L10n53

L10n52

L10n54

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Link Presentations

[edit Notes on L10n53's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X20,13,7,14 X9,15,10,14 X19,11,20,10 X5,16,6,17 X15,18,16,19 X2738 X4,11,5,12 X17,6,18,1
Gauss code	{1, -8, 2, -9, -6, 10}, {8, -1, -4, 5, 9, -2, 3, 4, -7, 6, -10, 7, -5, -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(2)^2 t(1)^2-3 t(2) t(1)^2+t(1)^2-3 t(2)^2 t(1)+5 t(2) t(1)-3 t(1)+t(2)^2-3 t(2)+1}{t(1) t(2)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{6}{q^{5/2}}-\frac{8}{q^{7/2}}+\frac{7}{q^{9/2}}-\frac{6}{q^{11/2}}+\frac{4}{q^{13/2}}-\frac{2}{q^{15/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ 2 a^7 z+a^7 z^{-1} -3 a^5 z^3-5 a^5 z-a^5 z^{-1} +a^3 z^5+2 a^3 z^3+a^3 z-a z^3-a z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 3 a^9 z^3-3 a^9 z+a^8 z^6+2 a^8 z^4-2 a^8 z^2+2 a^7 z^7-2 a^7 z^5+3 a^7 z^3+a^7 z-a^7 z^{-1} +a^6 z^8+2 a^6 z^6-2 a^6 z^4-a^6 z^2+a^6+5 a^5 z^7-8 a^5 z^5+2 a^5 z^3+3 a^5 z-a^5 z^{-1} +a^4 z^8+4 a^4 z^6-11 a^4 z^4+4 a^4 z^2+3 a^3 z^7-5 a^3 z^5+3 a^2 z^6-7 a^2 z^4+3 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           4 3     -1 -6         4 2       2 -8       3 4         1 -10     3 4           -1 -12   1 3             2 -14 1 3               -2 -16 2                 2

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{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{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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