L11n138

L11n137

L11n139

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Link Presentations

[edit Notes on L11n138's Link Presentations]

Planar diagram presentation	X8192 X11,19,12,18 X3,10,4,11 X17,3,18,2 X5,13,6,12 X6718 X9,16,10,17 X15,20,16,21 X13,22,14,7 X21,14,22,15 X19,4,20,5
Gauss code	{1, 4, -3, 11, -5, -6}, {6, -1, -7, 3, -2, 5, -9, 10, -8, 7, -4, 2, -11, 8, -10, 9}

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(2) t(1)-t(1)+1) (t(1) t(2)-t(2)+1)}{t(1) t(2)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{3}{q^{9/2}}+\frac{2}{q^{7/2}}-\frac{3}{q^{5/2}}+\frac{1}{q^{3/2}}-\frac{1}{q^{17/2}}+\frac{2}{q^{15/2}}-\frac{2}{q^{13/2}}+\frac{3}{q^{11/2}}-\frac{1}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 z^3+2 a^7 z-a^5 z^5-4 a^5 z^3-4 a^5 z+a^3 z^3+2 a^3 z+a^3 z^{-1} -a z-a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^7 a^9+5 z^5 a^9-6 z^3 a^9+z a^9-2 z^8 a^8+11 z^6 a^8-17 z^4 a^8+8 z^2 a^8-z^9 a^7+4 z^7 a^7-2 z^5 a^7-z^3 a^7-z a^7-3 z^8 a^6+16 z^6 a^6-24 z^4 a^6+12 z^2 a^6-z^9 a^5+5 z^7 a^5-8 z^5 a^5+9 z^3 a^5-5 z a^5-z^8 a^4+5 z^6 a^4-7 z^4 a^4+4 z^2 a^4-z^5 a^3+4 z^3 a^3-4 z a^3+a^3 z^{-1} -a^2-z a+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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ 0                 1 1 -2               1 1 0 -4             2     2 -6           1 2     1 -8         2 1       1 -10       1 1         0 -12     1 2           -1 -14   1 1             0 -16   1               -1 -18 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=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/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}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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