L10n95

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Link Presentations

[edit Notes on L10n95's Link Presentations]

Planar diagram presentation	X8192 X18,12,19,11 X3,10,4,11 X19,2,20,3 X16,8,17,7 X9,13,10,20 X12,18,7,17 X4,13,5,14 X14,5,15,6 X6,15,1,16
Gauss code	{1, 4, -3, -8, 9, -10}, {5, -1, -6, 3, 2, -7}, {8, -9, 10, -5, 7, -2, -4, 6}

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) t(3)^3+t(1)^2 t(3)^2-t(1) t(2)^2 t(3)^2-t(1) t(3)^2-t(1)^2 t(2) t(3)^2+2 t(1) t(2) t(3)^2-t(2) t(3)^2+t(1) t(2)^2 t(3)-t(2)^2 t(3)+t(1) t(3)+t(1)^2 t(2) t(3)-2 t(1) t(2) t(3)+t(2) t(3)+t(1) t(2)}{t(1) t(2) t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-7} -2 q^{-6} +4 q^{-5} -5 q^{-4} +6 q^{-3} -5 q^{-2} +2 q+5 q^{-1} -2 }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^6 z^2+a^6-a^4 z^4-a^4 z^2+a^4 z^{-2} +a^4-2 a^2 z^4-6 a^2 z^2-2 a^2 z^{-2} -6 a^2+2 z^2+ z^{-2} +4 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^8-2 z^2 a^8+2 z^5 a^7-3 z^3 a^7+3 z^6 a^6-7 z^4 a^6+7 z^2 a^6-2 a^6+2 z^7 a^5-3 z^5 a^5+3 z^3 a^5+z^8 a^4-z^6 a^4+3 z^4 a^4-3 z^2 a^4-a^4 z^{-2} +3 a^4+3 z^7 a^3-8 z^5 a^3+13 z^3 a^3-9 z a^3+2 a^3 z^{-1} +z^8 a^2-4 z^6 a^2+14 z^4 a^2-21 z^2 a^2-2 a^2 z^{-2} +11 a^2+z^7 a-3 z^5 a+7 z^3 a-9 z a+2 a z^{-1} +3 z^4-9 z^2- z^{-2} +7 }[/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 3                 2 2 1               1 1 0 -1             4 1   3 -3           3 3     0 -5         3 2       1 -7       2 3         1 -9     2 3           -1 -11     2             2 -13 1 2               -1 -15 1                 1

Integral Khovanov Homology (db, data source)

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