L10n19

L10n18

L10n20

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Link Presentations

[edit Notes on L10n19's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X11,14,12,15 X3849 X5,13,6,12 X13,5,14,20 X9,16,10,17 X15,10,16,11 X17,2,18,3
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{2 t(1) t(2)^3-4 t(1) t(2)^2+t(2)^2+t(1) t(2)-4 t(2)+2}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{5}{q^{9/2}}-\frac{5}{q^{7/2}}+\frac{4}{q^{5/2}}-\frac{3}{q^{3/2}}-\frac{2}{q^{15/2}}+\frac{3}{q^{13/2}}-\frac{4}{q^{11/2}}-\sqrt{q}+\frac{1}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 z^{-1} -a^7 z^3-3 a^7 z-2 a^7 z^{-1} +a^5 z^5+3 a^5 z^3+2 a^5 z+a^5 z^{-1} +a^3 z^5+3 a^3 z^3+2 a^3 z+a^3 z^{-1} -a z^3-3 a z-a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 3 a^9 z^3-5 a^9 z+a^9 z^{-1} +a^8 z^6-2 a^8 z^2+a^8+2 a^7 z^7-8 a^7 z^5+18 a^7 z^3-13 a^7 z+2 a^7 z^{-1} +a^6 z^8-3 a^6 z^6+8 a^6 z^4-7 a^6 z^2+3 a^6+3 a^5 z^7-10 a^5 z^5+16 a^5 z^3-9 a^5 z+a^5 z^{-1} +a^4 z^8-3 a^4 z^6+6 a^4 z^4-6 a^4 z^2+2 a^4+a^3 z^7-a^3 z^5-3 a^3 z^3+3 a^3 z-a^3 z^{-1} +a^2 z^6-2 a^2 z^4-a^2 z^2+a^2+a z^5-4 a z^3+4 a z-a 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   \ -6 -5 -4 -3 -2 -1 0 1 2 χ 2                 1 1 0                   0 -2             3 1   2 -4           2 1     -1 -6         3 2       1 -8       2 2         0 -10     2 3           -1 -12   2 3             1 -14   1               -1 -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{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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