L10n82

L10n81

L10n83

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Link Presentations

[edit Notes on L10n82's Link Presentations]

Planar diagram presentation	X6172 X14,4,15,3 X11,20,12,13 X7,18,8,19 X9,16,10,17 X17,8,18,9 X13,10,14,11 X19,12,20,5 X2536 X4,16,1,15
Gauss code	{1, -9, 2, -10}, {9, -1, -4, 6, -5, 7, -3, 8}, {-7, -2, 10, 5, -6, 4, -8, 3}

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^2 w^2-u v^2 w-u v w^2+u v w+u w^3-v^3-v^2 w^2+v^2 w+v w^2-v w}{\sqrt{u} v^{3/2} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-8} + q^{-6} + q^{-5} -2 q^{-4} +3 q^{-3} -2 q^{-2} +q+3 q^{-1} -2 }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^8 z^{-2} +a^8-a^6 z^2-2 a^6 z^{-2} -3 a^6+a^4 z^2+a^4 z^{-2} +a^4-a^2 z^4-2 a^2 z^2+z^2+1 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^8 z^8-8 a^8 z^6+21 a^8 z^4-22 a^8 z^2-a^8 z^{-2} +8 a^8+a^7 z^7-8 a^7 z^5+17 a^7 z^3-11 a^7 z+2 a^7 z^{-1} +a^6 z^8-9 a^6 z^6+25 a^6 z^4-27 a^6 z^2-2 a^6 z^{-2} +13 a^6+a^5 z^7-8 a^5 z^5+19 a^5 z^3-11 a^5 z+2 a^5 z^{-1} +3 a^4 z^4-6 a^4 z^2-a^4 z^{-2} +5 a^4+2 a^3 z^5-2 a^3 z^3+a^2 z^6-3 a^2 z^2+2 a z^5-4 a z^3+z^4-2 z^2+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 3                     1 1 1                   1   -1 -1                 2 1   1 -3               2 3     1 -5           1 2         1 -7           1 2         1 -9       1 2 2           -1 -11         2             2 -13     1                 1 -15 1                     1 -17 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{ i=1 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}_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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