L10n3

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Link Presentations

[edit Notes on L10n3's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X8493 X11,19,12,18 X17,5,18,20 X19,13,20,12 X9,16,10,17 X2,14,3,13
Gauss code	{1, -10, 5, -3}, {-4, -1, 2, -5, -9, 4, -6, 8, 10, -2, 3, 9, -7, 6, -8, 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{(t(1)-1) (t(2)-1) \left(t(2)^2-t(2)+1\right)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{1}{q^{9/2}}+q^{7/2}-\frac{2}{q^{7/2}}-2 q^{5/2}+\frac{3}{q^{5/2}}+3 q^{3/2}-\frac{4}{q^{3/2}}-5 \sqrt{q}+\frac{3}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^3 z^3-2 a^3 z+z a^{-3} -a^3 z^{-1} + a^{-3} z^{-1} +a z^5+4 a z^3-2 z^3 a^{-1} +6 a z-5 z a^{-1} +4 a z^{-1} -4 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^4 z^6-4 a^4 z^4+4 a^4 z^2+z^2 a^{-4} -a^4- a^{-4} +2 a^3 z^7-8 a^3 z^5+8 a^3 z^3+2 z^3 a^{-3} -4 a^3 z-2 z a^{-3} +a^3 z^{-1} + a^{-3} z^{-1} +a^2 z^8-a^2 z^6+z^6 a^{-2} -8 a^2 z^4-3 z^4 a^{-2} +11 a^2 z^2+7 z^2 a^{-2} -4 a^2-4 a^{-2} +4 a z^7+2 z^7 a^{-1} -16 a z^5-8 z^5 a^{-1} +19 a z^3+13 z^3 a^{-1} -11 a z-9 z a^{-1} +4 a z^{-1} +4 a^{-1} z^{-1} +z^8-z^6-7 z^4+13 z^2-7 }[/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   \ -5 -4 -3 -2 -1 0 1 2 3 χ 8                 1 -1 6               1   1 4             2 1   -1 2           3 1     2 0         2 4       2 -2       2 2 1       1 -4     1 2           1 -6   1 2             -1 -8   1               1 -10 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=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\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}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}\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}_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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