L10n7

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Link Presentations

[edit Notes on L10n7's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X9,14,10,15 X3849 X5,11,6,10 X11,5,12,20 X13,19,14,18 X19,13,20,12 X15,2,16,3
Gauss code	{1, 10, -5, -3}, {-6, -1, 2, 5, -4, 6, -7, 9, -8, 4, -10, -2, 3, 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{2 u v^3-3 u v^2-3 v+2}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{2}{q^{9/2}}-\frac{3}{q^{7/2}}-q^{5/2}+\frac{3}{q^{5/2}}+q^{3/2}-\frac{3}{q^{3/2}}-\frac{2}{q^{11/2}}-2 \sqrt{q}+\frac{3}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 z^{-1} -z^3 a^5-3 z a^5-a^5 z^{-1} +z^5 a^3+3 z^3 a^3-a^3 z^{-1} +z^5 a+4 z^3 a+4 z a+2 a z^{-1} -z^3 a^{-1} -3 z a^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 3 a^7 z-a^7 z^{-1} +a^6 z^4+a^6+2 a^5 z^5-4 a^5 z^3+3 a^5 z-a^5 z^{-1} +2 a^4 z^6-4 a^4 z^4-a^4 z^2+2 a^4+2 a^3 z^7-7 a^3 z^5+8 a^3 z^3-7 a^3 z+a^3 z^{-1} +a^2 z^8-3 a^2 z^6+2 a^2 z^4-4 a^2 z^2+3 a^2+3 a z^7+z^7 a^{-1} -15 a z^5-6 z^5 a^{-1} +23 a z^3+11 z^3 a^{-1} -13 a z-6 z a^{-1} +2 a z^{-1} + a^{-1} z^{-1} +z^8-5 z^6+7 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   \ -4 -3 -2 -1 0 1 2 3 4 χ 6                 1 1 4                   0 2             2 1   1 0           1       -1 -2         2 2       0 -4       2 2         0 -6     1 1           0 -8   1 2             1 -10 1 1               0 -12 2                 2

Integral Khovanov Homology (db, data source)

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