L11n183

L11n182

L11n184

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Link Presentations

[edit Notes on L11n183's Link Presentations]

Planar diagram presentation	X8192 X20,9,21,10 X21,1,22,6 X7,18,8,19 X3,10,4,11 X15,12,16,13 X5,14,6,15 X13,4,14,5 X11,16,12,17 X17,22,18,7 X2,20,3,19
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(2)^2 t(1)^2-3 t(2) t(1)^2+t(1)^2-4 t(2)^2 t(1)+7 t(2) t(1)-4 t(1)+t(2)^2-3 t(2)+2}{t(1) t(2)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{2}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{8}{q^{9/2}}-\frac{9}{q^{11/2}}+\frac{9}{q^{13/2}}-\frac{7}{q^{15/2}}+\frac{5}{q^{17/2}}-\frac{3}{q^{19/2}}+\frac{1}{q^{21/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 \left(-z^3\right)-a^9 z+a^7 z^5+2 a^7 z^3+a^7 z-a^7 z^{-1} +a^5 z^5+2 a^5 z^3+3 a^5 z+3 a^5 z^{-1} -2 a^3 z^3-4 a^3 z-2 a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^6 a^{12}+3 z^4 a^{12}-2 z^2 a^{12}-3 z^7 a^{11}+10 z^5 a^{11}-8 z^3 a^{11}+z a^{11}-3 z^8 a^{10}+8 z^6 a^{10}-4 z^4 a^{10}+z^2 a^{10}-z^9 a^9-3 z^7 a^9+14 z^5 a^9-9 z^3 a^9+z a^9-5 z^8 a^8+12 z^6 a^8-8 z^4 a^8+2 z^2 a^8+a^8-z^9 a^7-2 z^7 a^7+7 z^5 a^7-6 z^3 a^7+2 z a^7-a^7 z^{-1} -2 z^8 a^6+2 z^6 a^6-z^4 a^6-4 z^2 a^6+3 a^6-2 z^7 a^5+3 z^5 a^5-8 z^3 a^5+7 z a^5-3 a^5 z^{-1} -z^6 a^4-3 z^2 a^4+3 a^4-3 z^3 a^3+5 z a^3-2 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               5 1   4 -8             4 3     -1 -10           5 4       1 -12         4 4         0 -14       3 5           -2 -16     2 4             2 -18   1 3               -2 -20   2                 2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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

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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