Threading a link by a polynomial: Difference between revisions

From Knot Atlas
Jump to navigationJump to search
No edit summary
Tag: Manual revert
No edit summary
Tag: Manual revert
 
(8 intermediate revisions by the same user not shown)
Line 1: Line 1:
<code>CableLink[link,poly,strandList,vars]</code>, whose code is available [[cableLink.m|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.
<code>CableLink[link,poly,strandList,vars]</code>, whose code is available [[cableLink.m|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:
As an example, we can verify some formulas from {{ref|Masbaum}}, after importing KnotTheory` and the CableLink code:

<!--$$Import["http://katlas.org/w/index.php?title=CableLink.m&action=raw"];$$-->
<!--END-->
<!--$$hopfLink=PD[X[3,1,4,2],X[2,4,1,3]]; //
<!--$$hopfLink=PD[X[3,1,4,2],X[2,4,1,3]]; //
bracket[n_]:=a^n-a^(-n); //
bracket[n_]:=a^n-a^(-n); //
bracketFact[n_]:=Product[bracket[i],{i,1,n}]; //
bracketFact[n_]:=Product[bracket[i],{i,1,n}]; //
lambda[n_] := A^(2*n + 2) + A^(-2*n - 2); //
R[z_, n_] := Product[z + lambda[2*i], {i, 0, n - 1}]; //
R[z_, n_] := Product[z + lambda[2*i], {i, 0, n - 1}]; //
cheb[0, z_] = 1;
cheb[0, z_] = 1;
Line 11: Line 11:
cheb[n_, z_] := cheb[n, z] = z*cheb[n - 1, z] - cheb[n - 2, z];
cheb[n_, z_] := cheb[n, z] = z*cheb[n - 1, z] - cheb[n - 2, z];
$$-->
$$-->
<!--Robot Land, no human edits to "END"-->
{{In|
n = 3 |
in = <nowiki>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}];
lambda[n_] := A^(2*n + 2) + A^(-2*n - 2);
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];</nowiki> |
}}
<!--END-->
<!--END-->

<!--$$Expand[CableLink[hopfLink,
<!--$$Expand[CableLink[hopfLink,
R[Subscript[z, 1], 1]*cheb[2, Subscript[z, 2]], {1, 3}, {Subscript[
R[Subscript[z, 1], 1]*cheb[2, Subscript[z, 2]], {1, 3}, {Subscript[
Line 17: Line 30:
Expand[(-1)^1*bracketFact[3]/bracket[1]]
Expand[(-1)^1*bracketFact[3]/bracket[1]]
$$-->
$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 4 |
in = <nowiki>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]]</nowiki> |
out= <nowiki>-1/a^5 + 1/a + a - a^5
-1/a^5 + 1/a + a - a^5</nowiki>}}
<!--END-->
<!--END-->
<!--$$Expand[CableLink[hopfLink,
<!--$$Expand[CableLink[hopfLink,
Line 23: Line 45:
Expand[(-1)^2*bracketFact[5]/bracket[1]] //
Expand[(-1)^2*bracketFact[5]/bracket[1]] //
$$-->
$$-->
{{InOut|
n = 5 |
in = <nowiki>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)^2*bracketFact[5]/bracket[1]]</nowiki> |
out= <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
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-->

{{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

Latest revision as of 19:05, 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}]; lambda[n_] := A^(2*n + 2) + A^(-2*n - 2); 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)^2*bracketFact[5]/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

Retrieved from "https://katlas.org/index.php?title=Threading_a_link_by_a_polynomial&oldid=1725866"