L11n224

L11n223

L11n225

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Link Presentations

[edit Notes on L11n224's Link Presentations]

Planar diagram presentation	X10,1,11,2 X3,12,4,13 X16,9,17,10 X20,12,21,11 X22,15,9,16 X5,14,6,15 X7,18,8,19 X13,4,14,5 X17,6,18,7 X19,8,20,1 X2,21,3,22
Gauss code	{1, -11, -2, 8, -6, 9, -7, 10}, {3, -1, 4, 2, -8, 6, 5, -3, -9, 7, -10, -4, 11, -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{t(1)^3 t(2)^3+t(1)^2 t(2)^3-t(1) t(2)^3-t(1)^2 t(2)^2+2 t(1) t(2)^2+2 t(1)^2 t(2)-t(1) t(2)-t(1)^2+t(1)+1}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{7/2}}-\frac{2}{q^{13/2}}+\frac{3}{q^{15/2}}-\frac{3}{q^{17/2}}+\frac{3}{q^{19/2}}-\frac{3}{q^{21/2}}+\frac{2}{q^{23/2}}-\frac{1}{q^{25/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^{11}+z a^{11}+2 z^3 a^9+4 z a^9+a^9 z^{-1} -z^7 a^7-7 z^5 a^7-14 z^3 a^7-9 z a^7-a^7 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{15} z^5-3 a^{15} z^3+2 a^{15} z+2 a^{14} z^6-6 a^{14} z^4+3 a^{14} z^2+a^{13} z^7-a^{13} z^5-3 a^{13} z^3+a^{13} z+2 a^{12} z^6-3 a^{12} z^4-a^{12} z^2+3 a^{11} z^5-6 a^{11} z^3+3 a^{11} z+a^{10} z^4+a^{10} z^2-2 a^9 z^5+8 a^9 z^3-5 a^9 z+a^9 z^{-1} -2 a^8 z^4+5 a^8 z^2-a^8+a^7 z^7-7 a^7 z^5+14 a^7 z^3-9 a^7 z+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   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -6                     1 1 -8                     1 1 -10               1 1     0 -12             2         2 -14           2 2 1       -1 -16         2 2           0 -18       1 2 1           0 -20     2 2               0 -22   1 2                 1 -24   1                   -1 -26 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=-10 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/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{ 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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