L10n98

L10n97

L10n99

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Link Presentations

[edit Notes on L10n98's Link Presentations]

Planar diagram presentation	X6172 X2536 X13,15,14,20 X10,3,11,4 X4,9,1,10 X16,7,17,8 X8,15,5,16 X11,19,12,18 X19,13,20,12 X17,9,18,14
Gauss code	{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, -8, 9, -3, 10}, {7, -6, -10, 8, -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{t(1) t(4)^2 t(3)^2+t(2) t(4)^2 t(3)^2-t(4)^2 t(3)^2-t(2) t(4) t(3)^2-t(1) t(4)^2 t(3)-t(2) t(3)+t(1) t(4) t(3)+t(2) t(4) t(3)+t(1)-t(1) t(2)+t(2)-t(1) t(4)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{1}{q^{9/2}}-\frac{4}{q^{7/2}}-2 q^{5/2}+\frac{2}{q^{5/2}}+2 q^{3/2}-\frac{5}{q^{3/2}}-\frac{1}{q^{11/2}}-4 \sqrt{q}+\frac{3}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z^{-3} +a^5 z+3 a^5 z^{-1} -3 a^3 z^3-3 a^3 z^{-3} -11 a^3 z-10 a^3 z^{-1} +2 a z^5+10 a z^3+3 a z^{-3} -2 z^3 a^{-1} - a^{-1} z^{-3} +16 a z+11 a z^{-1} -6 z a^{-1} -4 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^4 z^8-a^2 z^8-a^5 z^7-5 a^3 z^7-4 a z^7+3 a^4 z^6-a^2 z^6-4 z^6+6 a^5 z^5+25 a^3 z^5+17 a z^5-2 z^5 a^{-1} +4 a^4 z^4+19 a^2 z^4-z^4 a^{-2} +14 z^4-13 a^5 z^3-39 a^3 z^3-24 a z^3+2 z^3 a^{-1} -17 a^4 z^2-33 a^2 z^2-16 z^2+13 a^5 z+28 a^3 z+21 a z+3 z a^{-1} -3 z a^{-3} +13 a^4+24 a^2- a^{-2} +11-6 a^5 z^{-1} -14 a^3 z^{-1} -12 a z^{-1} -3 a^{-1} z^{-1} + a^{-3} z^{-1} -3 a^4 z^{-2} -6 a^2 z^{-2} -3 z^{-2} +a^5 z^{-3} +3 a^3 z^{-3} +3 a z^{-3} + a^{-1} z^{-3} }[/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 6                 2 2 4               3 3 0 2             1   1 2 0           3 4     1 -2         2 1 1     2 -4       1 4         3 -6     3 1           2 -8   1 4             3 -10                   0 -12 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{ i=2 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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