L10n85

L10n84

L10n86

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Link L10n85.

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Link Presentations

[edit Notes on L10n85's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X20,14,15,13 X7,16,8,17 X15,8,16,9 X18,12,19,11 X12,20,13,19 X14,18,5,17 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{(w-1) \left(u v^2 w-u v w+2 u v-u-v^2 w+2 v w-v+1\right)}{\sqrt{u} v w} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^4+2 q^{-4} -3 q^3-3 q^{-3} +5 q^2+6 q^{-2} -6 q-6 q^{-1} +8 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^6+a^2 z^4+z^4 a^{-2} -4 z^4+a^2 z^2+2 z^2 a^{-2} -5 z^2+a^4-2 a^2+ a^{-2} +a^4 z^{-2} -2 a^2 z^{-2} + z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 3 a^4 z^4+z^4 a^{-4} -7 a^4 z^2-z^2 a^{-4} -a^4 z^{-2} +4 a^4+a^3 z^7-a^3 z^5+3 z^5 a^{-3} +2 a^3 z^3-4 z^3 a^{-3} -5 a^3 z+2 a^3 z^{-1} +a^2 z^8-a^2 z^6+4 z^6 a^{-2} +4 a^2 z^4-6 z^4 a^{-2} -8 a^2 z^2+3 z^2 a^{-2} -2 a^2 z^{-2} +6 a^2- a^{-2} +4 a z^7+3 z^7 a^{-1} -7 a z^5-3 z^5 a^{-1} +7 a z^3+z^3 a^{-1} -5 a z+2 a z^{-1} +z^8+3 z^6-6 z^4+3 z^2- z^{-2} +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   \ -4 -3 -2 -1 0 1 2 3 4 χ 9                 1 1 7               2   -2 5             3 1   2 3           4 3     -1 1         4 2       2 -1       3 5         2 -3     3 3           0 -5   1 4             3 -7 1 2               -1 -9 2                 2

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=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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=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}_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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