Threading a link by a polynomial
From Knot Atlas
Jump to navigationJump to search
CableLink[link,poly,strandList,vars], whose code is available here, computes the Kauffman bracket of link (given as a PD) with components L1,L2,...,Ln, cabled by the polynomial poly in the variables z1,z2,...,zn. strandList is a list of strand labels of length n, where the ith element is the first strand label corresponding to component Li.
As an example, we can verify some formulas from Mausbaum, after importing KnotTheory` and the CableLink code:
In[3]:=
|
hopfLink=PD[X[3,1,4,2],X[2,4,1,3]];
bracket[n_]:=a^n-a^(-n);
bracketFact[n_]:=Product[bracket[i],{i,1,n}];
R[z_, n_] := Product[z + lambda[2*i], {i, 0, n - 1}];
cheb[0, z_] = 1;
cheb[1, z_] = z;
cheb[n_, z_] := cheb[n, z] = z*cheb[n - 1, z] - cheb[n - 2, z];
|
In[4]:=
|
Expand[CableLink[hopfLink,
R[Subscript[z, 1], 1]*cheb[2, Subscript[z, 2]], {1, 3}, {Subscript[
z, 1], Subscript[z, 2]}] /. {A -> a^(1/2)}]
Expand[(-1)^1*bracketFact[3]/bracket[1]]
|
Out[4]=
|
-1/a^5 + 1/a + a - a^5
-1/a^5 + 1/a + a - a^5
|
In[5]:=
|
Expand[CableLink[hopfLink,
R[Subscript[z, 1], 2]*cheb[4, Subscript[z, 2]], {1, 3}, {Subscript[
z, 1], Subscript[z, 2]}] /. {A -> a^(1/2)}]
Expand[(-1)^1*bracketFact[3]/bracket[1]]
|
Out[5]=
|
2 + 1/a^14 - 1/a^10 - 1/a^8 - 1/a^6 + 1/a^2 + a^2 - a^6 - a^8 - a^10 + a^14
2 + 1/a^14 - 1/a^10 - 1/a^8 - 1/a^6 + 1/a^2 + a^2 - a^6 - a^8 - a^10 + a^14
|
