L10n87

L10n86

L10n88

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Link Presentations

[edit Notes on L10n87's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,19,12,18 X17,9,18,8 X7,17,8,16 X13,15,14,20 X15,5,16,14 X19,13,20,12 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {-7, 5, -4, 3, -8, 6}, {9, -1, -5, 4, 10, -2, -3, 8, -6, 7}

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{(w-1) \left(u v w^2+u w+u+v^2 w^3+v^2 w^2+v w\right)}{\sqrt{u} v w^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^7-q^6+2 q^5-2 q^4+2 q^3-q^2+q+1+ q^{-2} }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^6 a^{-2} -7 z^4 a^{-2} +z^4-14 z^2 a^{-2} +2 z^2 a^{-4} +z^2 a^{-6} +5 z^2-10 a^{-2} +3 a^{-4} + a^{-6} +6-2 a^{-2} z^{-2} + a^{-4} z^{-2} + z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-8} -3 z^2 a^{-8} + a^{-8} +z^5 a^{-7} -2 z^3 a^{-7} +z^6 a^{-6} -3 z^4 a^{-6} +3 z^2 a^{-6} - a^{-6} +z^5 a^{-5} -z^3 a^{-5} +z^4 a^{-4} -2 z^2 a^{-4} - a^{-4} z^{-2} +4 a^{-4} +z^7 a^{-3} -8 z^5 a^{-3} +19 z^3 a^{-3} -13 z a^{-3} +2 a^{-3} z^{-1} +z^8 a^{-2} -9 z^6 a^{-2} +26 z^4 a^{-2} -30 z^2 a^{-2} -2 a^{-2} z^{-2} +14 a^{-2} +z^7 a^{-1} -8 z^5 a^{-1} +18 z^3 a^{-1} -13 z a^{-1} +2 a^{-1} z^{-1} +z^8-8 z^6+21 z^4-22 z^2- z^{-2} +9 }[/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 5 6 χ 15                     1 1 13                   1 1 0 11                 1     1 9               1 1     0 7           1 2 1       0 5           1 2         1 3       1 2 1           0 1         2             2 -1     1                 1 -3 1                     1 -5 1                     1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ i=3 }[/math] [math]\displaystyle{ i=5 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/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=6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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