L11n141

L11n140

L11n142

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Link Presentations

[edit Notes on L11n141's Link Presentations]

Planar diagram presentation	X8192 X18,11,19,12 X3,10,4,11 X2,17,3,18 X12,5,13,6 X6718 X9,16,10,17 X15,20,16,21 X13,22,14,7 X21,14,22,15 X19,4,20,5
Gauss code	{1, -4, -3, 11, 5, -6}, {6, -1, -7, 3, 2, -5, -9, 10, -8, 7, 4, -2, -11, 8, -10, 9}

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{2 t(1)^2 t(2)^4-t(1)^2 t(2)^3+t(1) t(2)^2-t(2)+2}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{1}{q^{9/2}}-\frac{1}{q^{7/2}}+\frac{1}{q^{21/2}}-\frac{1}{q^{19/2}}+\frac{1}{q^{17/2}}-\frac{2}{q^{15/2}}+\frac{1}{q^{13/2}}-\frac{2}{q^{11/2}} }[/math] (db)
Signature	-7 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^{13}-a^{13} z^{-1} +z^5 a^{11}+6 z^3 a^{11}+8 z a^{11}+2 a^{11} z^{-1} -z^7 a^9-6 z^5 a^9-10 z^3 a^9-5 z a^9-z^7 a^7-6 z^5 a^7-10 z^3 a^7-5 z a^7-a^7 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^2 a^{14}-2 a^{14}+z^3 a^{13}-2 z a^{13}+a^{13} z^{-1} -z^8 a^{12}+7 z^6 a^{12}-16 z^4 a^{12}+16 z^2 a^{12}-5 a^{12}-z^9 a^{11}+7 z^7 a^{11}-17 z^5 a^{11}+20 z^3 a^{11}-10 z a^{11}+2 a^{11} z^{-1} -2 z^8 a^{10}+12 z^6 a^{10}-21 z^4 a^{10}+14 z^2 a^{10}-3 a^{10}-z^9 a^9+6 z^7 a^9-11 z^5 a^9+9 z^3 a^9-3 z a^9-z^8 a^8+5 z^6 a^8-5 z^4 a^8-z^2 a^8+a^8-z^7 a^7+6 z^5 a^7-10 z^3 a^7+5 z a^7-a^7 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   \ -7 -6 -5 -4 -3 -2 -1 0 χ -6               1 1 -8             1 1 0 -10           1     1 -12       1 1 1     1 -14       2 1       1 -16   1 1 1         1 -18   2 2           0 -20                 0 -22 1               -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-8 }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\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] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}\oplus{\mathbb Z}_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} }[/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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