SterProj[{{z1_,z2_},{w1_,w2_}}]:={(z1/(Sqrt[2]-w2)),(z2/(Sqrt[2]-w2)),(w1/(Sqrt[2]-w2))};
knotpairs={ζ Exp[q I θ],η Exp[p I θ+((π I)/q)]};Factor[ComplexExpand[{Re[#],Im[#]}&/@(knotpairs/.{q->2,p->3,ζ-> 1,η-> 1})]];
SterProj[%];
knotplot=ParametricPlot3D[%,{θ,0,2π},Boxed->False,Axes->False]

knotpoly=z^2+w^3/.{z->zr+I zi,w->wr+I wi}

ComplexExpand[{Re[knotpoly],Im[knotpoly]}]

#==0&/@ComplexExpand[{Re[knotpoly],Im[knotpoly]}]

InverseStereographicProjection={zr->((4zr)/(zr^2+zi^2+wr^2+2)),zi->((4zi)/(zr^2+zi^2+wr^2+2)),wr->((4wr)/(zr^2+zi^2+wr^2+2)),wi->(( Sqrt[2](zr^2+zi^2+wr^2-2))/(zr^2+zi^2+wr^2+2))}
(*InverseStereographicProjection={zr->((2zr)/(zr^2+zi^2+wr^2+1)),zi->((2zi)/(zr^2+zi^2+wr^2+1)),wr->((2wr)/(zr^2+zi^2+wr^2+1)),wi->(( zr^2+zi^2+wr^2-1)/(zr^2+zi^2+wr^2+1))}*)

Factor[ComplexExpand[{Re[knotpoly],Im[knotpoly]}]/.InverseStereographicProjection]
{f,g}=Simplify[%*(((2+(wr^2)+(zi^2)+(zr^2)))^3)]

function=Cos[θ]f+Sin[θ]g/.θ->Pi/2

Show[ContourPlot3D[{function==0}, {zr,-4, 4}, {zi, -4, 4}, {wr, -4, 4},PlotPoints->40,Mesh->False,Boxed->False,Axes->False],knotplot,ViewPoint->{5, 0, 0}]

zr^2+zi^2+wr^2+wi^2==1

{zr^2+zi^2+wr^2+wi^2==1 ,-3 (wi^2) wr+(wr^3)-(zi^2)+(zr^2)==0,-(wi^3)+3 wi (wr^2)+2 zi zr==0}

Solve[{(wi^2)+(wr^2)+(zi^2)+(zr^2)==1,-3 (wi^2) wr+(wr^3)-(zi^2)+(zr^2)==0,-(wi^3)+3 wi (wr^2)+2 zi zr==0},{wi,wr,zi,zr}]

{(wi^2)+(wr^2)+(zi^2)+(zr^2)==Sqrt[2],-3 (wi^2) wr+(wr^3)-(zi^2)+(zr^2)==0,-(wi^3)+3 wi (wr^2)+2 zi zr==0}/.InverseStereographicProjection

Solve[%23,{wr,zi,zr}]

Solve[%14,{wr,zi,zr}]

RegionPlot3D[cond,{zr,-100,100},{zi,-100,100},{wr,-100,100}]

cond=((wi^2)+(wr^2)+(zi^2)+(zr^2)==1&&-3 (wi^2) wr+(wr^3)-(zi^2)+(zr^2)==0&&-(wi^3)+3 wi (wr^2)+2 zi zr==0)/.InverseStereographicProjection

Solve[cond,{wr,zi,zr}]

Simplify[cond]

Solve[%43,{wr,zi,zr}]

Simplify[Factor[#==0&/@ComplexExpand[{Re[knotpoly],Im[knotpoly]}]/.InverseStereographicProjection]]

cond=(1/(1+(wr^2)+(zi^2)+(zr^2)))(3 (wr^5)+2 (wr^2) ((zi^2)-(zr^2))+3 wr (((-1+(zi^2)+(zr^2)))^2)+2 (wr^3) (-5+3 (zi^2)+3 (zr^2))+2 ((zi^2)+(zi^4)-(zr^2) (1+(zr^2))))==0&&(1/(1+(wr^2)+(zi^2)+(zr^2)))((wr^6)+(zi^6)-8 (zi^3) zr+3 (zi^4) (-1+(zr^2))+3 (zi^2) (((-1+(zr^2)))^2)+(((-1+(zr^2)))^3)+3 (wr^4) (-5+(zi^2)+(zr^2))-8 zi (zr+(zr^3))+(wr^2) (3 (zi^4)-8 zi zr+6 (zi^2) (-3+(zr^2))+3 (5-6 (zr^2)+(zr^4))))==0&&((wi^2)+(wr^2)+(zi^2)+(zr^2))==1/.InverseStereographicProjection

cond={(wi^2)+(wr^2)+(zi^2)+(zr^2)==Sqrt[2],-3 (wi^2) wr+(wr^3)-(zi^2)+(zr^2)==0,-(wi^3)+3 wi (wr^2)+2 zi zr==0}/.InverseStereographicProjection

#==0&/@eqns

#==0&/@eqns

-8 (3 (wr^5)+2 (wr^2) ((zi^2)-(zr^2))+3 wr (((-2+(zi^2)+(zr^2)))^2)+(wr^3) (-20+6 (zi^2)+6 (zr^2))+2 (2 (zi^2)+(zi^4)-(zr^2) (2+(zr^2))))==0&&-2 ((Sqrt[2]) (wr^6)+(Sqrt[2]) (zi^6)-16 (zi^3) zr+3 (Sqrt[2]) (zi^4) (-2+(zr^2))+3 (Sqrt[2]) (zi^2) (((-2+(zr^2)))^2)+(Sqrt[2]) (((-2+(zr^2)))^3)-16 zi zr (2+(zr^2))+3 (Sqrt[2]) (wr^4) (-10+(zi^2)+(zr^2))+(wr^2) (3 (Sqrt[2]) (zi^4)-16 zi zr+6 (Sqrt[2]) (zi^2) (-6+(zr^2))+3 (Sqrt[2]) (20-12 (zr^2)+(zr^4))))==0

RegionPlot3D[-8 (3 (wr^5)+2 (wr^2) ((zi^2)-(zr^2))+3 wr (((-2+(zi^2)+(zr^2)))^2)+(wr^3) (-20+6 (zi^2)+6 (zr^2))+2 (2 (zi^2)+(zi^4)-(zr^2) (2+(zr^2))))==0&&-2 ((Sqrt[2]) (wr^6)+(Sqrt[2]) (zi^6)-16 (zi^3) zr+3 (Sqrt[2]) (zi^4) (-2+(zr^2))+3 (Sqrt[2]) (zi^2) (((-2+(zr^2)))^2)+(Sqrt[2]) (((-2+(zr^2)))^3)-16 zi zr (2+(zr^2))+3 (Sqrt[2]) (wr^4) (-10+(zi^2)+(zr^2))+(wr^2) (3 (Sqrt[2]) (zi^4)-16 zi zr+6 (Sqrt[2]) (zi^2) (-6+(zr^2))+3 (Sqrt[2]) (20-12 (zr^2)+(zr^4))))==0,{zr,-1,1},{zi,-1,1},{wr,-1,1}]

ParametricPlot3D[{((0.5` Cos[2 θ])/(1-0.5` Cos[3 θ])),((0.5` Sin[2 θ])/(1-0.5` Cos[3 θ])),-((0.5` Sin[3 θ])/(1-0.5` Cos[3 θ]))},{θ,-26.389378290154262`,26.389378290154262`}]

RegionPlot3D@ImplicitRegion[0.8<x^2+y^2+z^2<1&&x>0,{x,y,z}]

ExpToTrig[Exp[I x]]