L10n99

L10n98

L10n100

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Link Presentations

[edit Notes on L10n99's Link Presentations]

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

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{(x-1) \left(u v w x-u v w-2 u v x-u w x+u x-v w x+v x+2 w x+x^2-x\right)}{\sqrt{u} \sqrt{v} \sqrt{w} x^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{7/2}+2 q^{5/2}-6 q^{3/2}+7 \sqrt{q}-\frac{9}{\sqrt{q}}+\frac{7}{q^{3/2}}-\frac{9}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{3}{q^{9/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z^{-3} +2 a^5 z^{-1} +a^3 z^3-3 a^3 z^{-3} -3 a^3 z-6 a^3 z^{-1} -z a^{-3} - a^{-3} z^{-1} -a z^5-a z^3+3 a z^{-3} +2 z^3 a^{-1} - a^{-1} z^{-3} +2 a z+5 a z^{-1} +2 z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 6 a^5 z^3-a^5 z^{-3} -10 a^5 z+6 a^5 z^{-1} +3 a^4 z^6-6 a^4 z^4+15 a^4 z^2+3 a^4 z^{-2} -13 a^4+4 a^3 z^7-9 a^3 z^5+z^5 a^{-3} +19 a^3 z^3-3 z^3 a^{-3} -3 a^3 z^{-3} -23 a^3 z+3 z a^{-3} +14 a^3 z^{-1} - a^{-3} z^{-1} +a^2 z^8+8 a^2 z^6+2 z^6 a^{-2} -24 a^2 z^4-3 z^4 a^{-2} +33 a^2 z^2+6 a^2 z^{-2} -24 a^2+ a^{-2} +7 a z^7+3 z^7 a^{-1} -15 a z^5-5 z^5 a^{-1} +19 a z^3+3 z^3 a^{-1} -3 a z^{-3} - a^{-1} z^{-3} -20 a z-4 z a^{-1} +12 a z^{-1} +3 a^{-1} z^{-1} +z^8+7 z^6-21 z^4+18 z^2+3 z^{-2} -11 }[/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 8                 1 1 6               2 1 -1 4             4     4 2           3 2     -1 0         6 4       2 -2       5 7         2 -4     4 2           2 -6     5             5 -8 3 4               -1 -10 3                 3

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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