L10n83

L10n82

L10n84

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Link Presentations

[edit Notes on L10n83's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X5,16,6,17 X8493 X17,20,18,15 X19,10,20,11 X9,18,10,19 X15,14,16,5 X2,12,3,11
Gauss code	{1, -10, 5, -3}, {-9, 4, -6, 8, -7, 6}, {-4, -1, 2, -5, -8, 7, 10, -2, 3, 9}

A Braid Representative	{{{braid_table}}}
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-1) (w-1)^2 (v w+1)}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 1-2 q^{-1} +5 q^{-2} -4 q^{-3} +6 q^{-4} -5 q^{-5} +5 q^{-6} -3 q^{-7} + q^{-8} }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^4 a^6+2 z^2 a^6+a^6 z^{-2} +a^6-z^6 a^4-4 z^4 a^4-5 z^2 a^4-2 a^4 z^{-2} -4 a^4+z^4 a^2+3 z^2 a^2+a^2 z^{-2} +3 a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^2 a^{10}-a^{10}+3 z^3 a^9-z a^9+2 z^6 a^8-5 z^4 a^8+9 z^2 a^8-3 a^8+3 z^7 a^7-9 z^5 a^7+10 z^3 a^7-3 z a^7+z^8 a^6+3 z^6 a^6-18 z^4 a^6+20 z^2 a^6+a^6 z^{-2} -7 a^6+5 z^7 a^5-15 z^5 a^5+9 z^3 a^5+z a^5-2 a^5 z^{-1} +z^8 a^4+2 z^6 a^4-17 z^4 a^4+18 z^2 a^4+2 a^4 z^{-2} -8 a^4+2 z^7 a^3-6 z^5 a^3+2 z^3 a^3+3 z a^3-2 a^3 z^{-1} +z^6 a^2-4 z^4 a^2+6 z^2 a^2+a^2 z^{-2} -4 a^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   \ -6 -5 -4 -3 -2 -1 0 1 2 χ 1                 1 1 -1               1   -1 -3             4 1   3 -5           2 3     1 -7         4 2       2 -9     1 2 2         1 -11     4 4           0 -13   1 3             2 -15   2               -2 -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=-5 }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}_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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