L10n2

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Link Presentations

[edit Notes on L10n2's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X15,1,16,4 X5,10,6,11 X3849 X11,19,12,18 X17,5,18,20 X19,13,20,12 X9,16,10,17 X2,14,3,13
Gauss code	{1, -10, -5, 3}, {-4, -1, 2, 5, -9, 4, -6, 8, 10, -2, -3, 9, -7, 6, -8, 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)-1) (t(2)-1)^3}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{1}{q^{9/2}}+2 q^{7/2}-\frac{2}{q^{7/2}}-4 q^{5/2}+\frac{3}{q^{5/2}}+4 q^{3/2}-\frac{5}{q^{3/2}}-6 \sqrt{q}+\frac{5}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^3 z^3-2 a^3 z+2 z a^{-3} -a^3 z^{-1} + a^{-3} z^{-1} +a z^5+4 a z^3-3 z^3 a^{-1} +7 a z-7 z a^{-1} +4 a z^{-1} -4 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^4 z^6-4 a^4 z^4+4 a^4 z^2+3 z^2 a^{-4} -a^4- a^{-4} +2 a^3 z^7-8 a^3 z^5+z^5 a^{-3} +9 a^3 z^3+3 z^3 a^{-3} -4 a^3 z-4 z a^{-3} +a^3 z^{-1} + a^{-3} z^{-1} +a^2 z^8+3 z^6 a^{-2} -10 a^2 z^4-7 z^4 a^{-2} +12 a^2 z^2+10 z^2 a^{-2} -4 a^2-4 a^{-2} +5 a z^7+3 z^7 a^{-1} -18 a z^5-9 z^5 a^{-1} +21 a z^3+15 z^3 a^{-1} -13 a z-13 z a^{-1} +4 a z^{-1} +4 a^{-1} z^{-1} +z^8+2 z^6-13 z^4+15 z^2-7 }[/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 8                 2 -2 6               2   2 4             2 2   0 2           4 2     2 0         3 4       1 -2       2 2         0 -4     1 3           2 -6   1 2             -1 -8   1               1 -10 1                 -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=0 }[/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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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=2 }[/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^{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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