L10n84

L10n83

L10n85

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Link L10n84.

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Link Presentations

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Planar diagram presentation	X6172 X10,3,11,4 X13,20,14,15 X7,16,8,17 X15,8,16,9 X11,18,12,19 X19,12,20,13 X17,14,18,5 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {-5, 4, -8, 6, -7, 3}, {9, -1, -4, 5, 10, -2, -6, 7, -3, 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{(w-1) \left(u v^2 w^2+u v^2 w+u v+v w^3+w^2+w\right)}{\sqrt{u} v w^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-12} - q^{-11} +2 q^{-10} - q^{-9} +2 q^{-8} - q^{-7} + q^{-6} + q^{-3} }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{12} z^{-2} +a^{12}-2 z^2 a^{10}-2 a^{10} z^{-2} -4 a^{10}-z^2 a^8+a^8 z^{-2} +z^6 a^6+6 z^4 a^6+9 z^2 a^6+3 a^6 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{14} z^6-5 a^{14} z^4+6 a^{14} z^2-2 a^{14}+a^{13} z^7-4 a^{13} z^5+2 a^{13} z^3+a^{13} z+a^{12} z^8-5 a^{12} z^6+7 a^{12} z^4-6 a^{12} z^2-a^{12} z^{-2} +4 a^{12}+2 a^{11} z^7-11 a^{11} z^5+17 a^{11} z^3-10 a^{11} z+2 a^{11} z^{-1} +a^{10} z^8-7 a^{10} z^6+17 a^{10} z^4-20 a^{10} z^2-2 a^{10} z^{-2} +10 a^{10}+a^9 z^7-7 a^9 z^5+14 a^9 z^3-10 a^9 z+2 a^9 z^{-1} -a^8 z^4+a^8 z^2-a^8 z^{-2} +2 a^8-a^7 z^3+a^7 z+a^6 z^6-6 a^6 z^4+9 a^6 z^2-3 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   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -5                     1 1 -7                     1 1 -9               1 1     0 -11             1         1 -13           2 3 1       0 -15         1             1 -17       1 3 1           1 -19     1 1 1             1 -21     1                 1 -23 1 1                   0 -25 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=-10 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/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} }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\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] [math]\displaystyle{ r=-1 }[/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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