L11n162

L11n161

L11n163

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Link Presentations

[edit Notes on L11n162's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X17,13,18,12 X14,5,15,6 X4,13,5,14 X11,19,12,18 X19,7,20,22 X15,21,16,20 X21,17,22,16 X2738 X6,9,1,10
Gauss code	{1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, -6, 3, 5, -4, -8, 9, -3, 6, -7, 8, -9, 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{t(1) t(2)^4+t(2)^4+t(1)^2 t(2)^3-2 t(1) t(2)^3-t(1)^2 t(2)^2+3 t(1) t(2)^2-t(2)^2-2 t(1) t(2)+t(2)+t(1)^2+t(1)}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-2 q^{7/2}+3 q^{5/2}-\frac{2}{q^{5/2}}-4 q^{3/2}+\frac{3}{q^{3/2}}-\frac{1}{q^{11/2}}+4 \sqrt{q}-\frac{4}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z a^5+a^5 z^{-1} -z^5 a-4 z^3 a-5 z a-2 a z^{-1} -z^5 a^{-1} -3 z^3 a^{-1} -z a^{-1} + a^{-1} z^{-1} +z^3 a^{-3} +2 z a^{-3} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^5 z^7-7 a^5 z^5+14 a^5 z^3-9 a^5 z+a^5 z^{-1} +z^6 a^{-4} -2 a^4 z^4-4 z^4 a^{-4} +4 a^4 z^2+3 z^2 a^{-4} -a^4+2 z^7 a^{-3} -2 a^3 z^5-8 z^5 a^{-3} +6 a^3 z^3+7 z^3 a^{-3} -3 a^3 z-2 z a^{-3} +a^2 z^8+2 z^8 a^{-2} -7 a^2 z^6-8 z^6 a^{-2} +15 a^2 z^4+8 z^4 a^{-2} -10 a^2 z^2-5 z^2 a^{-2} +3 a^2+2 a^{-2} +a z^9+z^9 a^{-1} -6 a z^7-3 z^7 a^{-1} +13 a z^5-13 a z^3+2 z^3 a^{-1} +8 a z-2 a z^{-1} - a^{-1} z^{-1} +3 z^8-16 z^6+29 z^4-22 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                   2 1   -1 4                 2 1     1 2               2 2       0 0           1 3 2         0 -2           2 3           1 -4       1 2 2             -1 -6         2               2 -8     1 1                 0 -10 1                       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=-2 }[/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{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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