L11n344

L11n343

L11n345

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Link Presentations

[edit Notes on L11n344's Link Presentations]

Planar diagram presentation	X6172 X2,16,3,15 X3,10,4,11 X5,14,6,15 X11,22,12,13 X13,12,14,5 X21,1,22,4 X20,17,21,18 X16,7,17,8 X8,20,9,19 X18,10,19,9
Gauss code	{1, -2, -3, 7}, {-4, -1, 9, -10, 11, 3, -5, 6}, {-6, 4, 2, -9, 8, -11, 10, -8, -7, 5}

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 v^2 w^3-3 u v^2 w^2+2 u v^2 w-u v w^3+3 u v w^2-3 u v w-u w^2+u w+v^3 \left(-w^2\right)+v^3 w+3 v^2 w^2-3 v^2 w+v^2-2 v w^2+3 v w-v}{\sqrt{u} v^{3/2} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 2 q-3+7 q^{-1} -9 q^{-2} +11 q^{-3} -9 q^{-4} +9 q^{-5} -6 q^{-6} +3 q^{-7} - q^{-8} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^6 \left(-z^4\right)-2 a^6 z^2-2 a^6+a^4 z^6+4 a^4 z^4+8 a^4 z^2+a^4 z^{-2} +7 a^4-3 a^2 z^4-9 a^2 z^2-2 a^2 z^{-2} -9 a^2+2 z^2+ z^{-2} +4 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^9-2 z^3 a^9+z a^9+3 z^6 a^8-6 z^4 a^8+3 z^2 a^8-a^8+4 z^7 a^7-6 z^5 a^7-z^3 a^7+z a^7+3 z^8 a^6-2 z^6 a^6-4 z^4 a^6+2 z^2 a^6+z^9 a^5+4 z^7 a^5-10 z^5 a^5+9 z^3 a^5-3 z a^5+5 z^8 a^4-12 z^6 a^4+21 z^4 a^4-21 z^2 a^4-a^4 z^{-2} +9 a^4+z^9 a^3+z^7 a^3-4 z^5 a^3+11 z^3 a^3-8 z a^3+2 a^3 z^{-1} +2 z^8 a^2-7 z^6 a^2+22 z^4 a^2-28 z^2 a^2-2 a^2 z^{-2} +13 a^2+z^7 a-z^5 a+3 z^3 a-5 z a+2 a z^{-1} +3 z^4-8 z^2- z^{-2} +6 }[/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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 3                   2 2 1                 1   -1 -1               6 2   4 -3             6 4     -2 -5           5 3       2 -7         4 6         2 -9       5 5           0 -11     2 5             3 -13   1 4               -3 -15   2                 2 -17 1                   -1

Integral Khovanov Homology (db, data source)

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

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