L10n65

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L10n66

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Link Presentations

[edit Notes on L10n65's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X11,16,12,17 X13,19,14,18 X17,20,18,9 X19,13,20,12 X15,8,16,5 X7,14,8,15 X2536 X9,1,10,4
Gauss code	{1, -9, -2, 10}, {9, -1, -8, 7}, {-10, 2, -3, 6, -4, 8, -7, 3, -5, 4, -6, 5}

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 w^3-3 u v w^2+3 u v w-u v-2 u w+u-v w^3+2 v w^2+w^3-3 w^2+3 w-1}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 3 q^{-5} -5 q^{-4} +q^3+7 q^{-3} -2 q^2-7 q^{-2} +5 q+8 q^{-1} -6 }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^6 z^{-2} +a^6-z^4 a^4-3 z^2 a^4-3 a^4 z^{-2} -5 a^4+z^6 a^2+4 z^4 a^2+7 z^2 a^2+4 a^2 z^{-2} +8 a^2-2 z^4-6 z^2-3 z^{-2} -6+z^2 a^{-2} + a^{-2} z^{-2} +2 a^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^2 z^8+z^8+5 a^3 z^7+7 a z^7+2 z^7 a^{-1} +7 a^4 z^6+10 a^2 z^6+z^6 a^{-2} +4 z^6+3 a^5 z^5-9 a^3 z^5-18 a z^5-6 z^5 a^{-1} -18 a^4 z^4-37 a^2 z^4-4 z^4 a^{-2} -23 z^4+4 a^3 z^3+8 a z^3+4 z^3 a^{-1} +6 a^6 z^2+24 a^4 z^2+40 a^2 z^2+6 z^2 a^{-2} +28 z^2+a^5 z+a^3 z+a z+z a^{-1} -4 a^6-14 a^4-21 a^2-4 a^{-2} -14-a^5 z^{-1} -a^3 z^{-1} -a z^{-1} - a^{-1} z^{-1} +a^6 z^{-2} +3 a^4 z^{-2} +4 a^2 z^{-2} + a^{-2} z^{-2} +3 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   \ -4 -3 -2 -1 0 1 2 3 4 χ 7                 1 1 5               1   -1 3             4 1   3 1           2 1     -1 -1         6 4       2 -3       4 5         1 -5     3 3           0 -7   2 4             2 -9 1 3               -2 -11 3                 3

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=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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