L11n129

L11n128

L11n130

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Link Presentations

[edit Notes on L11n129's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X22,10,7,9 X2738 X4,15,5,16 X5,13,6,12 X16,12,17,11 X17,6,18,1 X19,14,20,15 X13,20,14,21 X21,19,22,18
Gauss code	{1, -4, 2, -5, -6, 8}, {4, -1, 3, -2, 7, 6, -10, 9, 5, -7, -8, 11, -9, 10, -11, -3}

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^2 v^2-2 u^2 v+u^2-3 u v^2+7 u v-3 u+v^2-2 v+1}{u v} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-\frac{1}{q^{9/2}}-3 q^{7/2}+\frac{2}{q^{7/2}}+4 q^{5/2}-\frac{5}{q^{5/2}}-6 q^{3/2}+\frac{6}{q^{3/2}}+7 \sqrt{q}-\frac{7}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z^{-1} +z^3 a^{-3} -3 a^3 z-2 a^3 z^{-1} +z a^{-3} -z^5 a^{-1} +3 a z^3-3 z^3 a^{-1} +5 a z+2 a z^{-1} -4 z a^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a z^9-z^9 a^{-1} -2 a^2 z^8-3 z^8 a^{-2} -5 z^8-a^3 z^7-2 z^7 a^{-1} -3 z^7 a^{-3} +8 a^2 z^6+10 z^6 a^{-2} -z^6 a^{-4} +19 z^6+3 a^3 z^5+10 a z^5+18 z^5 a^{-1} +11 z^5 a^{-3} -2 a^4 z^4-15 a^2 z^4-7 z^4 a^{-2} +3 z^4 a^{-4} -23 z^4-a^5 z^3-9 a^3 z^3-21 a z^3-22 z^3 a^{-1} -9 z^3 a^{-3} +2 a^4 z^2+8 a^2 z^2+2 z^2 a^{-2} -z^2 a^{-4} +9 z^2+2 a^5 z+8 a^3 z+12 a z+8 z a^{-1} +2 z a^{-3} -a^2-a^5 z^{-1} -2 a^3 z^{-1} -2 a z^{-1} - a^{-1} 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 χ 10                   1 -1 8                 2   2 6               2 1   -1 4             4 2     2 2           3 2       -1 0         4 4         0 -2       3 4           1 -4     2 3             -1 -6   1 4               3 -8   1                 -1 -10 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=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/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}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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