L11n199

L11n198

L11n200

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Link Presentations

[edit Notes on L11n199's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,3,13,4 X5,14,6,15 X16,7,17,8 X20,15,21,16 X13,18,14,19 X21,6,22,7 X17,22,18,9 X4,19,5,20 X2,9,3,10 X8,11,1,12
Gauss code	{1, -10, 2, -9, -3, 7, 4, -11}, {10, -1, 11, -2, -6, 3, 5, -4, -8, 6, 9, -5, -7, 8}

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^3 v^4+u^2 v^5-u^2 v^4+u^2 v^2+u v^3-u v+u+v}{u^{3/2} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{2}{q^{11/2}}+\frac{1}{q^{13/2}}-\frac{1}{q^{15/2}}+\frac{1}{q^{17/2}}-\frac{1}{q^{19/2}} }[/math] (db)
Signature	-7 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^{13}-a^{13} z^{-1} +z^5 a^{11}+6 z^3 a^{11}+8 z a^{11}+3 a^{11} z^{-1} -z^7 a^9-6 z^5 a^9-10 z^3 a^9-6 z a^9-2 a^9 z^{-1} -z^7 a^7-6 z^5 a^7-10 z^3 a^7-5 z a^7 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^2 a^{14}-a^{14}+z^3 a^{13}-z a^{13}+a^{13} z^{-1} -z^8 a^{12}+7 z^6 a^{12}-15 z^4 a^{12}+13 z^2 a^{12}-3 a^{12}-z^9 a^{11}+7 z^7 a^{11}-17 z^5 a^{11}+22 z^3 a^{11}-14 z a^{11}+3 a^{11} z^{-1} -2 z^8 a^{10}+12 z^6 a^{10}-20 z^4 a^{10}+12 z^2 a^{10}-3 a^{10}-z^9 a^9+6 z^7 a^9-11 z^5 a^9+11 z^3 a^9-8 z a^9+2 a^9 z^{-1} -z^8 a^8+5 z^6 a^8-5 z^4 a^8-z^7 a^7+6 z^5 a^7-10 z^3 a^7+5 z a^7 }[/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 -6                 1 1 -8               1 1 0 -10             1     1 -12         1 1 1     1 -14       1 2 1       0 -16       1 1         0 -18     2 2           0 -20 1                 1 -22 1 1               0

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-8 }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}\oplus{\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] [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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