L10n17

L10n16

L10n18

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Link Presentations

[edit Notes on L10n17's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X5,12,6,13 X3849 X13,17,14,16 X9,15,10,14 X15,11,16,10 X11,20,12,5 X17,2,18,3
Gauss code	{1, 10, -5, -3}, {-4, -1, 2, 5, -7, 8, -9, 4, -6, 7, -8, 6, -10, -2, 3, 9}

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^5+2 u v^4-3 u v^3+u v^2+v^3-3 v^2+2 v-1}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{3/2}-2 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{5}{q^{7/2}}+\frac{4}{q^{9/2}}-\frac{3}{q^{11/2}}+\frac{1}{q^{13/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 (-z)-a^7 z^{-1} +a^5 z^5+5 a^5 z^3+8 a^5 z+4 a^5 z^{-1} -a^3 z^7-6 a^3 z^5-13 a^3 z^3-13 a^3 z-4 a^3 z^{-1} +a z^5+4 a z^3+4 a z+a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^8 z^2-a^8+3 a^7 z^3-2 a^7 z+a^7 z^{-1} +2 a^6 z^6-6 a^6 z^4+10 a^6 z^2-4 a^6+3 a^5 z^7-12 a^5 z^5+20 a^5 z^3-14 a^5 z+4 a^5 z^{-1} +a^4 z^8+a^4 z^6-13 a^4 z^4+18 a^4 z^2-7 a^4+5 a^3 z^7-20 a^3 z^5+26 a^3 z^3-17 a^3 z+4 a^3 z^{-1} +a^2 z^8-11 a^2 z^4+13 a^2 z^2-4 a^2+2 a z^7-8 a z^5+9 a z^3-5 a z+a z^{-1} +z^6-4 z^4+4 z^2-1 }[/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   \ -5 -4 -3 -2 -1 0 1 2 3 χ 4                 1 -1 2               1   1 0             2 1   -1 -2           3 1     2 -4         2 3       1 -6       3 2         1 -8     1 2           1 -10   2 3             -1 -12   2               2 -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{ r=-5 }[/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}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/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=2 }[/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 }[/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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