L10n75

L10n74

L10n76

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Link Presentations

[edit Notes on L10n75's Link Presentations]

Planar diagram presentation	X6172 X5,14,6,15 X3849 X2,16,3,15 X16,7,17,8 X9,18,10,19 X17,1,18,4 X19,12,20,5 X11,20,12,13 X13,10,14,11
Gauss code	{1, -4, -3, 7}, {-2, -1, 5, 3, -6, 10, -9, 8}, {-10, 2, 4, -5, -7, 6, -8, 9}

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 v^2 w^2+u v^2 w-u v w^3+2 u v w^2-u v w-u w^2+v^3 w+v^2 w^2-2 v^2 w+v^2-v w^2+v w}{\sqrt{u} v^{3/2} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 2 q^{-1} -3 q^{-2} +5 q^{-3} -4 q^{-4} +5 q^{-5} -4 q^{-6} +3 q^{-7} - q^{-8} + q^{-9} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^2 a^8+a^8 z^{-2} +2 a^8-z^4 a^6-3 z^2 a^6-2 a^6 z^{-2} -5 a^6-z^4 a^4-z^2 a^4+a^4 z^{-2} +a^4+2 z^2 a^2+2 a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{10}-5 z^4 a^{10}+7 z^2 a^{10}-2 a^{10}+z^7 a^9-3 z^5 a^9+z^3 a^9+z a^9+z^8 a^8-3 z^6 a^8+3 z^4 a^8-4 z^2 a^8-a^8 z^{-2} +3 a^8+3 z^7 a^7-9 z^5 a^7+9 z^3 a^7-8 z a^7+2 a^7 z^{-1} +z^8 a^6-2 z^6 a^6+6 z^4 a^6-14 z^2 a^6-2 a^6 z^{-2} +9 a^6+2 z^7 a^5-5 z^5 a^5+9 z^3 a^5-8 z a^5+2 a^5 z^{-1} +2 z^6 a^4-2 z^4 a^4-a^4 z^{-2} +3 a^4+z^5 a^3+z^3 a^3+z a^3+3 z^2 a^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 χ -1                 2 2 -3               3 2 -1 -5             2     2 -7           2 3     1 -9         3 2       1 -11       2 3         1 -13     1 2           -1 -15     2             2 -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=-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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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