L11n190

L11n189

L11n191

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Link Presentations

[edit Notes on L11n190's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X22,10,7,9 X10,14,11,13 X5,16,6,17 X20,15,21,16 X18,21,19,22 X14,19,15,20 X2738 X4,11,5,12 X17,6,18,1
Gauss code	{1, -9, 2, -10, -5, 11}, {9, -1, 3, -4, 10, -2, 4, -8, 6, 5, -11, -7, 8, -6, 7, -3}

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^3+4 u^2 v^2-3 u^2 v+u^2+4 u v^3-7 u v^2+4 u v+v^4-3 v^3+4 v^2-v}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{10}{q^{9/2}}-\frac{8}{q^{7/2}}+\frac{5}{q^{5/2}}-\frac{2}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{3}{q^{19/2}}+\frac{6}{q^{17/2}}-\frac{9}{q^{15/2}}+\frac{10}{q^{13/2}}-\frac{12}{q^{11/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{11} z^{-1} +4 a^9 z+3 a^9 z^{-1} -4 a^7 z^3-6 a^7 z-2 a^7 z^{-1} +a^5 z^5+a^5 z^3+a^5 z-2 a^3 z^3-2 a^3 z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^6 a^{12}+3 z^4 a^{12}-3 z^2 a^{12}+a^{12}-3 z^7 a^{11}+9 z^5 a^{11}-8 z^3 a^{11}+3 z a^{11}-a^{11} z^{-1} -3 z^8 a^{10}+4 z^6 a^{10}+8 z^4 a^{10}-11 z^2 a^{10}+3 a^{10}-z^9 a^9-7 z^7 a^9+26 z^5 a^9-24 z^3 a^9+12 z a^9-3 a^9 z^{-1} -6 z^8 a^8+8 z^6 a^8+7 z^4 a^8-9 z^2 a^8+3 a^8-z^9 a^7-7 z^7 a^7+19 z^5 a^7-16 z^3 a^7+7 z a^7-2 a^7 z^{-1} -3 z^8 a^6+2 z^6 a^6-2 z^4 a^6+2 z^2 a^6-3 z^7 a^5+2 z^5 a^5-3 z^3 a^5-z^6 a^4-4 z^4 a^4+3 z^2 a^4-3 z^3 a^3+2 z a^3 }[/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                 4 1 -3 -6               4 1   3 -8             6 4     -2 -10           6 4       2 -12         4 6         2 -14       5 6           -1 -16     2 5             3 -18   1 4               -3 -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}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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