L11n326

L11n325

L11n327

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Link Presentations

[edit Notes on L11n326's Link Presentations]

Planar diagram presentation	X6172 X5,14,6,15 X3849 X15,2,16,3 X16,7,17,8 X13,18,14,19 X9,21,10,20 X19,5,20,12 X11,13,12,22 X21,11,22,10 X4,17,1,18
Gauss code	{1, 4, -3, -11}, {-2, -1, 5, 3, -7, 10, -9, 8}, {-6, 2, -4, -5, 11, 6, -8, 7, -10, 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 v^3 w^3+u v^3 w^2+u v^2 w^3-2 u v^2 w^2+u v^2 w+u v w^2-v^2 w-v w^2+2 v w-v-w+1}{\sqrt{u} v^{3/2} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-6} +2 q^{-5} -2 q^{-4} +q^3+5 q^{-3} -2 q^2-4 q^{-2} +3 q+5 q^{-1} -3 }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^2 z^8+a^4 z^6-7 a^2 z^6+z^6+6 a^4 z^4-17 a^2 z^4+5 z^4-a^6 z^2+11 a^4 z^2-18 a^2 z^2+7 z^2-2 a^6+7 a^4-9 a^2+4+a^4 z^{-2} -2 a^2 z^{-2} + z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^3 z^9+a z^9+2 a^4 z^8+4 a^2 z^8+2 z^8+a^5 z^7-3 a^3 z^7-2 a z^7+2 z^7 a^{-1} -12 a^4 z^6-21 a^2 z^6+z^6 a^{-2} -8 z^6-5 a^5 z^5-3 a^3 z^5-6 a z^5-8 z^5 a^{-1} +2 a^6 z^4+28 a^4 z^4+37 a^2 z^4-4 z^4 a^{-2} +7 z^4+a^7 z^3+11 a^5 z^3+17 a^3 z^3+13 a z^3+6 z^3 a^{-1} -5 a^6 z^2-27 a^4 z^2-28 a^2 z^2+3 z^2 a^{-2} -3 z^2-2 a^7 z-7 a^5 z-12 a^3 z-8 a z-z a^{-1} +3 a^6+11 a^4+11 a^2- a^{-2} +3+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-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 7                   1 1 5                 1   -1 3               2 1   1 1           1 2 1     0 -1         1 5 2       2 -3         3 4         1 -5       3 2           1 -7   1 1 3             3 -9   2 2               0 -11   1                 1 -13 1                   -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/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}\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}_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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