L10n13

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Link Presentations

[edit Notes on L10n13's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X9,14,10,15 X3849 X5,13,6,12 X13,5,14,20 X11,16,12,17 X15,10,16,11 X17,2,18,3
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) t(2)^5-2 t(1) t(2)^4+t(1) t(2)^3-t(2)^3-t(1) t(2)^2+t(2)^2-2 t(2)+1}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{3}{q^{9/2}}+\frac{3}{q^{7/2}}-\frac{3}{q^{5/2}}+\frac{1}{q^{3/2}}-\frac{1}{q^{17/2}}+\frac{2}{q^{15/2}}-\frac{3}{q^{13/2}}+\frac{3}{q^{11/2}}-\frac{1}{\sqrt{q}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 z+a^9 z^{-1} -a^7 z^5-5 a^7 z^3-7 a^7 z-3 a^7 z^{-1} +a^5 z^7+6 a^5 z^5+12 a^5 z^3+11 a^5 z+4 a^5 z^{-1} -a^3 z^5-5 a^3 z^3-7 a^3 z-2 a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{11} z+2 a^{10} z^2-a^{10}+a^9 z^5-a^9 z^3-a^9 z+a^9 z^{-1} +3 a^8 z^6-11 a^8 z^4+10 a^8 z^2-3 a^8+3 a^7 z^7-13 a^7 z^5+17 a^7 z^3-11 a^7 z+3 a^7 z^{-1} +a^6 z^8-a^6 z^6-9 a^6 z^4+12 a^6 z^2-3 a^6+4 a^5 z^7-20 a^5 z^5+30 a^5 z^3-18 a^5 z+4 a^5 z^{-1} +a^4 z^8-4 a^4 z^6+2 a^4 z^4+4 a^4 z^2-2 a^4+a^3 z^7-6 a^3 z^5+12 a^3 z^3-9 a^3 z+2 a^3 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 0                 1 1 -2                   0 -4             3 1   2 -6           1 1     0 -8         2 2       0 -10     1 2 1         0 -12     2 2           0 -14   1 2             1 -16   1               -1 -18 1                 1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/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}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/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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