[θ],r Cos[θ],(1/3) (E^(-3 Im[θ])) ((Abs[r])^3) Arg[(E^(2 I θ)) (r^2)]},{r Sin[θ+π/2],r Cos[θ+π/2],π+(1/3) (E^(-3 Im[θ])) ((Abs[r])^3) Arg[(E^(2 I θ)) (r^2)]},{r Sin[θ],r Cos[θ],2 π+(1/3) (E^(-3 Im[θ])) ((Abs[r])^3) Arg[(E^(2 I θ)) (r^2)]}},{θ,0,2Pi},{r,0,1}],
ParametricPlot3D[knot,{θ,0,2π},PlotStyle->{Thick,Blue}]
]

Theta[{x_,y_,z_}]:=ArcTan[x,y]
Phi[{x_,y_,z_}]:=z
Torus[α_,θ_,ϕ_]:=ReIm/@{Sin[α]Exp[I θ], Cos[α] Exp[I ϕ]}

Show[
ParametricPlot3D[SterProj@(Torus[r,Theta[#],Phi[#]])&@{((Sin[θ+π/2])/r),((Cos[θ+π/2])/r),2π/3+(1/3) ((Abs[Exp[I θ]])^3) Arg[-((Exp[I θ])^2)]},{θ,0,2Pi},{r,0,1},PlotPoints->100],
ParametricPlot3D[SterProj@(Torus[r,Theta[#],Phi[#]])&@{((Sin[θ+3π/2])/r),((Cos[θ+3π/2])/r),2π/3+(1/3) ((Abs[Exp[I θ]])^3) Arg[-((Exp[I θ])^2)]},{θ,0,2Pi},{r,0,1},PlotPoints->100],

ParametricPlot3D[SterProj@(Torus[r,Theta[#],Phi[#]])&@{((Sin[θ+π])/r),((Cos[θ+π])/r),5π/3+(1/3) ((Abs[Exp[I θ]])^3) Arg[-((Exp[I θ])^2)]},{θ,0,2Pi},{r,0,1},PlotPoints->100],

ParametricPlot3D[SterProj@(Torus[r,Theta[#],Phi[#]])&@{((Sin[θ+2π])/r),((Cos[θ+2π])/r),5π/3+(1/3) ((Abs[Exp[I θ]])^3) Arg[-((Exp[I θ])^2)]},{θ,0,2Pi},{r,0,1},PlotPoints->100]
]

Theta@{r Sin[θ],r Cos[θ],π+(1/3) ((Abs[Cos[θ]+I Sin[θ]])^3) Arg[-(((Cos[θ]+I Sin[θ]))^2)]}

ParametricPlot3D[SterProj/@(Torus[r,Theta[#],Phi[#]]&/@surface),{θ,0,2Pi},{r,0.9,1},PlotPoints->100]

ParametricPlot3D[SterProj/@(Torus[r,Theta[#],Phi[#]]&/@knot),{θ,0,2Pi},{r,0.99,1},PlotPoints->50]

Show[
ParametricPlot3D[SterProj/@(Torus[r,Theta[#],Phi[#]]&/@knot),{θ,0,2Pi},{r,0.99,1},PlotPoints->50,MeshStyle->None,
PlotStyle->Red,PlotRange->{{-5,5},{-5,5},{-5,5}}],
ParametricPlot3D[SterProj/@(Torus[r,Theta[#],Phi[#]]&/@surface),{θ,0,2Pi},{r,0,1},PlotPoints->50,MeshStyle->None,PlotStyle->Blue],
ParametricPlot3D[SterProj@(Torus[r,Theta[#],Phi[#]])&@{((Sin[θ+π/2])/r),((Cos[θ+π/2])/r),2π/3+(1/3) ((Abs[Exp[I θ]])^3) Arg[-((Exp[I θ])^2)]},{θ,0,2Pi},{r,0,1},PlotPoints->100],
ParametricPlot3D[SterProj@(Torus[r,Theta[#],Phi[#]])&@{((Sin[θ+3π/2])/r),((Cos[θ+3π/2])/r),2π/3+(1/3) ((Abs[Exp[I θ]])^3) Arg[-((Exp[I θ])^2)]},{θ,0,2Pi},{r,0,1},PlotPoints->100],

ParametricPlot3D[SterProj@(Torus[r,Theta[#],Phi[#]])&@{((Sin[θ+π])/r),((Cos[θ+π])/r),5π/3+(1/3) ((Abs[Exp[I θ]])^3) Arg[-((Exp[I θ])^2)]},{θ,0,2Pi},{r,0,1},PlotPoints->100],

ParametricPlot3D[SterProj@(Torus[r,Theta[#],Phi[#]])&@{((Sin[θ+2π])/r),((Cos[θ+2π])/r),5π/3+(1/3) ((Abs[Exp[I θ]])^3) Arg[-((Exp[I θ])^2)]},{θ,0,2Pi},{r,0,1},PlotPoints->100]
]

Show[
ParametricPlot3D[SterProj/@(Torus[r,Theta[#],Phi[#]]&/@knot),{θ,0,2Pi},{r,0.99,1},PlotPoints->50,MeshStyle->None,
PlotStyle->Red,PlotRange->{{-2,2},{-2,2},{-2,2}}],
ParametricPlot3D[SterProj/@(Torus[r,Theta[#],Phi[#]]&/@surface),{θ,0,2Pi},{r,0,1},PlotPoints->50,MeshStyle->None,PlotStyle->Blue],

ParametricPlot3D[SterProj@(Torus[r,Theta[#],Phi[#]])&@{((Sin[θ+π/2])/r),((Cos[θ+π/2])/r),2π/3+(1/3) ((Abs[Exp[I θ]])^3) Arg[-((Exp[I θ])^2)]},{θ,0,2Pi},{r,0,1},PlotPoints->100],
ParametricPlot3D[SterProj@(Torus[r,Theta[#],Phi[#]])&@{((Sin[θ+3π/2])/r),((Cos[θ+3π/2])/r),2π/3+(1/3) ((Abs[Exp[I θ]])^3) Arg[-((Exp[I θ])^2)]},{θ,0,2Pi},{r,0,1},PlotPoints->100],

ParametricPlot3D[SterProj@(Torus[r,Theta[#],Phi[#]])&@{((Sin[θ+π])/r),((Cos[θ+π])/r),5π/3+(1/3) ((Abs[Exp[I θ]])^3) Arg[-((Exp[I θ])^2)]},{θ,0,2Pi},{r,0,1},PlotPoints->100],

ParametricPlot3D[SterProj@(Torus[r,Theta[#],Phi[#]])&@{((Sin[θ+2π])/r),((Cos[θ+2π])/r),5π/3+(1/3) ((Abs[Exp[I θ]])^3) Arg[-((Exp[I θ])^2)]},{θ,0,2Pi},{r,0,1},PlotPoints->100]

]