L11n109

L11n108

L11n110

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Link Presentations

[edit Notes on L11n109's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X7,18,8,19 X19,22,20,5 X13,20,14,21 X21,14,22,15 X9,16,10,17 X15,10,16,11 X17,8,18,9 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, -7, 8, 11, -2, -5, 6, -8, 7, -9, 3, -4, 5, -6, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ -\frac{2 u v^3-u v^2-u v+u+v^3-v^2-v+2}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{2}{q^{9/2}}+\frac{1}{q^{11/2}}-\frac{2}{q^{13/2}}+\frac{1}{q^{15/2}}-\frac{1}{q^{19/2}}+\frac{1}{q^{21/2}}-\frac{1}{q^{23/2}}+\frac{1}{q^{25/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{13} z^{-1} +2 z a^{11}+2 a^{11} z^{-1} +z a^9-a^9 z^{-1} -z^5 a^7-4 z^3 a^7-2 z a^7+a^7 z^{-1} -z^5 a^5-4 z^3 a^5-3 z a^5-a^5 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^8 a^{14}+7 z^6 a^{14}-15 z^4 a^{14}+11 z^2 a^{14}-3 a^{14}-z^9 a^{13}+7 z^7 a^{13}-15 z^5 a^{13}+12 z^3 a^{13}-4 z a^{13}+a^{13} z^{-1} -2 z^8 a^{12}+15 z^6 a^{12}-34 z^4 a^{12}+28 z^2 a^{12}-7 a^{12}-z^9 a^{11}+8 z^7 a^{11}-20 z^5 a^{11}+21 z^3 a^{11}-10 z a^{11}+2 a^{11} z^{-1} -z^8 a^{10}+8 z^6 a^{10}-18 z^4 a^{10}+14 z^2 a^{10}-4 a^{10}+2 z^3 a^9-4 z a^9+a^9 z^{-1} -z^6 a^8+4 z^4 a^8-3 z^2 a^8-z^7 a^7+4 z^5 a^7-3 z^3 a^7-z a^7+a^7 z^{-1} -z^6 a^6+3 z^4 a^6-a^6-z^5 a^5+4 z^3 a^5-3 z a^5+a^5 z^{-1} }[/math] (db)

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]).

\   r    \    j   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                       1 1 -6                     1 1 0 -8                   1     1 -10               1 1 1     1 -12             1 3 1       1 -14               1         1 -16         1 2 2           -1 -18       1                 1 -20       1 1               0 -22   1 1                   0 -24                         0 -26 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-11 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-10 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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