L11n350

L11n349

L11n351

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Link Presentations

[edit Notes on L11n350's Link Presentations]

Planar diagram presentation	X6172 X14,4,15,3 X11,21,12,20 X7,18,8,19 X9,16,10,17 X17,8,18,9 X19,13,20,22 X13,10,14,11 X21,5,22,12 X2536 X4,16,1,15
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 6, -5, 8, -3, 9}, {-8, -2, 11, 5, -6, 4, -7, 3, -9, 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{-t(3) t(2)^3+t(2)^3-t(1) t(3)^2 t(2)^2+t(3)^2 t(2)^2+t(1) t(3) t(2)^2-t(3) t(2)^2-t(2)^2+t(1) t(3)^3 t(2)+t(1) t(3)^2 t(2)-t(3)^2 t(2)-t(1) t(3) t(2)+t(3) t(2)-t(1) t(3)^3+t(1) t(3)^2}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^5+2 q^4-2 q^3+2 q^2-q+2+ q^{-1} - q^{-2} +2 q^{-3} - q^{-4} + q^{-5} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^2 a^4+a^4 z^{-2} +2 a^4-z^4 a^2-4 z^2 a^2-2 a^2 z^{-2} -5 a^2+ z^{-2} +2+z^4 a^{-2} +3 z^2 a^{-2} +2 a^{-2} -z^2 a^{-4} - a^{-4} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-5} -3 z^3 a^{-5} +z a^{-5} +a^4 z^8-7 a^4 z^6+2 z^6 a^{-4} +16 a^4 z^4-7 z^4 a^{-4} -15 a^4 z^2+4 z^2 a^{-4} -a^4 z^{-2} +6 a^4- a^{-4} +a^3 z^9-6 a^3 z^7+z^7 a^{-3} +9 a^3 z^5-3 z^5 a^{-3} -a^3 z^3-5 a^3 z+z a^{-3} +2 a^3 z^{-1} +3 a^2 z^8-21 a^2 z^6+z^6 a^{-2} +44 a^2 z^4-4 z^4 a^{-2} -36 a^2 z^2+2 z^2 a^{-2} -2 a^2 z^{-2} +13 a^2+a z^9-6 a z^7+z^7 a^{-1} +6 a z^5-7 z^5 a^{-1} +6 a z^3+10 z^3 a^{-1} -8 a z-3 z a^{-1} +2 a z^{-1} +2 z^8-15 z^6+31 z^4-23 z^2- z^{-2} +9 }[/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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     1   1 7                   1 1   0 5               1 2 1     0 3             1 1 1       1 1           1 4 2         1 -1         1 2 4           3 -3       1 1               0 -5     1 1 1               1 -7   1 2                   1 -9                         0 -11 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=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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