L11a186

L11a185

L11a187

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Link Presentations

[edit Notes on L11a186's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X22,18,7,17 X16,5,17,6 X4,15,5,16 X14,22,15,21 X18,12,19,11 X20,14,21,13 X12,20,13,19 X2738 X6,9,1,10
Gauss code	{1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, 7, -9, 8, -6, 5, -4, 3, -7, 9, -8, 6, -3}

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^5-2 u^2 v^4+2 u^2 v^3-2 u^2 v^2+2 u^2 v+u v^6-3 u v^5+5 u v^4-5 u v^3+5 u v^2-3 u v+u+2 v^5-2 v^4+2 v^3-2 v^2+v}{u v^3} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -3 q^{9/2}+\frac{2}{q^{9/2}}+6 q^{7/2}-\frac{5}{q^{7/2}}-9 q^{5/2}+\frac{7}{q^{5/2}}+12 q^{3/2}-\frac{11}{q^{3/2}}+q^{11/2}-\frac{1}{q^{11/2}}-13 \sqrt{q}+\frac{12}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^3 z^5+z^5 a^{-3} +4 a^3 z^3+3 z^3 a^{-3} +5 a^3 z+2 z a^{-3} +3 a^3 z^{-1} -a z^7-z^7 a^{-1} -5 a z^5-4 z^5 a^{-1} -10 a z^3-4 z^3 a^{-1} -11 a z+z a^{-1} -5 a z^{-1} +2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-6} -z^2 a^{-6} +a^5 z^7-5 a^5 z^5+3 z^5 a^{-5} +7 a^5 z^3-3 z^3 a^{-5} -2 a^5 z+2 a^4 z^8-8 a^4 z^6+5 z^6 a^{-4} +8 a^4 z^4-6 z^4 a^{-4} -a^4 z^2+3 z^2 a^{-4} - a^{-4} +3 a^3 z^9-14 a^3 z^7+6 z^7 a^{-3} +26 a^3 z^5-9 z^5 a^{-3} -28 a^3 z^3+7 z^3 a^{-3} +15 a^3 z-2 z a^{-3} -3 a^3 z^{-1} +a^2 z^{10}+2 a^2 z^8+5 z^8 a^{-2} -18 a^2 z^6-6 z^6 a^{-2} +28 a^2 z^4-20 a^2 z^2+4 z^2 a^{-2} +5 a^2+6 a z^9+3 z^9 a^{-1} -22 a z^7-z^7 a^{-1} +36 a z^5-7 z^5 a^{-1} -40 a z^3+5 z^3 a^{-1} +22 a z+3 z a^{-1} -5 a z^{-1} -2 a^{-1} z^{-1} +z^{10}+5 z^8-21 z^6+27 z^4-19 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 12                       1 -1 10                     2   2 8                   4 1   -3 6                 5 2     3 4               7 4       -3 2             6 5         1 0           7 8           1 -2         4 5             -1 -4       3 7               4 -6     2 4                 -2 -8     3                   3 -10 1 2                     -1 -12 1                       1

Integral Khovanov Homology (db, data source)

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