L10n46

L10n45

L10n47

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Link Presentations

[edit Notes on L10n46's Link Presentations]

Planar diagram presentation	X8192 X9,19,10,18 X6718 X13,20,14,7 X12,5,13,6 X3,10,4,11 X4,15,5,16 X16,12,17,11 X19,14,20,15 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+u v^4-2 u v^3+u v^2-2 u v+u-v^2}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{4}{q^{9/2}}-\frac{3}{q^{7/2}}+\frac{2}{q^{5/2}}-\frac{2}{q^{3/2}}-\frac{2}{q^{15/2}}+\frac{2}{q^{13/2}}-\frac{3}{q^{11/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 z^{-1} -a^7 z^3-3 a^7 z-2 a^7 z^{-1} +a^5 z^5+4 a^5 z^3+5 a^5 z+2 a^5 z^{-1} -2 a^3 z^3-5 a^3 z-a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -3 z^3 a^9+6 z a^9-a^9 z^{-1} -z^6 a^8+2 z^4 a^8-2 z^2 a^8-z^7 a^7+4 z^5 a^7-11 z^3 a^7+10 z a^7-2 a^7 z^{-1} -2 z^6 a^6+4 z^4 a^6-4 z^2 a^6-a^6-z^7 a^5+4 z^5 a^5-11 z^3 a^5+10 z a^5-2 a^5 z^{-1} -z^6 a^4+2 z^4 a^4-2 z^2 a^4-3 z^3 a^3+6 z a^3-a^3 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 χ -2             2 2 -4           1 1 0 -6         2 1   1 -8       2 1     -1 -10     1 2       -1 -12   1 2         1 -14 1 1           0 -16 2             2

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}\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 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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