Threading a link by a polynomial: Difference between revisions
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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. |
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As an example, we can verify some formulas from Mausbaum: |
As an example, we can verify some formulas from Mausbaum, after importing KnotTheory` and the CableLink code: |
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{{Startup Note}} |
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<!--$$Import["http://katlas.org/w/index.php?title=CableLink.m&action=raw"];$$--> |
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<!--$$Import["http://katlas.org/w/index.php?title=CableLink.m&action=raw"];$$--> |
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<!--Robot Land, no human edits to "END"--> |
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{{In| |
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n = 2 | |
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in = <nowiki>Import["http://katlas.org/w/index.php?title=CableComponent.m&action=raw"];</nowiki>}} |
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<!--END--> |
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<!--$$hopfLink=PD[X[3,1,4,2],X[2,4,1,3]]; // |
<!--$$hopfLink=PD[X[3,1,4,2],X[2,4,1,3]]; // |
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bracket[n_]:=a^n-a^(-n); // |
bracket[n_]:=a^n-a^(-n); // |
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Revision as of 18:35, 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, after importing KnotTheory` and the CableLink code:
In[3]:=
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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];
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In[4]:=
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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]]
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Out[4]=
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-1/a^5 + 1/a + a - a^5
-1/a^5 + 1/a + a - a^5
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In[5]:=
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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]]
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Out[5]=
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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
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