L10n28

L10n27

L10n29

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Link Presentations

[edit Notes on L10n28's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X7,14,8,15 X11,19,12,18 X15,20,16,5 X19,16,20,17 X17,13,18,12 X13,8,14,9 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, -3, 8, 10, -2, -4, 7, -8, 3, -5, 6, -7, 4, -6, 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) t(2)^3+t(2)^3-2 t(1) t(2)^2-t(2)^2-t(1) t(2)-2 t(2)+t(1)+1}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\sqrt{q}+\frac{1}{\sqrt{q}}-\frac{2}{q^{3/2}}+\frac{1}{q^{5/2}}-\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{1}{q^{13/2}}+\frac{1}{q^{15/2}}-\frac{1}{q^{17/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 z^{-1} -2 a^7 z-2 a^7 z^{-1} +a^5 z^{-1} +a^3 z^5+4 a^3 z^3+3 a^3 z+a^3 z^{-1} -a z^3-3 a z-a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^7 a^9+6 z^5 a^9-10 z^3 a^9+6 z a^9-a^9 z^{-1} -z^8 a^8+6 z^6 a^8-9 z^4 a^8+4 z^2 a^8-a^8-2 z^7 a^7+14 z^5 a^7-26 z^3 a^7+14 z a^7-2 a^7 z^{-1} -z^8 a^6+7 z^6 a^6-13 z^4 a^6+8 z^2 a^6-3 a^6-z^7 a^5+7 z^5 a^5-13 z^3 a^5+8 z a^5-a^5 z^{-1} -z^4 a^4+4 z^2 a^4-2 a^4-2 z^5 a^3+7 z^3 a^3-4 z a^3+a^3 z^{-1} -z^6 a^2+3 z^4 a^2-a^2-z^5 a+4 z^3 a-4 z a+a 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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                     1 1 0                       0 -2                 2 1   1 -4             1 1 1     1 -6             1 1       0 -8         1 2 1         0 -10       1 1 1           -1 -12       1 2             1 -14   1 1                 0 -16                       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=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_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}\oplus{\mathbb Z}_2 }[/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}_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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