L11n231

L11n230

L11n232

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Link Presentations

[edit Notes on L11n231's Link Presentations]

Planar diagram presentation	X10,1,11,2 X16,11,17,12 X5,21,6,20 X12,4,13,3 X14,8,15,7 X6,14,7,13 X17,9,18,22 X21,19,22,18 X8,9,1,10 X19,5,20,4 X2,16,3,15
Gauss code	{1, -11, 4, 10, -3, -6, 5, -9}, {9, -1, 2, -4, 6, -5, 11, -2, -7, 8, -10, 3, -8, 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{u^3 v^3-2 u^3 v^2+u^3 v-u^2 v^3+5 u^2 v^2-3 u^2 v-3 u v^2+5 u v-u+v^2-2 v+1}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{15/2}+3 q^{13/2}-6 q^{11/2}+8 q^{9/2}-9 q^{7/2}+8 q^{5/2}-8 q^{3/2}+5 \sqrt{q}-\frac{3}{\sqrt{q}}+\frac{1}{q^{3/2}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^7 a^{-3} -z^5 a^{-1} +5 z^5 a^{-3} -z^5 a^{-5} -3 z^3 a^{-1} +9 z^3 a^{-3} -3 z^3 a^{-5} -2 z a^{-1} +7 z a^{-3} -3 z a^{-5} + a^{-3} z^{-1} - a^{-5} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^9 a^{-3} -z^9 a^{-5} -3 z^8 a^{-2} -5 z^8 a^{-4} -2 z^8 a^{-6} -3 z^7 a^{-1} -4 z^7 a^{-3} -2 z^7 a^{-5} -z^7 a^{-7} +7 z^6 a^{-2} +12 z^6 a^{-4} +4 z^6 a^{-6} -z^6+10 z^5 a^{-1} +20 z^5 a^{-3} +10 z^5 a^{-5} -6 z^4 a^{-4} -6 z^4 a^{-6} -3 z^4 a^{-8} +3 z^4-9 z^3 a^{-1} -20 z^3 a^{-3} -13 z^3 a^{-5} -3 z^3 a^{-7} -z^3 a^{-9} -2 z^2 a^{-2} -z^2 a^{-4} +z^2 a^{-6} +2 z^2 a^{-8} -2 z^2+3 z a^{-1} +9 z a^{-3} +6 z a^{-5} +z a^{-7} +z a^{-9} + a^{-4} - a^{-3} z^{-1} - a^{-5} z^{-1} }[/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   \ -3 -2 -1 0 1 2 3 4 5 6 χ 16                   1 1 14                 2   -2 12               4 1   3 10             5 3     -2 8           4 3       1 6         4 5         1 4       4 4           0 2     2 5             3 0   1 3               -2 -2   2                 2 -4 1                   -1

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