L10n41

L10n40

L10n42

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Link Presentations

[edit Notes on L10n41's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X20,10,7,9 X2738 X4,15,5,16 X5,13,6,12 X11,16,12,17 X17,6,18,1 X14,19,15,20 X18,13,19,14
Gauss code	{1, -4, 2, -5, -6, 8}, {4, -1, 3, -2, -7, 6, 10, -9, 5, 7, -8, -10, 9, -3}

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^2 v^2-2 u^2 v+u^2-u v^2+3 u v-u+v^2-2 v+1}{u v} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{4}{q^{9/2}}+\frac{4}{q^{7/2}}-\frac{5}{q^{5/2}}+\frac{3}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{2}{q^{13/2}}+\frac{3}{q^{11/2}}+\sqrt{q}-\frac{3}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^7-a^7 z^{-1} +2 z^3 a^5+5 z a^5+3 a^5 z^{-1} -z^5 a^3-4 z^3 a^3-6 z a^3-2 a^3 z^{-1} +z^3 a+z a }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^8 z^6-4 a^8 z^4+4 a^8 z^2-a^8+2 a^7 z^7-8 a^7 z^5+8 a^7 z^3-3 a^7 z+a^7 z^{-1} +a^6 z^8-a^6 z^6-7 a^6 z^4+9 a^6 z^2-3 a^6+4 a^5 z^7-15 a^5 z^5+17 a^5 z^3-10 a^5 z+3 a^5 z^{-1} +a^4 z^8-a^4 z^6-4 a^4 z^4+6 a^4 z^2-3 a^4+2 a^3 z^7-7 a^3 z^5+12 a^3 z^3-9 a^3 z+2 a^3 z^{-1} +a^2 z^6-a^2 z^4+2 a^2 z^2+3 a z^3-2 a z+z^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   \ -7 -6 -5 -4 -3 -2 -1 0 1 χ 2                 1 -1 0               2   2 -2             2 2   0 -4           3 1     2 -6         2 3       1 -8       2 2         0 -10     1 2           1 -12   1 2             -1 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/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}_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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