L10n108

L10n107

L10n109

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Link Presentations

[edit Notes on L10n108's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X15,2,16,3 X16,7,17,8 X19,11,20,14 X13,15,14,20 X9,18,10,19 X11,10,12,5 X4,17,1,18
Gauss code	{1, 4, -3, -10}, {-9, 2, -7, 6}, {-2, -1, 5, 3, -8, 9}, {-4, -5, 10, 8, -6, 7}

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 v w^2 x^2-u v w^2 x-u w^2 x^2+u w x^2+v w-v-x+1}{\sqrt{u} \sqrt{v} w x} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{3}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{1}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}}-\frac{1}{q^{17/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z a^9+a^9 z^{-1} +a^9 z^{-3} -z^5 a^7-5 z^3 a^7-7 z a^7-6 a^7 z^{-1} -3 a^7 z^{-3} +z^7 a^5+6 z^5 a^5+12 z^3 a^5+13 z a^5+9 a^5 z^{-1} +3 a^5 z^{-3} -z^5 a^3-5 z^3 a^3-7 z a^3-4 a^3 z^{-1} -a^3 z^{-3} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{11} z+a^{10} z^2+3 a^9 z^3-a^9 z^{-3} -8 a^9 z+5 a^9 z^{-1} +2 a^8 z^6-9 a^8 z^4+13 a^8 z^2+3 a^8 z^{-2} -10 a^8+3 a^7 z^7-16 a^7 z^5+28 a^7 z^3-3 a^7 z^{-3} -25 a^7 z+12 a^7 z^{-1} +a^6 z^8-2 a^6 z^6-9 a^6 z^4+23 a^6 z^2+6 a^6 z^{-2} -19 a^6+4 a^5 z^7-22 a^5 z^5+37 a^5 z^3-3 a^5 z^{-3} -27 a^5 z+12 a^5 z^{-1} +a^4 z^8-4 a^4 z^6+11 a^4 z^2+3 a^4 z^{-2} -10 a^4+a^3 z^7-6 a^3 z^5+12 a^3 z^3-a^3 z^{-3} -11 a^3 z+5 a^3 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   \ -6 -5 -4 -3 -2 -1 0 1 2 χ 0                 1 1 -2                   0 -4             3 1   2 -6         1 1 2     2 -8         4 1       3 -10     2 1 2         3 -12     4 2           2 -14 1   1             2 -16 2 2               0 -18 1                 1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/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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