L10n91

L10n90

L10n92

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Link Presentations

[edit Notes on L10n91's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X13,20,14,17 X7,18,8,19 X17,10,18,11 X9,15,10,14 X15,9,16,8 X19,16,20,5 X2536 X4,11,1,12
Gauss code	{1, -9, 2, -10}, {-5, 4, -8, 3}, {9, -1, -4, 7, -6, 5, 10, -2, -3, 6, -7, 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{(t(3)-1) (t(3)+1) (-2 t(1) t(2)+t(1) t(3) t(2)-2 t(3)+1)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 1- q^{-1} + q^{-2} - q^{-3} + q^{-5} + q^{-6} +2 q^{-7} - q^{-8} + q^{-9} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{10} z^{-2} -2 a^8 z^{-2} -4 a^8+z^4 a^6+6 z^2 a^6+a^6 z^{-2} +6 a^6-z^6 a^4-5 z^4 a^4-6 z^2 a^4-4 a^4+z^4 a^2+4 z^2 a^2+2 a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{10} z^6-5 a^{10} z^4+6 a^{10} z^2+a^{10} z^{-2} -4 a^{10}+a^9 z^7-4 a^9 z^5+4 a^9 z-2 a^9 z^{-1} +a^8 z^8-5 a^8 z^6+3 a^8 z^4+6 a^8 z^2+2 a^8 z^{-2} -5 a^8+2 a^7 z^7-12 a^7 z^5+16 a^7 z^3-4 a^7 z-2 a^7 z^{-1} +a^6 z^8-5 a^6 z^6+2 a^6 z^4+6 a^6 z^2+a^6 z^{-2} -2 a^6+2 a^5 z^7-13 a^5 z^5+22 a^5 z^3-12 a^5 z+2 a^4 z^6-11 a^4 z^4+12 a^4 z^2-2 a^4+a^3 z^7-5 a^3 z^5+6 a^3 z^3-4 a^3 z+a^2 z^6-5 a^2 z^4+6 a^2 z^2-2 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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                     1 1 -1                       0 -3               1 2 1   0 -5               1 1     0 -7           2 2 1       -1 -9         1 1 1         1 -11       1 4 1           2 -13     1 1 3             3 -15     1                 1 -17 1 1                   0 -19 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=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/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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