L11n184

L11n183

L11n185

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Link Presentations

[edit Notes on L11n184's Link Presentations]

Planar diagram presentation	X8192 X9,21,10,20 X6,21,1,22 X18,8,19,7 X3,10,4,11 X15,12,16,13 X5,14,6,15 X13,4,14,5 X11,16,12,17 X22,18,7,17 X19,2,20,3
Gauss code	{1, 11, -5, 8, -7, -3}, {4, -1, -2, 5, -9, 6, -8, 7, -6, 9, 10, -4, -11, 2, 3, -10}

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{2 t(1) t(2)^4-t(2)^4-2 t(1) t(2)^3+2 t(2)^3+t(1) t(2)^2+2 t(1)^2 t(2)-2 t(1) t(2)-t(1)^2+2 t(1)}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{5}{q^{9/2}}+\frac{4}{q^{7/2}}-\frac{5}{q^{5/2}}+\frac{3}{q^{3/2}}-\frac{1}{q^{17/2}}+\frac{2}{q^{15/2}}-\frac{3}{q^{13/2}}+\frac{4}{q^{11/2}}+\sqrt{q}-\frac{2}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^7+2 z a^7-z^5 a^5-3 z^3 a^5-z a^5+2 a^5 z^{-1} -z^5 a^3-4 z^3 a^3-6 z a^3-3 a^3 z^{-1} +z^3 a+2 z a+a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^9 z^7-5 a^9 z^5+7 a^9 z^3-3 a^9 z+2 a^8 z^8-10 a^8 z^6+14 a^8 z^4-6 a^8 z^2+a^7 z^9-2 a^7 z^7-6 a^7 z^5+10 a^7 z^3-2 a^7 z+4 a^6 z^8-18 a^6 z^6+22 a^6 z^4-9 a^6 z^2+a^5 z^9-a^5 z^7-9 a^5 z^5+14 a^5 z^3-7 a^5 z+2 a^5 z^{-1} +2 a^4 z^8-7 a^4 z^6+5 a^4 z^4+2 a^4 z^2-3 a^4+2 a^3 z^7-8 a^3 z^5+13 a^3 z^3-10 a^3 z+3 a^3 z^{-1} +a^2 z^6-3 a^2 z^4+6 a^2 z^2-3 a^2+2 a z^3-2 a z+a z^{-1} +z^2-1 }[/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 1 χ 2                   1 -1 0                 1   1 -2               3 2   -1 -4             2       2 -6           2 3       1 -8         3 2         1 -10       1 2           1 -12     2 3             -1 -14   1 2               1 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/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

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