L11n146

L11n145

L11n147

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Link Presentations

[edit Notes on L11n146's Link Presentations]

Planar diagram presentation	X8192 X9,19,10,18 X6718 X19,7,20,22 X5,13,6,12 X3,10,4,11 X15,5,16,4 X11,16,12,17 X13,21,14,20 X21,15,22,14 X17,2,18,3
Gauss code	{1, 11, -6, 7, -5, -3}, {3, -1, -2, 6, -8, 5, -9, 10, -7, 8, -11, 2, -4, 9, -10, 4}

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{(t(1)+t(2)-1) (t(2) t(1)-t(1)-t(2)) \left(t(2)^2-t(2)+1\right)}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{9/2}+\frac{1}{q^{9/2}}+4 q^{7/2}-\frac{3}{q^{7/2}}-6 q^{5/2}+\frac{5}{q^{5/2}}+8 q^{3/2}-\frac{8}{q^{3/2}}-10 \sqrt{q}+\frac{8}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^3 z^3-z^3 a^{-3} -a^3 z+ a^{-3} z^{-1} +a z^5+z^5 a^{-1} +2 a z^3+z^3 a^{-1} +2 a z-2 z a^{-1} +2 a z^{-1} -3 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^3 a^{-5} +a^4 z^6-3 a^4 z^4+4 z^4 a^{-4} +2 a^4 z^2-2 z^2 a^{-4} - a^{-4} +3 a^3 z^7+z^7 a^{-3} -10 a^3 z^5+2 z^5 a^{-3} +9 a^3 z^3-z^3 a^{-3} -3 a^3 z+ a^{-3} z^{-1} +3 a^2 z^8+2 z^8 a^{-2} -7 a^2 z^6-3 z^6 a^{-2} +5 z^4 a^{-2} +3 a^2 z^2-3 a^{-2} +a z^9+z^9 a^{-1} +4 a z^7+2 z^7 a^{-1} -19 a z^5-7 z^5 a^{-1} +18 a z^3+7 z^3 a^{-1} -8 a z-5 z a^{-1} +2 a z^{-1} +3 a^{-1} z^{-1} +5 z^8-11 z^6+4 z^4+3 z^2-3 }[/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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 10                   1 1 8                 3   -3 6               3 1   2 4             5 3     -2 2           5 3       2 0         4 6         2 -2       4 4           0 -4     2 5             3 -6   1 3               -2 -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}^{3}\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^{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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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