L11n392

L11n391

L11n393

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Link Presentations

[edit Notes on L11n392's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X5,12,6,13 X8493 X22,14,19,13 X20,10,21,9 X10,20,11,19 X14,22,15,21 X11,18,12,5 X2,16,3,15
Gauss code	{1, -11, 5, -3}, {8, -7, 9, -6}, {-4, -1, 2, -5, 7, -8, -10, 4, 6, -9, 11, -2, 3, 10}

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{(t(1)-1) (t(3)-1)^2 (t(3) t(2)-2 t(2)-2 t(3)+1)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-5} +3 q^4+4 q^{-4} -6 q^3-9 q^{-3} +13 q^2+13 q^{-2} -14 q-16 q^{-1} +17 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^4 z^2+2 a^{-4} z^{-2} +2 a^{-4} +2 a^2 z^4+z^4 a^{-2} +a^2 z^2-3 z^2 a^{-2} -a^2 z^{-2} -5 a^{-2} z^{-2} -2 a^2-8 a^{-2} -z^6-z^4+3 z^2+4 z^{-2} +8 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^5 z^5-a^5 z^3+4 a^4 z^6-5 a^4 z^4+6 z^4 a^{-4} +2 a^4 z^2-11 z^2 a^{-4} -2 a^{-4} z^{-2} +8 a^{-4} +8 a^3 z^7+3 z^7 a^{-3} -14 a^3 z^5-3 z^5 a^{-3} +9 a^3 z^3+10 z^3 a^{-3} -3 a^3 z-13 z a^{-3} +a^3 z^{-1} +5 a^{-3} z^{-1} +7 a^2 z^8+5 z^8 a^{-2} -6 a^2 z^6-8 z^6 a^{-2} -4 a^2 z^4+17 z^4 a^{-2} -24 z^2 a^{-2} -a^2 z^{-2} -5 a^{-2} z^{-2} +2 a^2+16 a^{-2} +2 a z^9+2 z^9 a^{-1} +15 a z^7+10 z^7 a^{-1} -38 a z^5-26 z^5 a^{-1} +29 a z^3+29 z^3 a^{-1} -14 a z-24 z a^{-1} +5 a z^{-1} +9 a^{-1} z^{-1} +12 z^8-18 z^6+12 z^4-15 z^2-4 z^{-2} +11 }[/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                   3 3 7                 5 2 -3 5               8 1   7 3             6 5     -1 1           11 8       3 -1         7 8         1 -3       6 9           -3 -5     3 7             4 -7   1 6               -5 -9   3                 3 -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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\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^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\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^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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