L11a164

L11a163

L11a165

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Link Presentations

[edit Notes on L11a164's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X20,16,21,15 X18,12,19,11 X12,20,13,19 X22,18,7,17 X16,22,17,21 X6718 X4,13,5,14
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 6, -7, 11, -4, 5, -9, 8, -6, 7, -5, 9, -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{u^2 v^6-2 u^2 v^5+3 u^2 v^4-3 u^2 v^3+2 u^2 v^2+2 u v^5-4 u v^4+5 u v^3-4 u v^2+2 u v+2 v^4-3 v^3+3 v^2-2 v+1}{u v^3} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-2 q^{7/2}+5 q^{5/2}-8 q^{3/2}+10 \sqrt{q}-\frac{13}{\sqrt{q}}+\frac{12}{q^{3/2}}-\frac{11}{q^{5/2}}+\frac{8}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a z^9+a^3 z^7-8 a z^7+z^7 a^{-1} +6 a^3 z^5-25 a z^5+6 z^5 a^{-1} +13 a^3 z^3-37 a z^3+13 z^3 a^{-1} +11 a^3 z-25 a z+11 z a^{-1} +3 a^3 z^{-1} -5 a z^{-1} +2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^5-3 a^7 z^3+2 a^7 z+2 a^6 z^6-4 a^6 z^4+a^6 z^2+3 a^5 z^7-6 a^5 z^5+4 a^5 z^3-2 a^5 z+3 a^4 z^8-5 a^4 z^6+z^6 a^{-4} +4 a^4 z^4-4 z^4 a^{-4} -2 a^4 z^2+4 z^2 a^{-4} - a^{-4} +3 a^3 z^9-9 a^3 z^7+2 z^7 a^{-3} +20 a^3 z^5-6 z^5 a^{-3} -22 a^3 z^3+3 z^3 a^{-3} +13 a^3 z-3 a^3 z^{-1} +a^2 z^{10}+3 a^2 z^8+3 z^8 a^{-2} -17 a^2 z^6-9 z^6 a^{-2} +34 a^2 z^4+7 z^4 a^{-2} -23 a^2 z^2-3 z^2 a^{-2} +5 a^2+6 a z^9+3 z^9 a^{-1} -24 a z^7-10 z^7 a^{-1} +49 a z^5+16 z^5 a^{-1} -51 a z^3-19 z^3 a^{-1} +28 a z+11 z a^{-1} -5 a z^{-1} -2 a^{-1} z^{-1} +z^{10}+3 z^8-20 z^6+37 z^4-27 z^2+5 }[/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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     1   1 6                   4 1   -3 4                 4 1     3 2               6 4       -2 0             7 4         3 -2           6 7           1 -4         5 6             -1 -6       3 6               3 -8     2 5                 -3 -10     3                   3 -12 1 2                     -1 -14 1                       1

Integral Khovanov Homology (db, data source)

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