L10n31

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Link Presentations

[edit Notes on L10n31's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X11,14,12,15 X3,10,4,11 X5,13,6,12 X13,5,14,20 X16,9,17,10 X15,2,16,3 X8,17,9,18
Gauss code	{1, 9, -5, -3}, {-6, -1, 2, -10, 8, 5, -4, 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{(v-2) (2 v-1) (u v+1)}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{7}{q^{9/2}}-\frac{6}{q^{7/2}}+\frac{5}{q^{5/2}}-\frac{5}{q^{3/2}}-\frac{2}{q^{15/2}}+\frac{3}{q^{13/2}}-\frac{5}{q^{11/2}}-\sqrt{q}+\frac{2}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 z^{-1} -z^3 a^7-3 z a^7-2 a^7 z^{-1} +z^5 a^5+3 z^3 a^5+3 z a^5+a^5 z^{-1} +z^5 a^3+2 z^3 a^3+a^3 z^{-1} -z^3 a-2 z a-a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 3 a^9 z^3-5 a^9 z+a^9 z^{-1} +a^8 z^6+a^8+2 a^7 z^7-6 a^7 z^5+15 a^7 z^3-12 a^7 z+2 a^7 z^{-1} +a^6 z^8+3 a^6+4 a^5 z^7-9 a^5 z^5+11 a^5 z^3-8 a^5 z+a^5 z^{-1} +a^4 z^8+a^4 z^6-4 a^4 z^4+2 a^4+2 a^3 z^7-2 a^3 z^5-4 a^3 z^3+2 a^3 z-a^3 z^{-1} +2 a^2 z^6-4 a^2 z^4+a^2+a z^5-3 a z^3+3 a z-a z^{-1} }[/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 χ 2                 1 1 0               1   -1 -2             4 1   3 -4           2 2     0 -6         4 3       1 -8       3 2         -1 -10     2 4           -2 -12   2 4             2 -14   1               -1 -16 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=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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]

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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