L11n143

L11n142

L11n144

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Link Presentations

[edit Notes on L11n143's Link Presentations]

Planar diagram presentation	X8192 X11,19,12,18 X3,10,4,11 X17,3,18,2 X12,5,13,6 X6718 X16,10,17,9 X20,16,21,15 X22,14,7,13 X14,22,15,21 X4,20,5,19
Gauss code	{1, 4, -3, -11, 5, -6}, {6, -1, 7, 3, -2, -5, 9, -10, 8, -7, -4, 2, 11, -8, 10, -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^2 v^2-2 u^2 v+u v-2 v+2}{u v} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{13/2}-q^{11/2}+2 q^{9/2}-3 q^{7/2}+2 q^{5/2}-3 q^{3/2}+2 \sqrt{q}-\frac{2}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{1}{q^{5/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^{-5} +3 z a^{-5} + a^{-5} z^{-1} -z^5 a^{-3} -4 z^3 a^{-3} -4 z a^{-3} -2 a^{-3} z^{-1} -z^5 a^{-1} +a z^3-4 z^3 a^{-1} +3 a z-3 z a^{-1} +a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^8 a^{-6} -7 z^6 a^{-6} +16 z^4 a^{-6} -13 z^2 a^{-6} +2 a^{-6} +z^9 a^{-5} -6 z^7 a^{-5} +11 z^5 a^{-5} -8 z^3 a^{-5} +4 z a^{-5} - a^{-5} z^{-1} +3 z^8 a^{-4} -18 z^6 a^{-4} +33 z^4 a^{-4} -22 z^2 a^{-4} +5 a^{-4} +z^9 a^{-3} -4 z^7 a^{-3} +2 z^5 a^{-3} +a^3 z+5 z a^{-3} -2 a^{-3} z^{-1} +2 z^8 a^{-2} -10 z^6 a^{-2} +14 z^4 a^{-2} +a^2 z^2-9 z^2 a^{-2} +3 a^{-2} +2 z^7 a^{-1} -9 z^5 a^{-1} +2 a z^3+10 z^3 a^{-1} -4 a z-4 z a^{-1} +a z^{-1} +z^6-3 z^4+z^2-1 }[/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   \ -2 -1 0 1 2 3 4 5 6 7 χ 14                   1 -1 12                     0 10               2 1   -1 8             1       1 6           1 2       1 4         2 1         1 2         1           1 0     2 2             0 -2     1               1 -4 1 1                 0 -6 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{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=7 }[/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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