L11n208

L11n207

L11n209

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Link Presentations

[edit Notes on L11n208's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X5,14,6,15 X22,18,9,17 X19,5,20,4 X21,6,22,7 X7,17,8,16 X8,9,1,10 X18,14,19,13 X15,21,16,20
Gauss code	{1, -2, 3, 6, -4, 7, -8, -9}, {9, -1, 2, -3, 10, 4, -11, 8, 5, -10, -6, 11, -7, -5}

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^3 v^3-u^3 v^2+u^2 v^2+u v-v+1}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{1}{q^{9/2}}+q^{7/2}-\frac{1}{q^{7/2}}-q^{5/2}+\frac{1}{q^{5/2}}+q^{3/2}-\frac{2}{q^{3/2}}-\frac{1}{q^{11/2}}-2 \sqrt{q}+\frac{1}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^3 z^5+5 a^3 z^3+6 a^3 z+2 a^3 z^{-1} -a z^7-7 a z^5+z^5 a^{-1} -16 a z^3+5 z^3 a^{-1} -14 a z+6 z a^{-1} -3 a z^{-1} + a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a z^9-z^9 a^{-1} -a^2 z^8-z^8 a^{-2} -2 z^8+7 a z^7+7 z^7 a^{-1} +6 a^2 z^6+7 z^6 a^{-2} +13 z^6-2 a^3 z^5-18 a z^5-16 z^5 a^{-1} -a^4 z^4-12 a^2 z^4-15 z^4 a^{-2} -26 z^4-a^5 z^3+8 a^3 z^3+24 a z^3+15 z^3 a^{-1} -a^6 z^2+2 a^4 z^2+11 a^2 z^2+10 z^2 a^{-2} +18 z^2-a^7 z+a^5 z-7 a^3 z-16 a z-7 z a^{-1} -3 a^2- a^{-2} -3+2 a^3 z^{-1} +3 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 8                   1 -1 6                     0 4               1 1   0 2             1       1 0             1       1 -2         2 1         1 -4         1           1 -6     1 1             0 -8                     0 -10 1 1                 0 -12 1                   1

Integral Khovanov Homology (db, data source)

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