L11n153

L11n152

L11n154

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Link Presentations

[edit Notes on L11n153's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X7,14,8,15 X13,20,14,21 X19,1,20,6 X18,11,19,12 X5,12,6,13 X15,22,16,7 X4,18,5,17 X21,16,22,17
Gauss code	{1, -2, 3, -10, -8, 6}, {-4, -1, 2, -3, 7, 8, -5, 4, -9, 11, 10, -7, -6, 5, -11, 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{2 t(2)^2 t(1)^2-2 t(2) t(1)^2-4 t(2)^2 t(1)+7 t(2) t(1)-4 t(1)-2 t(2)+2}{t(1) t(2)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{7}{q^{9/2}}-\frac{6}{q^{7/2}}+\frac{3}{q^{5/2}}-\frac{2}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{2}{q^{19/2}}+\frac{4}{q^{17/2}}-\frac{6}{q^{15/2}}+\frac{7}{q^{13/2}}-\frac{8}{q^{11/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 \left(-z^3\right)-2 a^9 z-a^9 z^{-1} +a^7 z^5+3 a^7 z^3+4 a^7 z+2 a^7 z^{-1} +a^5 z^5+2 a^5 z^3+a^5 z-2 a^3 z^3-4 a^3 z-a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{12} z^6-4 a^{12} z^4+4 a^{12} z^2+2 a^{11} z^7-7 a^{11} z^5+6 a^{11} z^3-a^{11} z+2 a^{10} z^8-6 a^{10} z^6+6 a^{10} z^4-6 a^{10} z^2+2 a^{10}+a^9 z^9-a^9 z^7-5 a^9 z^3+3 a^9 z-a^9 z^{-1} +4 a^8 z^8-13 a^8 z^6+20 a^8 z^4-18 a^8 z^2+5 a^8+a^7 z^9-a^7 z^7+3 a^7 z^5-7 a^7 z^3+7 a^7 z-2 a^7 z^{-1} +2 a^6 z^8-5 a^6 z^6+10 a^6 z^4-7 a^6 z^2+3 a^6+2 a^5 z^7-4 a^5 z^5+7 a^5 z^3-2 a^5 z+a^4 z^6+a^4 z^2-a^4+3 a^3 z^3-5 a^3 z+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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -2                   2 2 -4                 2 1 -1 -6               4 1   3 -8             4 3     -1 -10           4 3       1 -12         3 4         1 -14       3 4           -1 -16     1 3             2 -18   1 3               -2 -20   1                 1 -22 1                   -1

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=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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=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

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