L11n320

L11n319

L11n321

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Link Presentations

[edit Notes on L11n320's Link Presentations]

Planar diagram presentation	X6172 X5,14,6,15 X3849 X15,2,16,3 X16,7,17,8 X9,18,10,19 X4,17,1,18 X19,13,20,22 X13,10,14,11 X21,5,22,12 X11,21,12,20
Gauss code	{1, 4, -3, -7}, {-2, -1, 5, 3, -6, 9, -11, 10}, {-9, 2, -4, -5, 7, 6, -8, 11, -10, 8}

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)^3 t(2)^3-t(1) t(3)^2 t(2)^3-t(1) t(3)^3 t(2)^2+2 t(1) t(3)^2 t(2)^2-2 t(3) t(2)+t(2)+t(3)-1}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q+2-2 q^{-1} +3 q^{-2} -2 q^{-3} +4 q^{-4} -2 q^{-5} +2 q^{-6} - q^{-7} + q^{-8} }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^8 z^2+a^8 z^{-2} +2 a^8-a^6 z^6-6 a^6 z^4-11 a^6 z^2-2 a^6 z^{-2} -9 a^6+a^4 z^8+7 a^4 z^6+17 a^4 z^4+19 a^4 z^2+a^4 z^{-2} +9 a^4-a^2 z^6-5 a^2 z^4-6 a^2 z^2-2 a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^2 a^{10}-a^{10}+z^3 a^9-z a^9+z^4 a^8-2 z^2 a^8-a^8 z^{-2} +3 a^8+z^7 a^7-5 z^5 a^7+9 z^3 a^7-8 z a^7+2 a^7 z^{-1} +2 z^8 a^6-12 z^6 a^6+25 z^4 a^6-25 z^2 a^6-2 a^6 z^{-2} +11 a^6+z^9 a^5-3 z^7 a^5-5 z^5 a^5+17 z^3 a^5-12 z a^5+2 a^5 z^{-1} +4 z^8 a^4-23 z^6 a^4+41 z^4 a^4-32 z^2 a^4-a^4 z^{-2} +11 a^4+z^9 a^3-3 z^7 a^3-5 z^5 a^3+15 z^3 a^3-7 z a^3+2 z^8 a^2-11 z^6 a^2+17 z^4 a^2-10 z^2 a^2+3 a^2+z^7 a-5 z^5 a+6 z^3 a-2 z a }[/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 χ 3                   1 -1 1                 1   1 -1               1 1   0 -3           1 3 1     1 -5         1 2 2       1 -7         4 2         2 -9     2 2 2           2 -11   1 4 3             0 -13   1 2               1 -15 1 1                 0 -17 1                   1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/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^{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}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/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 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=3 }[/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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