L10n94

L10n93

L10n95

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Link Presentations

[edit Notes on L10n94's Link Presentations]

Planar diagram presentation	X8192 X18,10,19,9 X6,18,1,17 X7,17,8,16 X3,10,4,11 X14,6,15,5 X4,14,5,13 X11,13,12,20 X15,7,16,12 X19,3,20,2
Gauss code	{1, 10, -5, -7, 6, -3}, {-4, -1, 2, 5, -8, 9}, {7, -6, -9, 4, 3, -2, -10, 8}

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{(t(1) t(2) t(3)+1) \left(t(3)^2-t(1) t(2)\right)}{t(1) t(2) t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^9+q^4+q^3+q^2 }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^2 a^{-8} + a^{-8} z^{-2} +3 a^{-8} -z^4 a^{-6} -6 z^2 a^{-6} -2 a^{-6} z^{-2} -9 a^{-6} +z^4 a^{-4} +5 z^2 a^{-4} + a^{-4} z^{-2} +6 a^{-4} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{-10} -6 z^4 a^{-10} +9 z^2 a^{-10} -2 a^{-10} -z^2 a^{-8} - a^{-8} z^{-2} +3 a^{-8} -z^5 a^{-7} +6 z^3 a^{-7} -9 z a^{-7} +2 a^{-7} z^{-1} -z^6 a^{-6} +7 z^4 a^{-6} -15 z^2 a^{-6} -2 a^{-6} z^{-2} +11 a^{-6} -z^5 a^{-5} +6 z^3 a^{-5} -9 z a^{-5} +2 a^{-5} z^{-1} +z^4 a^{-4} -5 z^2 a^{-4} - a^{-4} z^{-2} +7 a^{-4} }[/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 19                 1 1 17                 1 1 15           1 1     0 13                   0 11       1 2 1       0 9     1             1 7     3 1           2 5 1   1             2 3 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=3 }[/math] [math]\displaystyle{ i=5 }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ r=8 }[/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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