Threading a link by a polynomial: Difference between revisions

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(Created page with "<code>cableLink[link,poly,strandList,vars]</code>, 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: <!--$$Import["http://katlas.org/w/index.php?t...")
 
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<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 Mausbaum:
<!--$$Import["http://katlas.org/w/index.php?title=CableComponent.m&action=raw"];$$-->
<!--$$Import["http://katlas.org/w/index.php?title=CableLink.m&action=raw"];$$-->
<!--END-->
<!--END-->
<!--$$CableComponent[BR[3, {1, 2}], Knot[3, 1]] // DrawMorseLink$$-->
<!--$$hopfLink=PD[X[3,1,4,2],X[2,4,1,3]]; //
bracket[n_]:=a^n-a^(-n); //
bracketFact[n_]:=Product[bracket[i],{i,1,n}]; //
R[z_, n_] := Product[z + lambda[2*i], {i, 0, n - 1}]; //
cheb[0, z_] = 1;
cheb[1, z_] = z;
cheb[n_, z_] := cheb[n, z] = z*cheb[n - 1, z] - cheb[n - 2, z];
$$-->
<!--END-->
<!--$$Expand[cableLink[hopfLink,
R[Subscript[z, 1], 1]*cheb[2, Subscript[z, 2]], {1, 3}, {Subscript[
z, 1], Subscript[z, 2]}] /. {A -> a^(1/2)}] //
Expand[(-1)^1*bracketFact[3]/bracket[1]]
$$-->
<!--END-->
<!--$$Expand[cableLink[hopfLink,
R[Subscript[z, 1], 2]*cheb[4, Subscript[z, 2]], {1, 3}, {Subscript[
z, 1], Subscript[z, 2]}] /. {A -> a^(1/2)}]
Expand[(-1)^2*bracketFact[5]/bracket[1]] //
$$-->
<!--END-->
<!--END-->

Revision as of 16:28, 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 Mausbaum:

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