L11n330

L11n329

L11n331

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Link Presentations

[edit Notes on L11n330's Link Presentations]

Planar diagram presentation	X6172 X5,14,6,15 X3849 X2,16,3,15 X16,7,17,8 X13,18,14,19 X9,13,10,22 X11,21,12,20 X19,5,20,12 X21,11,22,10 X17,1,18,4
Gauss code	{1, -4, -3, 11}, {-2, -1, 5, 3, -7, 10, -8, 9}, {-6, 2, 4, -5, -11, 6, -9, 8, -10, 7}

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^2-2 u v^2 w+u v w^3-2 u v w^2+u v w-u w^3+u w^2+v^3 (-w)+v^3-v^2 w^2+2 v^2 w-v^2+2 v w^2-v w}{\sqrt{u} v^{3/2} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 2 q^4-2 q^3+4 q^2-5 q+7-5 q^{-1} +5 q^{-2} -3 q^{-3} +2 q^{-4} - q^{-5} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^4 \left(-z^2\right)+ a^{-4} z^{-2} -a^4+2 a^{-4} +a^2 z^4+2 a^2 z^2-3 z^2 a^{-2} -2 a^{-2} z^{-2} +2 a^2-5 a^{-2} +z^4+z^2+ z^{-2} +2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^5 z^5-3 a^5 z^3+a^5 z+2 a^4 z^6-6 a^4 z^4+3 z^4 a^{-4} +3 a^4 z^2-10 z^2 a^{-4} - a^{-4} z^{-2} -a^4+6 a^{-4} +2 a^3 z^7+z^7 a^{-3} -5 a^3 z^5-3 z^5 a^{-3} +a^3 z^3+4 z^3 a^{-3} +a^3 z-5 z a^{-3} +2 a^{-3} z^{-1} +2 a^2 z^8+2 z^8 a^{-2} -7 a^2 z^6-11 z^6 a^{-2} +9 a^2 z^4+27 z^4 a^{-2} -4 a^2 z^2-30 z^2 a^{-2} -2 a^{-2} z^{-2} +13 a^{-2} +a z^9+z^9 a^{-1} -2 a z^7-3 z^7 a^{-1} -2 a z^5+z^5 a^{-1} +9 a z^3+9 z^3 a^{-1} -3 a z-8 z a^{-1} +2 a^{-1} z^{-1} +4 z^8-20 z^6+39 z^4-27 z^2- z^{-2} +9 }[/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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                   2 2 7                 1 1 0 5               3 1   2 3             2 1     -1 1           5 3       2 -1         3 5         2 -3       2 2           0 -5     1 3             2 -7   1 2               -1 -9   1                 1 -11 1                   -1

Integral Khovanov Homology (db, data source)

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

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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