L10a141

L10a140

L10a142

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Link Presentations

[edit Notes on L10a141's Link Presentations]

Planar diagram presentation	X6172 X14,6,15,5 X10,4,11,3 X2,16,3,15 X18,7,19,8 X16,9,17,10 X8,17,9,18 X20,12,13,11 X12,14,5,13 X4,19,1,20
Gauss code	{1, -4, 3, -10}, {2, -1, 5, -7, 6, -3, 8, -9}, {9, -2, 4, -6, 7, -5, 10, -8}

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