L10n69

L10n68

L10n70

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Link Presentations

[edit Notes on L10n69's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X2,14,3,13 X14,7,15,8 X9,18,10,19 X11,16,12,17 X17,20,18,11 X15,1,16,4 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 u v w^2-2 u v w-u w^2+2 u w-2 v^2 w+v^2+2 v w-2 v}{\sqrt{u} v w} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-9} -2 q^{-8} +3 q^{-7} -4 q^{-6} +5 q^{-5} -4 q^{-4} +5 q^{-3} -2 q^{-2} +2 q^{-1} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^2 a^8+a^8-z^4 a^6-2 z^2 a^6+a^6 z^{-2} -a^6-z^4 a^4-2 z^2 a^4-2 a^4 z^{-2} -3 a^4+2 z^2 a^2+a^2 z^{-2} +3 a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{10} z^6-4 a^{10} z^4+4 a^{10} z^2-a^{10}+2 a^9 z^7-8 a^9 z^5+8 a^9 z^3-2 a^9 z+a^8 z^8-a^8 z^6-6 a^8 z^4+6 a^8 z^2-a^8+4 a^7 z^7-14 a^7 z^5+14 a^7 z^3-6 a^7 z+a^6 z^8-7 a^6 z^4+8 a^6 z^2+a^6 z^{-2} -3 a^6+2 a^5 z^7-5 a^5 z^5+5 a^5 z^3-2 a^5 z^{-1} +2 a^4 z^6-5 a^4 z^4+9 a^4 z^2+2 a^4 z^{-2} -6 a^4+a^3 z^5-a^3 z^3+4 a^3 z-2 a^3 z^{-1} +3 a^2 z^2+a^2 z^{-2} -4 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               2 2 0 -5             3     3 -7           2 3     1 -9         3 2       1 -11       1 2         1 -13     2 3           -1 -15   1 2             1 -17   1               -1 -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{ 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}^{2} }[/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}^{2}\oplus{\mathbb Z}_2^{3} }[/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}^{3} }[/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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