L10n4

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Link Presentations

[edit Notes on L10n4's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X9,14,10,15 X3849 X5,11,6,10 X13,5,14,20 X11,19,12,18 X19,13,20,12 X15,2,16,3
Gauss code	{1, 10, -5, -3}, {-6, -1, 2, 5, -4, 6, -8, 9, -7, 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{u v^5-2 u v^4+2 u v^3-2 u v^2-2 v^3+2 v^2-2 v+1}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{7/2}+2 q^{5/2}-3 q^{3/2}+4 \sqrt{q}-\frac{5}{\sqrt{q}}+\frac{5}{q^{3/2}}-\frac{4}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{2}{q^{9/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a z^7-a^3 z^5+6 a z^5-z^5 a^{-1} -5 a^3 z^3+12 a z^3-4 z^3 a^{-1} +a^5 z-8 a^3 z+9 a z-4 z a^{-1} +2 a^5 z^{-1} -4 a^3 z^{-1} +3 a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 3 a^5 z^3-7 a^5 z+2 a^5 z^{-1} +a^4 z^6-2 a^4 z^4+3 a^4 z^2-2 a^4+2 a^3 z^7-9 a^3 z^5+z^5 a^{-3} +20 a^3 z^3-3 z^3 a^{-3} -16 a^3 z+z a^{-3} +4 a^3 z^{-1} +a^2 z^8-2 a^2 z^6+2 z^6 a^{-2} -6 z^4 a^{-2} +7 a^2 z^2+3 z^2 a^{-2} -3 a^2- a^{-2} +4 a z^7+2 z^7 a^{-1} -16 a z^5-6 z^5 a^{-1} +25 a z^3+5 z^3 a^{-1} -13 a z-3 z a^{-1} +3 a z^{-1} + a^{-1} z^{-1} +z^8-z^6-4 z^4+7 z^2-3 }[/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 8                 1 1 6               1   -1 4             2 1   1 2           2 1     -1 0         3 2       1 -2       3 3         0 -4     1 2           -1 -6   1 3             2 -8 1 1               0 -10 2                 2

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