L11n31

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L11n32

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Link Presentations

[edit Notes on L11n31's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X5,12,6,13 X8493 X9,16,10,17 X13,22,14,5 X15,10,16,11 X21,14,22,15 X11,20,12,21 X2,18,3,17
Gauss code	{1, -11, 5, -3}, {-4, -1, 2, -5, -6, 8, -10, 4, -7, 9, -8, 6, 11, -2, 3, 10, -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(1)-1) (t(2)-1) \left(2 t(2)^2-3 t(2)+2\right)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{7}{q^{5/2}}-\frac{10}{q^{7/2}}+\frac{9}{q^{9/2}}-\frac{9}{q^{11/2}}+\frac{6}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{2}{q^{17/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^9 z^{-1} -z^3 a^7+a^7 z^{-1} +z^5 a^5+2 z^3 a^5+3 z a^5+2 a^5 z^{-1} +z^5 a^3+z^3 a^3-2 z a^3-2 a^3 z^{-1} -z^3 a-z a }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 3 a^{10} z^4-8 a^{10} z^2+4 a^{10}+a^9 z^7-a^9 z^5+a^9 z^3-2 a^9 z-a^9 z^{-1} +2 a^8 z^8-7 a^8 z^6+18 a^8 z^4-21 a^8 z^2+9 a^8+a^7 z^9+a^7 z^7-8 a^7 z^5+17 a^7 z^3-7 a^7 z-a^7 z^{-1} +5 a^6 z^8-14 a^6 z^6+20 a^6 z^4-11 a^6 z^2+4 a^6+a^5 z^9+4 a^5 z^7-16 a^5 z^5+22 a^5 z^3-11 a^5 z+2 a^5 z^{-1} +3 a^4 z^8-4 a^4 z^6-a^4 z^4+3 a^4 z^2-2 a^4+4 a^3 z^7-8 a^3 z^5+4 a^3 z^3-5 a^3 z+2 a^3 z^{-1} +3 a^2 z^6-6 a^2 z^4+a^2 z^2+a z^5-2 a z^3+a z }[/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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                   1 1 0                 2   -2 -2               4 1   3 -4             5 4     -1 -6           5 2       3 -8         4 5         1 -10       5 5           0 -12     1 4             3 -14   2 5               -3 -16   1                 1 -18 2                   -2

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/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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