L11n23

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L11n24

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Link Presentations

[edit Notes on L11n23's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X17,1,18,4 X9,21,10,20 X3849 X21,18,22,19 X11,14,12,15 X5,13,6,12 X13,5,14,22 X19,11,20,10 X2,16,3,15
Gauss code	{1, -11, -5, 3}, {-8, -1, 2, 5, -4, 10, -7, 8, -9, 7, 11, -2, -3, 6, -10, 4, -6, 9}

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) \left(v^2-4 v+1\right)}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{9/2}+\frac{1}{q^{9/2}}+3 q^{7/2}-\frac{3}{q^{7/2}}-6 q^{5/2}+\frac{4}{q^{5/2}}+7 q^{3/2}-\frac{7}{q^{3/2}}-8 \sqrt{q}+\frac{8}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ - a^{-5} z^{-1} -a^3 z^3-a^3 z+4 z a^{-3} +3 a^{-3} z^{-1} +a z^5+3 a z^3-4 z^3 a^{-1} +5 a z-8 z a^{-1} +2 a z^{-1} -4 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a z^9-z^9 a^{-1} -3 a^2 z^8-3 z^8 a^{-2} -6 z^8-3 a^3 z^7-4 a z^7-3 z^7 a^{-1} -2 z^7 a^{-3} -a^4 z^6+9 a^2 z^6+12 z^6 a^{-2} +22 z^6+11 a^3 z^5+26 a z^5+22 z^5 a^{-1} +7 z^5 a^{-3} +3 a^4 z^4-5 a^2 z^4-23 z^4 a^{-2} -3 z^4 a^{-4} -28 z^4-10 a^3 z^3-34 a z^3-40 z^3 a^{-1} -17 z^3 a^{-3} -z^3 a^{-5} -a^4 z^2+2 a^2 z^2+15 z^2 a^{-2} +3 z^2 a^{-4} +15 z^2+3 a^3 z+15 a z+24 z a^{-1} +14 z a^{-3} +2 z a^{-5} -a^2-3 a^{-2} - a^{-4} -2-2 a z^{-1} -4 a^{-1} z^{-1} -3 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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 10                   1 1 8                 2   -2 6               4 1   3 4             3 2     -1 2           5 4       1 0         5 5         0 -2       2 3           -1 -4     2 5             3 -6   1 2               -1 -8   2                 2 -10 1                   -1

Integral Khovanov Homology (db, data source)

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