L11n13

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Link Presentations

[edit Notes on L11n13's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X5,10,6,11 X3849 X11,22,12,5 X13,20,14,21 X19,14,20,15 X21,12,22,13 X9,18,10,19 X15,2,16,3
Gauss code	{1, 11, -5, -3}, {-4, -1, 2, 5, -10, 4, -6, 9, -7, 8, -11, -2, 3, 10, -8, 7, -9, 6}

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{3 u v^3-2 u v^2-2 v+3}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{2}{q^{9/2}}+\frac{1}{q^{7/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{23/2}}-\frac{1}{q^{21/2}}+\frac{2}{q^{19/2}}-\frac{3}{q^{17/2}}+\frac{3}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{2}{q^{11/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{13} z^{-1} +a^{11} z^3+4 a^{11} z+2 a^{11} z^{-1} -a^9 z^5-4 a^9 z^3-3 a^9 z-a^9 z^{-1} -a^7 z^5-3 a^7 z^3+a^7 z^{-1} -a^5 z^5-4 a^5 z^3-3 a^5 z-a^5 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^4 a^{14}+4 z^2 a^{14}-3 a^{14}-z^5 a^{13}+3 z^3 a^{13}-2 z a^{13}+a^{13} z^{-1} -z^8 a^{12}+7 z^6 a^{12}-19 z^4 a^{12}+21 z^2 a^{12}-7 a^{12}-z^9 a^{11}+7 z^7 a^{11}-18 z^5 a^{11}+18 z^3 a^{11}-8 z a^{11}+2 a^{11} z^{-1} -2 z^8 a^{10}+12 z^6 a^{10}-25 z^4 a^{10}+17 z^2 a^{10}-4 a^{10}-z^9 a^9+6 z^7 a^9-13 z^5 a^9+11 z^3 a^9-6 z a^9+a^9 z^{-1} -z^8 a^8+4 z^6 a^8-4 z^4 a^8-z^7 a^7+3 z^5 a^7-3 z a^7+a^7 z^{-1} -z^6 a^6+3 z^4 a^6-a^6-z^5 a^5+4 z^3 a^5-3 z a^5+a^5 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 -4                   1 1 -6                 1 1 0 -8               1     1 -10           1 2 1     0 -12           3 1       2 -14         1 2         1 -16       2 2           0 -18       1             1 -20   1 2               -1 -22                     0 -24 1                   -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-6 }[/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 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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