In[]:=
ClearAll[a1,a2,a3,b1,b2,b3,c1,c2,c3,d1,d2,d3];v1={a1,a2};v2={b1,b2};v3={c1,c2};v4={d1,d2};M={{0,1},{-1,0}}MatrixForm[M]n=5;(*"Trefoil Knot"e3=v1+(v1.M.v2)v2;e1=v2+(v2.M.v3)v3;e2=v3+(v3.M.v1)v1;soln=Solve[{v1e1,v2e2,v3e3},Modulusn];Length[soln];soln={{a1,a2},{b1,b2},{c1,c2}}/.soln;"Coloring Polynomial Treefoil"gPol=0;For[i=1,i≤Length[soln],gPol=gPol+q^(CountDistinct[soln[[i]]]);i=i+1]gPol*)"Connected sum of Hopy links"e2=v3+(v3.M.v1)v1;e3=v2+(v2.M.v4)v4;e1=v1+(v1.M.v3)v3;e4=v4+(v4.M.v3)v3;soln=Solve[{v1e1,v2e2,v3e3,v4e4},Modulusn];Length[soln]soln={{a1,a2},{b1,b2},{c1,c2},{d1,d2}}/.soln;"Note that v2 = v3 in all solutions""Coloring Polynomial Hopf"gPol=0;For[i=1,i≤Length[soln],gPol=gPol+q^(CountDistinct[soln[[i]]]);i=i+1]gPol"Color configurations Hopf"s1=Map[PositionIndex,soln];s1=Map[Values,s1];s1=DeleteDuplicates[s1]
Out[]=
{{0,1},{-1,0}}
Out[]//MatrixForm=
0 | 1 |
-1 | 0 |
Out[]=
Connected sum of Hopy links
Out[]=
1225
Out[]=
Note that v2 = v3 in all solutions
Out[]=
Coloring Polynomial Hopf
Out[]=
25q+360+840
2
q
3
q
Out[]=
Color configurations Hopf
Out[]=
{{{1,2,3,4}},{{1,2,3},{4}},{{1,4},{2,3}},{{1},{2,3,4}},{{1},{2,3},{4}}}