L10n14

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Link Presentations

[edit Notes on L10n14's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X9,14,10,15 X8493 X5,13,6,12 X13,5,14,20 X11,16,12,17 X15,10,16,11 X2,18,3,17
Gauss code	{1, -10, 5, -3}, {-6, -1, 2, -5, -4, 9, -8, 6, -7, 4, -9, 8, 10, -2, 3, 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)}{\sqrt{t(1)} \sqrt{t(2)}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{5/2}+q^{3/2}-\sqrt{q}-\frac{1}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{2}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{2}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^5+2 z a^5+a^5 z^{-1} -z^5 a^3-5 z^3 a^3-8 z a^3-3 a^3 z^{-1} +z^5 a+6 z^3 a+9 z a+4 a z^{-1} -z^3 a^{-1} -3 z a^{-1} -2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^5-3 a^7 z^3+a^7 z+2 a^6 z^6-7 a^6 z^4+4 a^6 z^2-a^6+a^5 z^7-3 a^5 z^5+a^5 z^3-2 a^5 z+a^5 z^{-1} +2 a^4 z^6-9 a^4 z^4+10 a^4 z^2-3 a^4+2 a^3 z^7-13 a^3 z^5+24 a^3 z^3-15 a^3 z+3 a^3 z^{-1} +a^2 z^8-6 a^2 z^6+6 a^2 z^4+5 a^2 z^2-3 a^2+2 a z^7+z^7 a^{-1} -15 a z^5-6 z^5 a^{-1} +30 a z^3+10 z^3 a^{-1} -19 a z-7 z a^{-1} +4 a z^{-1} +2 a^{-1} z^{-1} +z^8-6 z^6+8 z^4-z^2-2 }[/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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                     1 1 4                       0 2                 1 1   0 0             3 1       2 -2           1 2 1       0 -4         2 2 1         1 -6       1 1             0 -8     1 2 1             0 -10   1 1                 0 -12   1                   -1 -14 1                     1

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{ i=0 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/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=-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} }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_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}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=3 }[/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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