L10n68

L10n67

L10n69

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Link Presentations

[edit Notes on L10n68's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X13,2,14,3 X14,7,15,8 X9,18,10,19 X11,16,12,17 X17,20,18,11 X4,15,1,16 X19,10,20,5
Gauss code	{1, 4, -3, -9}, {-2, -1, 5, 3, -6, 10}, {-7, 2, -4, -5, 9, 7, -8, 6, -10, 8}

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{2 t(1) t(3)^2 t(2)^2-t(1) t(3) t(2)^2-t(1) t(3)^2 t(2)+t(1) t(3) t(2)-t(3) t(2)+t(2)+t(3)-2}{\sqrt{t(1)} t(2) t(3)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-3} - q^{-4} +3 q^{-5} -2 q^{-6} +4 q^{-7} -3 q^{-8} +3 q^{-9} -2 q^{-10} + q^{-11} }[/math] (db)
Signature	-6 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{12}-a^{10} z^4-4 a^{10} z^2+a^{10} z^{-2} -2 a^{10}+a^8 z^6+4 a^8 z^4+2 a^8 z^2-2 a^8 z^{-2} -3 a^8+a^6 z^6+5 a^6 z^4+7 a^6 z^2+a^6 z^{-2} +4 a^6 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{14} z^2-a^{14}+2 a^{13} z^3-2 a^{13} z+a^{12} z^6-3 a^{12} z^4+5 a^{12} z^2-a^{12}+2 a^{11} z^7-9 a^{11} z^5+14 a^{11} z^3-6 a^{11} z+a^{10} z^8-3 a^{10} z^6+a^{10} z^4+3 a^{10} z^2+a^{10} z^{-2} -3 a^{10}+3 a^9 z^7-12 a^9 z^5+11 a^9 z^3-2 a^9 z^{-1} +a^8 z^8-3 a^8 z^6-a^8 z^4+6 a^8 z^2+2 a^8 z^{-2} -6 a^8+a^7 z^7-3 a^7 z^5-a^7 z^3+4 a^7 z-2 a^7 z^{-1} +a^6 z^6-5 a^6 z^4+7 a^6 z^2+a^6 z^{-2} -4 a^6 }[/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 -5                 1 1 -7               1 1 0 -9             2     2 -11         1 1 1     1 -13         4 2       2 -15     1 1 1         1 -17     3 3           0 -19   1 2             1 -21   1               -1 -23 1                 1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-7 }[/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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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} }[/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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