L10n105

L10n104

L10n106

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Link Presentations

[edit Notes on L10n105's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X2,16,3,15 X7,17,8,16 X14,9,11,10 X20,13,15,14 X19,5,20,10 X11,18,12,19 X17,1,18,4
Gauss code	{1, -4, -3, 10}, {-9, 2, 7, -6}, {-2, -1, -5, 3, 6, 8}, {4, 5, -10, 9, -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) t(4) t(3)^2-t(4)^2 t(3)-t(1) t(2) t(3)-t(1) t(4) t(3)+t(1) t(2) t(4) t(3)-t(2) t(4) t(3)+t(4) t(3)+t(2) t(4)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{2}{q^{9/2}}+\frac{1}{q^{7/2}}-\frac{4}{q^{5/2}}-2 q^{3/2}+\frac{2}{q^{3/2}}+\sqrt{q}-\frac{4}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ 2 a^5 z^{-1} +a^5 z^{-3} -4 z a^3-7 a^3 z^{-1} -3 a^3 z^{-3} +2 z^3 a+6 z a+8 a z^{-1} +3 a z^{-3} -2 z a^{-1} -3 a^{-1} z^{-1} - a^{-1} z^{-3} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 3 a^5 z^3-a^5 z^{-3} -8 a^5 z+5 a^5 z^{-1} +a^4 z^6-4 a^4 z^4+10 a^4 z^2+3 a^4 z^{-2} -10 a^4+a^3 z^7-5 a^3 z^5+15 a^3 z^3-3 a^3 z^{-3} -19 a^3 z+12 a^3 z^{-1} +2 a^2 z^6-8 a^2 z^4+20 a^2 z^2+6 a^2 z^{-2} -19 a^2+a z^7-5 a z^5+15 a z^3+3 z^3 a^{-1} -3 a z^{-3} - a^{-1} z^{-3} -19 a z-8 z a^{-1} +12 a z^{-1} +5 a^{-1} z^{-1} +z^6-4 z^4+10 z^2+3 z^{-2} -10 }[/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 χ 4             2 2 2           1 2 1 0         3     3 -2       1 3     2 -4     3 1       2 -6     3         3 -8 2 1           1 -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{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{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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