L11n328

L11n327

L11n329

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Link Presentations

[edit Notes on L11n328's Link Presentations]

Planar diagram presentation	X6172 X5,14,6,15 X8493 X2,16,3,15 X16,7,17,8 X13,18,14,19 X9,21,10,20 X19,5,20,12 X11,13,12,22 X21,11,22,10 X4,17,1,18
Gauss code	{1, -4, 3, -11}, {-2, -1, 5, -3, -7, 10, -9, 8}, {-6, 2, 4, -5, 11, 6, -8, 7, -10, 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-1) (v-1) (w-1) (v w+1)}{\sqrt{u} v w} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^5-2 q^4- q^{-4} +4 q^3+2 q^{-3} -4 q^2-3 q^{-2} +6 q+5 q^{-1} -4 }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^2 a^{-4} + a^{-4} z^{-2} +2 a^{-4} -a^2 z^4-2 z^4 a^{-2} -3 a^2 z^2-7 z^2 a^{-2} -2 a^{-2} z^{-2} -2 a^2-7 a^{-2} +z^6+5 z^4+9 z^2+ z^{-2} +7 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a z^9+z^9 a^{-1} +2 a^2 z^8+2 z^8 a^{-2} +4 z^8+a^3 z^7-2 a z^7-z^7 a^{-1} +2 z^7 a^{-3} -10 a^2 z^6-7 z^6 a^{-2} +z^6 a^{-4} -18 z^6-5 a^3 z^5-6 a z^5-8 z^5 a^{-1} -7 z^5 a^{-3} +15 a^2 z^4+9 z^4 a^{-2} -2 z^4 a^{-4} +26 z^4+7 a^3 z^3+13 a z^3+14 z^3 a^{-1} +10 z^3 a^{-3} +2 z^3 a^{-5} -9 a^2 z^2-10 z^2 a^{-2} +3 z^2 a^{-4} +z^2 a^{-6} -21 z^2-2 a^3 z-7 a z-10 z a^{-1} -6 z a^{-3} -z a^{-5} +3 a^2+7 a^{-2} + a^{-4} - a^{-6} +9+2 a^{-1} z^{-1} +2 a^{-3} z^{-1} -2 a^{-2} z^{-2} - a^{-4} z^{-2} - z^{-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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 11                   1 1 9                 1   -1 7               3 1   2 5             3 3     0 3           3 2 1     2 1         3 5         2 -1       2 2 1         1 -3     1 3             2 -5   1 2               -1 -7   1                 1 -9 1                   -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ i=3 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/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}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=4 }[/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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