L10n33

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L10n34

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Link Presentations

[edit Notes on L10n33's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X7,16,8,17 X20,18,5,17 X18,11,19,12 X10,19,11,20 X9,14,10,15 X15,8,16,9 X2536 X4,14,1,13
Gauss code	{1, -9, 2, -10}, {9, -1, -3, 8, -7, -6, 5, -2, 10, 7, -8, 3, 4, -5, 6, -4}

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)-1) (t(2)-1) \left(t(2)^2-4 t(2)+1\right)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{4}{q^{9/2}}+\frac{6}{q^{7/2}}+q^{5/2}-\frac{9}{q^{5/2}}-4 q^{3/2}+\frac{8}{q^{3/2}}+\frac{2}{q^{11/2}}+6 \sqrt{q}-\frac{8}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -2 a^5 z-a^5 z^{-1} +3 a^3 z^3+5 a^3 z+3 a^3 z^{-1} -a z^5-2 a z^3+z^3 a^{-1} -3 a z-2 a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 3 a^6 z^4-5 a^6 z^2+a^6+a^5 z^7+a^5 z^5-4 a^5 z^3+4 a^5 z-a^5 z^{-1} +a^4 z^8+a^4 z^6+a^4 z^4-6 a^4 z^2+3 a^4+5 a^3 z^7-4 a^3 z^5-6 a^3 z^3+9 a^3 z-3 a^3 z^{-1} +a^2 z^8+7 a^2 z^6-11 a^2 z^4+z^4 a^{-2} +3 a^2+4 a z^7-a z^5+4 z^5 a^{-1} -6 a z^3-4 z^3 a^{-1} +5 a z-2 a z^{-1} +6 z^6-8 z^4+z^2 }[/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   \ -5 -4 -3 -2 -1 0 1 2 3 χ 6                 1 -1 4               3   3 2             3 1   -2 0           5 3     2 -2         5 5       0 -4       4 3         1 -6     2 5           3 -8   2 4             -2 -10   2               2 -12 2                 -2

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/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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