L10n20

L10n19

L10n21

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Link Presentations

[edit Notes on L10n20's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X11,14,12,15 X8493 X5,13,6,12 X13,5,14,20 X9,16,10,17 X15,10,16,11 X2,18,3,17
Gauss code	{1, -10, 5, -3}, {-6, -1, 2, -5, -8, 9, -4, 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)-1) (t(2)-1) \left(t(2)^2-3 t(2)+1\right)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{2}{q^{9/2}}-q^{7/2}+\frac{4}{q^{7/2}}+2 q^{5/2}-\frac{6}{q^{5/2}}-5 q^{3/2}+\frac{7}{q^{3/2}}+6 \sqrt{q}-\frac{7}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z^{-1} +z^3 a^3-z a^3-2 a^3 z^{-1} -z^5 a-2 z^3 a-z a+a z^{-1} +2 z^3 a^{-1} +3 z a^{-1} + a^{-1} z^{-1} -z a^{-3} - a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^2 z^8-z^8-2 a^3 z^7-4 a z^7-2 z^7 a^{-1} -a^4 z^6-a^2 z^6-2 z^6 a^{-2} -2 z^6+3 a^3 z^5+5 a z^5+z^5 a^{-1} -z^5 a^{-3} -2 a^4 z^4-2 a^2 z^4+4 z^4 a^{-2} +4 z^4-3 a^5 z^3-9 a^3 z^3-3 a z^3+6 z^3 a^{-1} +3 z^3 a^{-3} +2 a^4 z^2+6 a^2 z^2-z^2 a^{-2} +3 z^2+3 a^5 z+9 a^3 z+4 a z-5 z a^{-1} -3 z a^{-3} -a^4-3 a^2- a^{-2} -2-a^5 z^{-1} -2 a^3 z^{-1} -a z^{-1} + a^{-1} z^{-1} + 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   \ -4 -3 -2 -1 0 1 2 3 4 χ 8                 1 1 6               1   -1 4             4 1   3 2           2 1     -1 0         5 4       1 -2       4 4         0 -4     2 3           -1 -6   2 4             2 -8   2               -2 -10 2                 2

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=-4 }[/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}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/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 }[/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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