L10n44

L10n43

L10n45

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Link Presentations

[edit Notes on L10n44's Link Presentations]

Planar diagram presentation	X8192 X9,19,10,18 X6718 X13,20,14,7 X5,13,6,12 X3,10,4,11 X15,5,16,4 X11,16,12,17 X19,14,20,15 X17,2,18,3
Gauss code	{1, 10, -6, 7, -5, -3}, {3, -1, -2, 6, -8, 5, -4, 9, -7, 8, -10, 2, -9, 4}

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-1) (u v-u-v)}{u v} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{3}{q^{9/2}}+\frac{3}{q^{7/2}}-\frac{3}{q^{5/2}}+\frac{2}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{2}{q^{13/2}}+\frac{2}{q^{11/2}}-\frac{2}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 (-z)+a^5 z^3+a^5 z+a^3 z^3+a^3 z+a^3 z^{-1} -2 a z-a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^8 z^6-4 a^8 z^4+3 a^8 z^2+2 a^7 z^7-9 a^7 z^5+10 a^7 z^3-3 a^7 z+a^6 z^8-3 a^6 z^6+a^6 z^2+3 a^5 z^7-13 a^5 z^5+15 a^5 z^3-5 a^5 z+a^4 z^8-4 a^4 z^6+5 a^4 z^4-3 a^4 z^2+a^3 z^7-4 a^3 z^5+5 a^3 z^3-a^3 z^{-1} +a^2 z^4-a^2 z^2+a^2+2 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   \ -7 -6 -5 -4 -3 -2 -1 0 χ 0               2 2 -2             1 1 0 -4           2 1   1 -6         2 2     0 -8       1 1       0 -10     1 2         1 -12   1 1           0 -14   1             1 -16 1               -1

Integral Khovanov Homology (db, data source)

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