L11n99

L11n98

L11n100

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Link Presentations

[edit Notes on L11n99's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X13,21,14,20 X7,17,8,16 X19,9,20,8 X9,19,10,18 X17,11,18,10 X15,5,16,22 X21,15,22,14 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, -6, 7, 11, -2, -3, 9, -8, 4, -7, 6, -5, 3, -9, 8}

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{u v^3-u v^2-u v+2 u+2 v^5-v^4-v^3+v^2}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{15/2}-q^{13/2}+q^{11/2}-2 q^{3/2}+\sqrt{q}-\frac{2}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{1}{q^{5/2}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^5 a^{-1} -z^5 a^{-3} +a z^3-4 z^3 a^{-1} -5 z^3 a^{-3} +3 a z-2 z a^{-1} -6 z a^{-3} +2 z a^{-5} +z a^{-7} +a z^{-1} + a^{-1} z^{-1} -4 a^{-3} z^{-1} +2 a^{-5} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^9 a^{-1} -z^9 a^{-3} -2 z^8 a^{-2} -z^8 a^{-4} -z^8-a z^7+6 z^7 a^{-1} +8 z^7 a^{-3} -z^7 a^{-7} +13 z^6 a^{-2} +8 z^6 a^{-4} -z^6 a^{-6} -z^6 a^{-8} +5 z^6+6 a z^5-11 z^5 a^{-1} -22 z^5 a^{-3} +5 z^5 a^{-7} -25 z^4 a^{-2} -19 z^4 a^{-4} +6 z^4 a^{-6} +5 z^4 a^{-8} -5 z^4-10 a z^3+10 z^3 a^{-1} +29 z^3 a^{-3} +4 z^3 a^{-5} -5 z^3 a^{-7} +19 z^2 a^{-2} +19 z^2 a^{-4} -7 z^2 a^{-6} -6 z^2 a^{-8} -z^2+5 a z-5 z a^{-1} -16 z a^{-3} -6 z a^{-5} -5 a^{-2} -6 a^{-4} + a^{-6} +2 a^{-8} +1-a z^{-1} + a^{-1} z^{-1} +4 a^{-3} z^{-1} +2 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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 16                       1 -1 14                         0 12                   1 1   0 10               2 1       -1 8               1 1       0 6           2 3 1         0 4         1   1           2 2       1 3 1             1 0     1 1 1               1 -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=0 }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ i=4 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\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}^{3} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/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}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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