L10n81

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Link Presentations

[edit Notes on L10n81's Link Presentations]

Planar diagram presentation	X6172 X2,16,3,15 X3,10,4,11 X5,14,6,15 X11,20,12,13 X13,12,14,5 X19,1,20,4 X8,17,9,18 X16,7,17,8 X18,9,19,10
Gauss code	{1, -2, -3, 7}, {-4, -1, 9, -8, 10, 3, -5, 6}, {-6, 4, 2, -9, 8, -10, -7, 5}

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(3)^2 t(2)^3-t(1) t(3)^3 t(2)^2+2 t(1) t(3)^2 t(2)^2-2 t(3)^2 t(2)^2-t(1) t(3) t(2)^2+2 t(3) t(2)^2-2 t(1) t(3)^2 t(2)+t(3)^2 t(2)+2 t(1) t(3) t(2)-2 t(3) t(2)+t(2)-t(1) t(3)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-10} -2 q^{-9} +5 q^{-8} -6 q^{-7} +6 q^{-6} -6 q^{-5} +6 q^{-4} -2 q^{-3} +2 q^{-2} }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^8 z^4+3 a^8 z^2+a^8 z^{-2} +4 a^8-a^6 z^6-5 a^6 z^4-11 a^6 z^2-2 a^6 z^{-2} -11 a^6+2 a^4 z^4+7 a^4 z^2+a^4 z^{-2} +7 a^4 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{12} z^4-2 a^{12} z^2+a^{12}+2 a^{11} z^5-2 a^{11} z^3+3 a^{10} z^6-4 a^{10} z^4+2 a^{10} z^2+2 a^9 z^7-2 a^9 z^3+a^8 z^8+4 a^8 z^4-7 a^8 z^2-a^8 z^{-2} +5 a^8+3 a^7 z^7-5 a^7 z^5+9 a^7 z^3-11 a^7 z+2 a^7 z^{-1} +a^6 z^8-3 a^6 z^6+12 a^6 z^4-20 a^6 z^2-2 a^6 z^{-2} +13 a^6+a^5 z^7-3 a^5 z^5+9 a^5 z^3-11 a^5 z+2 a^5 z^{-1} +3 a^4 z^4-9 a^4 z^2-a^4 z^{-2} +8 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             4     4 -9           2 2     0 -11         4 4       0 -13       3 3         0 -15     2 3           -1 -17     3             3 -19 1 2               -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{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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^{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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