L10n90

L10n89

L10n91

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Link Presentations

[edit Notes on L10n90's Link Presentations]

Planar diagram presentation	X6172 X3,13,4,12 X13,20,14,17 X7,18,8,19 X17,10,18,11 X9,15,10,14 X15,9,16,8 X19,16,20,5 X2536 X11,1,12,4
Gauss code	{1, -9, -2, 10}, {-5, 4, -8, 3}, {9, -1, -4, 7, -6, 5, -10, 2, -3, 6, -7, 8}

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(3)-1) \left(t(2) t(3)^2-2 t(2) t(3)+2 t(3)-1\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^3-3 q^2+5 q-7+9 q^{-1} -7 q^{-2} +8 q^{-3} -5 q^{-4} +3 q^{-5} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^6 z^{-2} +a^6-z^4 a^4-3 z^2 a^4-2 a^4 z^{-2} -5 a^4+z^6 a^2+4 z^4 a^2+7 z^2 a^2+a^2 z^{-2} +6 a^2-2 z^4-5 z^2-3+z^2 a^{-2} + a^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 2 a^2 z^8+2 z^8+6 a^3 z^7+9 a z^7+3 z^7 a^{-1} +7 a^4 z^6+5 a^2 z^6+z^6 a^{-2} -z^6+3 a^5 z^5-12 a^3 z^5-25 a z^5-10 z^5 a^{-1} -15 a^4 z^4-23 a^2 z^4-3 z^4 a^{-2} -11 z^4+7 a^3 z^3+16 a z^3+9 z^3 a^{-1} +6 a^6 z^2+16 a^4 z^2+18 a^2 z^2+3 z^2 a^{-2} +11 z^2+3 a^5 z+a^3 z-3 a z-z a^{-1} -4 a^6-8 a^4-7 a^2- a^{-2} -3-2 a^5 z^{-1} -2 a^3 z^{-1} +a^6 z^{-2} +2 a^4 z^{-2} +a^2 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   \ -4 -3 -2 -1 0 1 2 3 4 χ 7                 1 1 5               2   -2 3             3 1   2 1           4 2     -2 -1         5 3       2 -3       4 6         2 -5     4 3           1 -7   1 4             3 -9 2 4               -2 -11 3                 3

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=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/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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