L10n47

L10n46

L10n48

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Link Presentations

[edit Notes on L10n47's Link Presentations]

Planar diagram presentation	X8192 X9,19,10,18 X6718 X20,14,7,13 X12,5,13,6 X3,10,4,11 X4,15,5,16 X16,12,17,11 X14,20,15,19 X17,2,18,3
Gauss code	{1, 10, -6, -7, 5, -3}, {3, -1, -2, 6, 8, -5, 4, -9, 7, -8, -10, 2, 9, -4}

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^2 v^2+2 u^2 v+u v^4-2 u v^3+3 u v^2-2 u v+u+2 v^3-v^2}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{3/2}+3 \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{5}{q^{3/2}}-\frac{6}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^5+2 z a^5+2 a^5 z^{-1} -z^5 a^3-4 z^3 a^3-7 z a^3-3 a^3 z^{-1} +2 z^3 a+3 z a+a z^{-1} -z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^4 z^8-a^2 z^8-3 a^5 z^7-4 a^3 z^7-a z^7-2 a^6 z^6+2 a^2 z^6-a^7 z^5+11 a^5 z^5+14 a^3 z^5+2 a z^5+5 a^6 z^4+6 a^4 z^4-2 a^2 z^4-3 z^4+3 a^7 z^3-14 a^5 z^3-21 a^3 z^3-5 a z^3-z^3 a^{-1} -a^6 z^2-8 a^4 z^2-4 a^2 z^2+3 z^2-a^7 z+9 a^5 z+12 a^3 z+3 a z+z a^{-1} +3 a^4+3 a^2+1-2 a^5 z^{-1} -3 a^3 z^{-1} -a 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   \ -6 -5 -4 -3 -2 -1 0 1 2 χ 4                 1 1 2               2   -2 0             2 1   1 -2           4 3     -1 -4         2 1       1 -6       2 4         2 -8     2 2           0 -10     2             2 -12 1 2               -1 -14 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{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/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}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/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}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/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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