L11n292

L11n291

L11n293

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Link Presentations

[edit Notes on L11n292's Link Presentations]

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

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(3)^4+t(1) t(2) t(3)^4-t(2)^2 t(3)^3+2 t(1) t(3)^3-2 t(1) t(2) t(3)^3+t(2) t(3)^3+2 t(2)^2 t(3)^2-2 t(1) t(3)^2+t(1) t(2) t(3)^2-t(2) t(3)^2-2 t(2)^2 t(3)+t(1) t(3)-t(1) t(2) t(3)+2 t(2) t(3)+t(2)^2-t(2)}{\sqrt{t(1)} t(2) t(3)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^5-3 q^4- q^{-4} +5 q^3+3 q^{-3} -6 q^2-4 q^{-2} +8 q+7 q^{-1} -6 }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^2 a^{-4} + a^{-4} -a^2 z^4-2 z^4 a^{-2} -2 a^2 z^2-5 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -2 a^{-2} +z^6+4 z^4+5 z^2-2 z^{-2} +1 }[/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} +4 z^7 a^{-3} -14 a^2 z^6-21 z^6 a^{-2} +z^6 a^{-4} -36 z^6-4 a^3 z^5-6 a z^5-16 z^5 a^{-1} -14 z^5 a^{-3} +19 a^2 z^4+30 z^4 a^{-2} +z^4 a^{-4} +48 z^4+4 a^3 z^3+14 a z^3+24 z^3 a^{-1} +17 z^3 a^{-3} +3 z^3 a^{-5} -9 a^2 z^2-20 z^2 a^{-2} -3 z^2 a^{-4} +z^2 a^{-6} -25 z^2-a^3 z-3 a z-7 z a^{-1} -7 z a^{-3} -2 z a^{-5} +4 a^{-2} +2 a^{-4} +3-2 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 χ 11                   1 1 9                 2   -2 7               3 1   2 5             4 3     -1 3           4 3 1     2 1         4 6         2 -1       3 3 1         1 -3     2 5             3 -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=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}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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^{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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