L11n108

L11n107

L11n109

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Link Presentations

[edit Notes on L11n108's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X13,19,14,18 X17,11,18,10 X21,9,22,8 X7,17,8,16 X9,21,10,20 X15,5,16,22 X19,15,20,14 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -6, 5, -7, 4, 11, -2, -3, 9, -8, 6, -4, 3, -9, 7, -5, 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^4+3 u v^3-2 u v^2-u v+2 u+2 v^5-v^4-2 v^3+3 v^2-v}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -3 q^{9/2}+3 q^{7/2}-2 q^{5/2}-\frac{1}{q^{5/2}}+q^{3/2}+\frac{1}{q^{3/2}}+q^{15/2}-2 q^{13/2}+3 q^{11/2}-\sqrt{q}-\frac{2}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^5 a^{-1} +a z^3-5 z^3 a^{-1} -z^3 a^{-3} -z^3 a^{-5} +3 a z-5 z a^{-1} -z a^{-3} +z a^{-7} +2 a z^{-1} -2 a^{-1} z^{-1} - a^{-3} z^{-1} + a^{-5} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^9 a^{-1} -z^9 a^{-3} -2 z^8 a^{-2} -2 z^8 a^{-4} -z^8 a^{-6} -z^8-a z^7+6 z^7 a^{-1} +6 z^7 a^{-3} -3 z^7 a^{-5} -2 z^7 a^{-7} +14 z^6 a^{-2} +12 z^6 a^{-4} +2 z^6 a^{-6} -z^6 a^{-8} +5 z^6+6 a z^5-7 z^5 a^{-1} -8 z^5 a^{-3} +13 z^5 a^{-5} +8 z^5 a^{-7} -24 z^4 a^{-2} -21 z^4 a^{-4} +4 z^4 a^{-6} +4 z^4 a^{-8} -3 z^4-10 a z^3-2 z^3 a^{-1} +3 z^3 a^{-3} -12 z^3 a^{-5} -7 z^3 a^{-7} +10 z^2 a^{-2} +15 z^2 a^{-4} -3 z^2 a^{-6} -4 z^2 a^{-8} -4 z^2+7 a z+4 z a^{-1} -3 z a^{-3} +2 z a^{-5} +2 z a^{-7} -3 a^{-4} + a^{-8} +3-2 a z^{-1} -2 a^{-1} z^{-1} + a^{-3} z^{-1} + 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                     1   1 12                   2 1   -1 10               1 2 1     0 8               2 2       0 6           2 3 2         -1 4         1 2 2           1 2       1 3 2             0 0     1 1 3               3 -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}^{3} }[/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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/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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