L10n96

L10n95

L10n97

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Link Presentations

[edit Notes on L10n96's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
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^2 x-u v w x^2+2 u v w x-u v w-u v x+u v+u w x^2-u w x+u x+v w^2 x-v w x+v w+w^2 x^2-w^2 x-w x^2+2 w x-w-x}{\sqrt{u} \sqrt{v} w x} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{19/2}+2 q^{17/2}-5 q^{15/2}+5 q^{13/2}-8 q^{11/2}+6 q^{9/2}-7 q^{7/2}+3 q^{5/2}-3 q^{3/2} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -2 z^5 a^{-5} +3 z^3 a^{-3} -9 z^3 a^{-5} +3 z^3 a^{-7} +8 z a^{-3} -16 z a^{-5} +9 z a^{-7} -z a^{-9} +5 a^{-3} z^{-1} -12 a^{-5} z^{-1} +9 a^{-7} z^{-1} -2 a^{-9} z^{-1} + a^{-3} z^{-3} -3 a^{-5} z^{-3} +3 a^{-7} z^{-3} - a^{-9} z^{-3} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^8 a^{-6} -z^8 a^{-8} -4 z^7 a^{-5} -6 z^7 a^{-7} -2 z^7 a^{-9} -3 z^6 a^{-4} -4 z^6 a^{-6} -3 z^6 a^{-8} -2 z^6 a^{-10} +15 z^5 a^{-5} +18 z^5 a^{-7} +2 z^5 a^{-9} -z^5 a^{-11} +9 z^4 a^{-4} +18 z^4 a^{-6} +13 z^4 a^{-8} +4 z^4 a^{-10} -6 z^3 a^{-3} -31 z^3 a^{-5} -26 z^3 a^{-7} +2 z^3 a^{-9} +3 z^3 a^{-11} -17 z^2 a^{-4} -33 z^2 a^{-6} -16 z^2 a^{-8} +13 z a^{-3} +28 z a^{-5} +21 z a^{-7} +3 z a^{-9} -3 z a^{-11} +13 a^{-4} +24 a^{-6} +11 a^{-8} - a^{-10} -6 a^{-3} z^{-1} -14 a^{-5} z^{-1} -12 a^{-7} z^{-1} -3 a^{-9} z^{-1} + a^{-11} z^{-1} -3 a^{-4} z^{-2} -6 a^{-6} z^{-2} -3 a^{-8} z^{-2} + a^{-3} z^{-3} +3 a^{-5} z^{-3} +3 a^{-7} z^{-3} + a^{-9} z^{-3} }[/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   \ 0 1 2 3 4 5 6 7 8 χ 20                 1 1 18               1   -1 16             4 1   3 14           4 4     0 12         4 1       3 10       3 5         2 8     4 3           1 6     4             4 4 3 3               0 2 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=4 }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=8 }[/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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