L11n181

L11n180

L11n182

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Link Presentations

[edit Notes on L11n181's Link Presentations]

Planar diagram presentation	X8192 X3,10,4,11 X5,14,6,15 X16,8,17,7 X22,18,7,17 X15,13,16,12 X20,10,21,9 X11,19,12,18 X13,6,14,1 X19,4,20,5 X2,21,3,22
Gauss code	{1, -11, -2, 10, -3, 9}, {4, -1, 7, 2, -8, 6, -9, 3, -6, -4, 5, 8, -10, -7, 11, -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^2 v^2+u^2 v+u^2-u v^3+3 u v^2-u v+v^4+v^3-v^2}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{7/2}+2 q^{5/2}-\frac{2}{q^{5/2}}-2 q^{3/2}+\frac{2}{q^{3/2}}-\frac{1}{q^{11/2}}+3 \sqrt{q}-\frac{3}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z+2 a^5 z^{-1} -a^3 z^3-5 a^3 z-3 a^3 z^{-1} -z a^{-3} +a z^3+z^3 a^{-1} +a z+a z^{-1} +z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^5 z^7-7 a^5 z^5+14 a^5 z^3-9 a^5 z+2 a^5 z^{-1} -2 a^4 z^4+6 a^4 z^2-3 a^4-2 a^3 z^5+z^5 a^{-3} +9 a^3 z^3-3 z^3 a^{-3} -9 a^3 z+z a^{-3} +3 a^3 z^{-1} +2 z^6 a^{-2} -7 z^4 a^{-2} +4 a^2 z^2+5 z^2 a^{-2} -3 a^2+z^7 a^{-1} +2 a z^5-2 z^5 a^{-1} -3 a z^3-z^3 a^{-1} +z a^{-1} +a z^{-1} +2 z^6-5 z^4+3 z^2-1 }[/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 χ 8                     1 1 6                   1   -1 4                 1 1   0 2               2 1     -1 0           1 2 1       0 -2           2 3         1 -4       1 2 1           0 -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}\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 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\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 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}_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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