L11n289

L11n288

L11n290

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Link Presentations

[edit Notes on L11n289's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X2,14,3,13 X14,7,15,8 X11,18,12,19 X9,21,10,20 X19,5,20,10 X15,1,16,4 X17,22,18,11 X21,16,22,17
Gauss code	{1, -4, -3, 9}, {-2, -1, 5, 3, -7, 8}, {-6, 2, 4, -5, -9, 11, -10, 6, -8, 7, -11, 10}

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 w^4-2 u v w^3+2 u v w^2-2 u v w-u w^4+u w^3-u w^2+u w-v^2 w^3+v^2 w^2-v^2 w+v^2+2 v w^3-2 v w^2+2 v w-v}{\sqrt{u} v w^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 2 q-2+6 q^{-1} -6 q^{-2} +8 q^{-3} -7 q^{-4} +6 q^{-5} -4 q^{-6} +2 q^{-7} - q^{-8} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^4 a^6-3 z^2 a^6-a^6 z^{-2} -3 a^6+z^6 a^4+5 z^4 a^4+11 z^2 a^4+4 a^4 z^{-2} +11 a^4-3 z^4 a^2-11 z^2 a^2-5 a^2 z^{-2} -13 a^2+2 z^2+2 z^{-2} +5 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^9-3 z^3 a^9+z a^9+2 z^6 a^8-5 z^4 a^8+z^2 a^8+3 z^7 a^7-9 z^5 a^7+8 z^3 a^7-4 z a^7+a^7 z^{-1} +3 z^8 a^6-11 z^6 a^6+17 z^4 a^6-12 z^2 a^6-a^6 z^{-2} +5 a^6+z^9 a^5+z^7 a^5-14 z^5 a^5+31 z^3 a^5-21 z a^5+5 a^5 z^{-1} +5 z^8 a^4-23 z^6 a^4+47 z^4 a^4-40 z^2 a^4-4 a^4 z^{-2} +18 a^4+z^9 a^3-z^7 a^3-7 z^5 a^3+28 z^3 a^3-29 z a^3+9 a^3 z^{-1} +2 z^8 a^2-10 z^6 a^2+28 z^4 a^2-37 z^2 a^2-5 a^2 z^{-2} +21 a^2+z^7 a-3 z^5 a+8 z^3 a-13 z a+5 a z^{-1} +3 z^4-10 z^2-2 z^{-2} +9 }[/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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 3                   2 2 1                 1 1 0 -1               5 1   4 -3             3 3     0 -5           5 3       2 -7         3 4         1 -9       3 4           -1 -11     1 3             2 -13   1 3               -2 -15   1                 1 -17 1                   -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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

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