L11n82

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L11n83

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Link Presentations

[edit Notes on L11n82's Link Presentations]

Planar diagram presentation	X6172 X20,7,21,8 X4,21,1,22 X5,14,6,15 X3,10,4,11 X11,16,12,17 X15,12,16,13 X13,22,14,5 X18,9,19,10 X17,2,18,3 X8,19,9,20
Gauss code	{1, 10, -5, -3}, {-4, -1, 2, -11, 9, 5, -6, 7, -8, 4, -7, 6, -10, -9, 11, -2, 3, 8}

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{4 u v^3-6 u v^2+4 u v-u-v^3+4 v^2-6 v+4}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{10}{q^{13/2}}+\frac{10}{q^{15/2}}-\frac{10}{q^{17/2}}+\frac{7}{q^{19/2}}-\frac{4}{q^{21/2}}+\frac{2}{q^{23/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{13} z^{-1} +z^3 a^{11}+3 z a^{11}+2 a^{11} z^{-1} -z^5 a^9-2 z^3 a^9-z a^9-a^9 z^{-1} -2 z^5 a^7-5 z^3 a^7-z a^7+a^7 z^{-1} -z^5 a^5-3 z^3 a^5-3 z a^5-a^5 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -3 z^4 a^{14}+6 z^2 a^{14}-3 a^{14}-z^7 a^{13}-z^5 a^{13}+3 z^3 a^{13}-3 z a^{13}+a^{13} z^{-1} -2 z^8 a^{12}+5 z^6 a^{12}-16 z^4 a^{12}+20 z^2 a^{12}-7 a^{12}-z^9 a^{11}-7 z^5 a^{11}+14 z^3 a^{11}-9 z a^{11}+2 a^{11} z^{-1} -4 z^8 a^{10}+5 z^6 a^{10}-8 z^4 a^{10}+11 z^2 a^{10}-4 a^{10}-z^9 a^9-2 z^7 a^9-z^5 a^9+8 z^3 a^9-5 z a^9+a^9 z^{-1} -2 z^8 a^8-2 z^6 a^8+8 z^4 a^8-3 z^2 a^8-3 z^7 a^7+4 z^5 a^7-2 z a^7+a^7 z^{-1} -2 z^6 a^6+3 z^4 a^6-a^6-z^5 a^5+3 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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                   1 1 -6                 2 1 -1 -8               4     4 -10             4 2     -2 -12           6 4       2 -14         5 5         0 -16       5 5           0 -18     2 5             3 -20   2 5               -3 -22   2                 2 -24 2                   -2

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{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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