L10n27

L10n26

L10n28

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Link Presentations

[edit Notes on L10n27's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,17,12,16 X14,7,15,8 X8,15,9,16 X17,5,18,20 X13,18,14,19 X19,12,20,13 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, 4, -5, 10, -2, -3, 8, -7, -4, 5, 3, -6, 7, -8, 6}

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{u v^3-4 u v^2+3 u v-u-v^3+3 v^2-4 v+1}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{6}{q^{9/2}}-\frac{7}{q^{7/2}}+\frac{5}{q^{5/2}}-\frac{5}{q^{3/2}}-\frac{1}{q^{15/2}}+\frac{3}{q^{13/2}}-\frac{5}{q^{11/2}}-\sqrt{q}+\frac{3}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z a^7-2 z^3 a^5-2 z a^5+a^5 z^{-1} +z^5 a^3+2 z^3 a^3-a^3 z^{-1} -z^3 a-z a }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^9 z^3-a^9 z+3 a^8 z^4-2 a^8 z^2+a^7 z^7-a^7 z^5+4 a^7 z^3-a^7 z+a^6 z^8-a^6 z^6+3 a^6 z^4-a^6 z^2+4 a^5 z^7-8 a^5 z^5+6 a^5 z^3-2 a^5 z+a^5 z^{-1} +a^4 z^8+2 a^4 z^6-7 a^4 z^4+3 a^4 z^2-a^4+3 a^3 z^7-6 a^3 z^5+a^3 z^3-a^3 z+a^3 z^{-1} +3 a^2 z^6-7 a^2 z^4+2 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 3     0 -6         4 2       2 -8       2 3         1 -10     3 4           -1 -12   1 3             2 -14   2               -2 -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{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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