Threading a link by a polynomial: Difference between revisions
From Knot Atlas
Jump to navigationJump to search
Sam.panitch (talk | contribs) No edit summary Tag: Manual revert |
Sam.panitch (talk | contribs) No edit summary Tag: Manual revert |
||
| Line 52: | Line 52: | ||
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</nowiki>}} |
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</nowiki>}} |
||
<!--END--> |
<!--END--> |
||
==References== |
|||
{{note|Masbaum}} Masbaum, Gregor. ''Skein-theoretical derivations of some formulas of Habiro.'' Alg. and Geo. Topology 3 (2003): 537–556. https://doi.org/10.2140/agt.2003.3.537 |
{{note|Masbaum}} Masbaum, Gregor. ''Skein-theoretical derivations of some formulas of Habiro.'' Alg. and Geo. Topology 3 (2003): 537–556. https://doi.org/10.2140/agt.2003.3.537 |
||
Revision as of 18:46, 5 August 2025
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 [Masbaum], 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
|
[Masbaum] ^ Masbaum, Gregor. Skein-theoretical derivations of some formulas of Habiro. Alg. and Geo. Topology 3 (2003): 537–556. https://doi.org/10.2140/agt.2003.3.537
