L11n247

L11n246

L11n248

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Link Presentations

[edit Notes on L11n247's Link Presentations]

Planar diagram presentation	X12,1,13,2 X3849 X14,6,15,5 X7,18,8,19 X9,21,10,20 X10,11,1,12 X6,14,7,13 X17,4,18,5 X15,11,16,22 X19,3,20,2 X21,17,22,16
Gauss code	{1, 10, -2, 8, 3, -7, -4, 2, -5, -6}, {6, -1, 7, -3, -9, 11, -8, 4, -10, 5, -11, 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], ...)	0 (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-2 q^{7/2}+q^{5/2}-q^{3/2}-\frac{1}{q^{3/2}}+\frac{1}{q^{5/2}}-\frac{2}{q^{7/2}}+\frac{1}{q^{9/2}} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^3 z^3+z^3 a^{-3} -2 a^3 z+2 z a^{-3} +a z^5-z^5 a^{-1} +5 a z^3-5 z^3 a^{-1} +6 a z-6 z a^{-1} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^4 z^6+z^6 a^{-4} -4 a^4 z^4-4 z^4 a^{-4} +2 a^4 z^2+2 z^2 a^{-4} +2 a^3 z^7+2 z^7 a^{-3} -10 a^3 z^5-10 z^5 a^{-3} +11 a^3 z^3+11 z^3 a^{-3} -4 a^3 z-4 z a^{-3} +a^2 z^8+z^8 a^{-2} -5 a^2 z^6-5 z^6 a^{-2} +4 a^2 z^4+4 z^4 a^{-2} +3 a z^7+3 z^7 a^{-1} -19 a z^5-19 z^5 a^{-1} +29 a z^3+29 z^3 a^{-1} -12 a z-12 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +2 z^8-12 z^6+16 z^4-4 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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                     1 -1 8                   1   1 6               1 1 1   1 4             1 2 1     0 2           2 2 1       1 0         2 4 2         0 -2       1 2 2           1 -4     1 2 1             0 -6   1 1 1               1 -8   1                   1 -10 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=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\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} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{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 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}_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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