L10n103

L10n102

L10n104

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Link Presentations

[edit Notes on L10n103's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X11,18,12,19 X20,16,17,15 X16,20,9,19 X17,12,18,13 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -4}, {-8, 5, 7, -6}, {10, -2, -5, 8, 4, -3, 6, -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) (t(3)-1) (t(4)-1)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} \sqrt{t(4)}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{4}{q^{9/2}}-\frac{6}{q^{11/2}}+\frac{1}{q^{13/2}}-\frac{1}{q^{15/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 z^{-3} -3 a^7 z^{-3} -3 a^7 z^{-1} -a^5 z^3+3 a^5 z^{-3} +2 a^5 z+6 a^5 z^{-1} +a^3 z^5+2 a^3 z^3-a^3 z^{-3} -a^3 z-3 a^3 z^{-1} -a z^3-a z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^9 z^3-a^9 z^{-3} -3 a^9 z+3 a^9 z^{-1} +a^8 z^4+2 a^8 z^2+3 a^8 z^{-2} -6 a^8+a^7 z^7-4 a^7 z^5+12 a^7 z^3-3 a^7 z^{-3} -9 a^7 z+6 a^7 z^{-1} +a^6 z^8-2 a^6 z^6+a^6 z^4+9 a^6 z^2+6 a^6 z^{-2} -11 a^6+4 a^5 z^7-12 a^5 z^5+16 a^5 z^3-3 a^5 z^{-3} -11 a^5 z+6 a^5 z^{-1} +a^4 z^8+a^4 z^6-7 a^4 z^4+8 a^4 z^2+3 a^4 z^{-2} -6 a^4+3 a^3 z^7-7 a^3 z^5+3 a^3 z^3-a^3 z^{-3} -4 a^3 z+3 a^3 z^{-1} +3 a^2 z^6-7 a^2 z^4+a^2 z^2+a z^5-2 a z^3+a z }[/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 2                 1 1 0               2   -2 -2             3 1   2 -4           3 4     1 -6         4 1       3 -8     1 1 3         3 -10     6 4           2 -12   1 6             5 -14                   0 -16 1                 1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ i=0 }[/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}^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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