L11n120

L11n119

L11n121

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Link Presentations

[edit Notes on L11n120's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X9,16,10,17 X11,21,12,20 X21,9,22,8 X7,19,8,18 X19,13,20,12 X15,10,16,11 X17,5,18,22 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{t(2)^5-t(1) t(2)^4-3 t(2)^4+2 t(1) t(2)^3+2 t(2)^2-3 t(1) t(2)-t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-2 q^{7/2}+2 q^{5/2}-q^{3/2}-\frac{2}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z+a^5 z^{-1} -2 a^3 z^3+z^3 a^{-3} -5 a^3 z-2 a^3 z^{-1} +2 z a^{-3} + a^{-3} z^{-1} +a z^5-z^5 a^{-1} +5 a z^3-5 z^3 a^{-1} +7 a z+3 a z^{-1} -7 z a^{-1} -3 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^5 z^7-5 a^5 z^5+7 a^5 z^3-4 a^5 z+a^5 z^{-1} +2 a^4 z^8-10 a^4 z^6+z^6 a^{-4} +12 a^4 z^4-4 z^4 a^{-4} -4 a^4 z^2+3 z^2 a^{-4} - a^{-4} +a^3 z^9-3 a^3 z^7+2 z^7 a^{-3} -6 a^3 z^5-9 z^5 a^{-3} +16 a^3 z^3+8 z^3 a^{-3} -10 a^3 z-3 z a^{-3} +2 a^3 z^{-1} + a^{-3} z^{-1} +3 a^2 z^8+z^8 a^{-2} -18 a^2 z^6-4 z^6 a^{-2} +27 a^2 z^4-z^4 a^{-2} -13 a^2 z^2+6 z^2 a^{-2} +2 a^2-2 a^{-2} +a z^9-4 a z^7+2 z^7 a^{-1} -6 a z^5-14 z^5 a^{-1} +23 a z^3+22 z^3 a^{-1} -16 a z-13 z a^{-1} +3 a z^{-1} +3 a^{-1} z^{-1} +2 z^8-13 z^6+18 z^4-6 z^2 }[/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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     1   1 6                   1 1   0 4               2 2 1     -1 2             1 1 1       1 0           2 4 2         0 -2         2 2 2           2 -4       1 2 1             0 -6     2 2 1               1 -8   1 2                   1 -10   1                     -1 -12 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=-6 }[/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=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/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} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/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^{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}\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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