L11n440

L11n439

L11n441

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Link Presentations

[edit Notes on L11n440's Link Presentations]

Planar diagram presentation	X6172 X2536 X18,11,19,12 X3,11,4,10 X9,1,10,4 X7,15,8,14 X13,5,14,8 X19,13,20,22 X15,21,16,20 X21,17,22,16 X12,17,9,18
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, 3, -11}, {-7, 6, -9, 10, 11, -3, -8, 9, -10, 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 w x^2-u v w x-2 u v x^2+2 u v x-u v+u w+u x^2-u x-v w x^2+v w x+v x^3-w x^3+2 w x^2-2 w x-x^2+x}{\sqrt{u} \sqrt{v} \sqrt{w} x^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{15/2}-2 q^{13/2}+5 q^{11/2}-6 q^{9/2}+5 q^{7/2}-7 q^{5/2}+3 q^{3/2}-5 \sqrt{q}-\frac{1}{\sqrt{q}}-\frac{1}{q^{5/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^5 a^{-1} +2 z^5 a^{-3} +a z^3-8 z^3 a^{-1} +10 z^3 a^{-3} -3 z^3 a^{-5} +4 a z-17 z a^{-1} +21 z a^{-3} -9 z a^{-5} +z a^{-7} +5 a z^{-1} -17 a^{-1} z^{-1} +21 a^{-3} z^{-1} -11 a^{-5} z^{-1} +2 a^{-7} z^{-1} +2 a z^{-3} -7 a^{-1} z^{-3} +9 a^{-3} z^{-3} -5 a^{-5} z^{-3} + a^{-7} z^{-3} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{-8} -4 z^4 a^{-8} +6 z^2 a^{-8} + a^{-8} z^{-2} -4 a^{-8} +2 z^7 a^{-7} -6 z^5 a^{-7} +4 z^3 a^{-7} - a^{-7} z^{-3} -2 z a^{-7} +2 a^{-7} z^{-1} +z^8 a^{-6} +4 z^6 a^{-6} -26 z^4 a^{-6} +34 z^2 a^{-6} +7 a^{-6} z^{-2} -24 a^{-6} +7 z^7 a^{-5} -24 z^5 a^{-5} +23 z^3 a^{-5} -5 a^{-5} z^{-3} -16 z a^{-5} +12 a^{-5} z^{-1} +z^8 a^{-4} +7 z^6 a^{-4} -43 z^4 a^{-4} +75 z^2 a^{-4} +18 a^{-4} z^{-2} -58 a^{-4} +5 z^7 a^{-3} -23 z^5 a^{-3} +41 z^3 a^{-3} -9 a^{-3} z^{-3} -37 z a^{-3} +24 a^{-3} z^{-1} +5 z^6 a^{-2} -32 z^4 a^{-2} +73 z^2 a^{-2} +19 a^{-2} z^{-2} -60 a^{-2} +a z^7+z^7 a^{-1} -7 a z^5-12 z^5 a^{-1} +15 a z^3+37 z^3 a^{-1} -2 a z^{-3} -7 a^{-1} z^{-3} -16 a z-39 z a^{-1} +9 a z^{-1} +23 a^{-1} z^{-1} +z^6-11 z^4+26 z^2+7 z^{-2} -23 }[/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                   4 1   -3 10                 2 1     1 8               4 5       1 6           1 4 1         2 4             4           4 2       1 7 4             2 0         6               6 -2     1                   1 -4 1                       1 -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{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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