L10n79

L10n78

L10n80

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Link Presentations

[edit Notes on L10n79's Link Presentations]

Planar diagram presentation	X6172 X5,14,6,15 X8493 X2,16,3,15 X16,7,17,8 X9,18,10,19 X11,20,12,13 X13,12,14,5 X4,17,1,18 X19,10,20,11
Gauss code	{1, -4, 3, -9}, {-2, -1, 5, -3, -6, 10, -7, 8}, {-8, 2, 4, -5, 9, 6, -10, 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{(u-1) (v-1) (w-1) (v w+1)}{\sqrt{u} v w} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 2 q^{-2} -2 q^{-3} +5 q^{-4} -5 q^{-5} +6 q^{-6} -5 q^{-7} +4 q^{-8} -2 q^{-9} + q^{-10} }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^4 a^8+3 z^2 a^8+a^8 z^{-2} +3 a^8-z^6 a^6-5 z^4 a^6-10 z^2 a^6-2 a^6 z^{-2} -9 a^6+2 z^4 a^4+7 z^2 a^4+a^4 z^{-2} +6 a^4 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{12} z^4-2 a^{12} z^2+a^{12}+2 a^{11} z^5-3 a^{11} z^3+2 a^{10} z^6-a^{10} z^4-2 a^{10} z^2+2 a^9 z^7-3 a^9 z^5+3 a^9 z^3+a^8 z^8-a^8 z^6+3 a^8 z^4-3 a^8 z^2-a^8 z^{-2} +3 a^8+3 a^7 z^7-8 a^7 z^5+13 a^7 z^3-9 a^7 z+2 a^7 z^{-1} +a^6 z^8-3 a^6 z^6+8 a^6 z^4-12 a^6 z^2-2 a^6 z^{-2} +9 a^6+a^5 z^7-3 a^5 z^5+7 a^5 z^3-9 a^5 z+2 a^5 z^{-1} +3 a^4 z^4-9 a^4 z^2-a^4 z^{-2} +6 a^4 }[/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 χ -3                 2 2 -5               2 2 0 -7             3     3 -9           2 2     0 -11         4 3       1 -13       1 2         1 -15     3 4           -1 -17   1 3             2 -19   1               -1 -21 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{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{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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