L10n5

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Link Presentations

[edit Notes on L10n5's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X17,1,18,4 X9,14,10,15 X3849 X5,11,6,10 X13,5,14,20 X11,19,12,18 X19,13,20,12 X2,16,3,15
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{(t(1)-1) (t(2)-1) \left(t(2)^2+1\right)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 2 q^{9/2}-2 q^{7/2}+2 q^{5/2}-\frac{1}{q^{5/2}}-3 q^{3/2}+\frac{1}{q^{3/2}}-q^{11/2}+2 \sqrt{q}-\frac{2}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^5 a^{-1} +a z^3-5 z^3 a^{-1} +2 z^3 a^{-3} +3 a z-8 z a^{-1} +6 z a^{-3} -z a^{-5} +2 a z^{-1} -4 a^{-1} z^{-1} +3 a^{-3} z^{-1} - a^{-5} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^8 a^{-2} -z^8-a z^7-3 z^7 a^{-1} -2 z^7 a^{-3} +3 z^6 a^{-2} -2 z^6 a^{-4} +5 z^6+6 a z^5+16 z^5 a^{-1} +9 z^5 a^{-3} -z^5 a^{-5} +2 z^4 a^{-2} +8 z^4 a^{-4} -6 z^4-11 a z^3-26 z^3 a^{-1} -13 z^3 a^{-3} +2 z^3 a^{-5} -7 z^2 a^{-2} -9 z^2 a^{-4} -2 z^2 a^{-6} +7 a z+17 z a^{-1} +11 z a^{-3} -z a^{-7} +3 a^{-2} +3 a^{-4} + a^{-6} +2-2 a z^{-1} -4 a^{-1} z^{-1} -3 a^{-3} z^{-1} - a^{-5} z^{-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 12                 1 1 10               1   -1 8             2 2   0 6           1 1     0 4         2 2 1     1 2       2 3         1 0       1 1         0 -2   1 2             1 -4                   0 -6 1                 1

Integral Khovanov Homology (db, data source)

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