L10n9

L10n8

L10n10

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Link Presentations

[edit Notes on L10n9's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X9,14,10,15 X8493 X5,11,6,10 X11,5,12,20 X13,19,14,18 X19,13,20,12 X2,16,3,15
Gauss code	{1, -10, 5, -3}, {-6, -1, 2, -5, -4, 6, -7, 9, -8, 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-1) (v-1)}{\sqrt{u} \sqrt{v}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{11/2}+q^{9/2}-q^{7/2}+q^{5/2}-q^{3/2}+\sqrt{q}-\frac{1}{\sqrt{q}}-\frac{1}{q^{5/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^{-5} - a^{-5} z^{-1} +z^3 a^{-3} +a^3 z^{-1} +3 z a^{-3} +2 a^{-3} z^{-1} -a z-a z^{-1} -z a^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^8 a^{-2} -z^8 a^{-4} -z^7 a^{-1} -2 z^7 a^{-3} -z^7 a^{-5} +6 z^6 a^{-2} +6 z^6 a^{-4} +6 z^5 a^{-1} +12 z^5 a^{-3} +6 z^5 a^{-5} -11 z^4 a^{-2} -10 z^4 a^{-4} -z^4-11 z^3 a^{-1} -21 z^3 a^{-3} -10 z^3 a^{-5} +9 z^2 a^{-2} +5 z^2 a^{-4} +4 z^2-a^3 z-a z+8 z a^{-1} +13 z a^{-3} +5 z a^{-5} -a^2-3 a^{-2} - a^{-4} -2+a^3 z^{-1} +a z^{-1} - a^{-1} z^{-1} -2 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   \ -2 -1 0 1 2 3 4 5 6 χ 12                 1 1 10                   0 8             1 1   0 6         1 1       0 4         1 1       0 2     1 2 1         0 0   1 2 1           0 -2   1 2             1 -4 1                 1 -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=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_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}\oplus{\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=4 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=6 }[/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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