L11n122

L11n121

L11n123

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Link Presentations

[edit Notes on L11n122's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X16,10,17,9 X11,21,12,20 X21,9,22,8 X7,19,8,18 X19,13,20,12 X10,16,11,15 X22,17,5,18 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, -6, 5, 3, -8, -4, 7, 11, -2, 8, -3, 9, 6, -7, 4, -5, -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 u v^2+2 u v+u+v^3+2 v^2-2 v}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{11/2}+2 q^{9/2}-2 q^{7/2}+3 q^{5/2}-2 q^{3/2}+\sqrt{q}-\frac{2}{\sqrt{q}}-\frac{1}{q^{7/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z a^3+2 a^3 z^{-1} -z^3 a-5 z a-3 a z^{-1} +z^3 a^{-1} +2 z a^{-1} + a^{-1} z^{-1} +z^3 a^{-3} +z a^{-3} -z a^{-5} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^9 a^{-1} -z^9 a^{-3} -3 z^8 a^{-2} -2 z^8 a^{-4} -z^8+5 z^7 a^{-1} +4 z^7 a^{-3} -z^7 a^{-5} +17 z^6 a^{-2} +11 z^6 a^{-4} +6 z^6-a^3 z^5-5 z^5 a^{-1} -z^5 a^{-3} +5 z^5 a^{-5} -26 z^4 a^{-2} -17 z^4 a^{-4} -9 z^4+5 a^3 z^3+4 a z^3+2 z^3 a^{-1} -3 z^3 a^{-3} -6 z^3 a^{-5} +3 a^2 z^2+13 z^2 a^{-2} +9 z^2 a^{-4} +7 z^2-5 a^3 z-7 a z-3 z a^{-1} +z a^{-5} -3 a^2- a^{-2} -3+2 a^3 z^{-1} +3 a z^{-1} + a^{-1} 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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 12                     1 1 10                   1   -1 8                 1 1   0 6               2 1     -1 4           1 1 1       -1 2           1 2         1 0       1 3 1           1 -2         2             2 -4     1 1               0 -6 1                     1 -8 1                     1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/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=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/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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