L11a200

L11a199

L11a201

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Link Presentations

[edit Notes on L11a200's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X16,5,17,6 X22,18,7,17 X18,12,19,11 X20,14,21,13 X12,20,13,19 X14,22,15,21 X6718 X4,15,5,16
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 6, -8, 7, -9, 11, -4, 5, -6, 8, -7, 9, -5}

A Braid Representative	{{{braid_table}}}
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^2 v^6-2 u^2 v^5+2 u^2 v^4-2 u^2 v^3+2 u^2 v^2+2 u v^5-3 u v^4+3 u v^3-3 u v^2+2 u v+2 v^4-2 v^3+2 v^2-2 v+1}{u v^3} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-\frac{4}{q^{9/2}}-2 q^{7/2}+\frac{6}{q^{7/2}}+4 q^{5/2}-\frac{9}{q^{5/2}}-6 q^{3/2}+\frac{9}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{2}{q^{11/2}}+8 \sqrt{q}-\frac{10}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a z^9+a^3 z^7-8 a z^7+z^7 a^{-1} +6 a^3 z^5-24 a z^5+6 z^5 a^{-1} +12 a^3 z^3-33 a z^3+12 z^3 a^{-1} +9 a^3 z-21 a z+9 z a^{-1} +3 a^3 z^{-1} -5 a z^{-1} +2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^5-3 a^7 z^3+a^7 z+2 a^6 z^6-5 a^6 z^4+a^6 z^2+3 a^5 z^7-9 a^5 z^5+8 a^5 z^3-3 a^5 z+3 a^4 z^8-9 a^4 z^6+z^6 a^{-4} +9 a^4 z^4-4 z^4 a^{-4} -a^4 z^2+4 z^2 a^{-4} - a^{-4} +3 a^3 z^9-13 a^3 z^7+2 z^7 a^{-3} +26 a^3 z^5-7 z^5 a^{-3} -22 a^3 z^3+5 z^3 a^{-3} +12 a^3 z-3 a^3 z^{-1} +a^2 z^{10}+2 z^8 a^{-2} -11 a^2 z^6-5 z^6 a^{-2} +29 a^2 z^4-20 a^2 z^2+2 z^2 a^{-2} +5 a^2+5 a z^9+2 z^9 a^{-1} -25 a z^7-7 z^7 a^{-1} +55 a z^5+12 z^5 a^{-1} -54 a z^3-16 z^3 a^{-1} +25 a z+9 z a^{-1} -5 a z^{-1} -2 a^{-1} z^{-1} +z^{10}-z^8-6 z^6+19 z^4-20 z^2+5 }[/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                   3 1   -2 4                 3 1     2 2               5 3       -2 0             5 3         2 -2           5 6           1 -4         4 4             0 -6       2 5               3 -8     2 4                 -2 -10     2                   2 -12 1 2                     -1 -14 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{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\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^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/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}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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