L10n86

L10n85

L10n87

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Link L10n86.

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Link Presentations

[edit Notes on L10n86's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X13,20,14,15 X16,8,17,7 X8,16,9,15 X11,18,12,19 X19,12,20,13 X17,14,18,5 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {5, -4, -8, 6, -7, 3}, {9, -1, 4, -5, 10, -2, -6, 7, -3, 8}

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(3)-1) \left(t(1) t(3) t(2)^2-t(3) t(2)^2-t(1) t(3) t(2)-t(2)-t(1)+1\right)}{\sqrt{t(1)} t(2) t(3)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 1-2 q^{-1} +3 q^{-2} -3 q^{-3} +5 q^{-4} -3 q^{-5} +4 q^{-6} -2 q^{-7} + q^{-8} }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^8 z^{-2} +z^4 a^6+2 z^2 a^6-2 a^6 z^{-2} -2 a^6-z^6 a^4-4 z^4 a^4-3 z^2 a^4+a^4 z^{-2} +a^4+z^4 a^2+3 z^2 a^2+a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^2 a^{10}+2 z^3 a^9+4 z^4 a^8-6 z^2 a^8-a^8 z^{-2} +4 a^8+z^7 a^7-2 z^5 a^7+4 z^3 a^7-5 z a^7+2 a^7 z^{-1} +z^8 a^6-3 z^6 a^6+6 z^4 a^6-10 z^2 a^6-2 a^6 z^{-2} +6 a^6+3 z^7 a^5-10 z^5 a^5+9 z^3 a^5-5 z a^5+2 a^5 z^{-1} +z^8 a^4-2 z^6 a^4-2 z^4 a^4+z^2 a^4-a^4 z^{-2} +2 a^4+2 z^7 a^3-8 z^5 a^3+7 z^3 a^3+z^6 a^2-4 z^4 a^2+4 z^2 a^2-a^2 }[/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 χ 1                 1 1 -1               1   -1 -3             2 1   1 -5           2 2     0 -7         3 1       2 -9       1 3         2 -11     3 2           1 -13     2             2 -15 1 2               -1 -17 1                 1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{3} }[/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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/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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