L10n49

L10n48

L10n50

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Link Presentations

[edit Notes on L10n49's Link Presentations]

Planar diagram presentation	X8192 X18,9,19,10 X6718 X20,14,7,13 X12,5,13,6 X3,10,4,11 X4,15,5,16 X11,16,12,17 X14,20,15,19 X17,2,18,3
Gauss code	{1, 10, -6, -7, 5, -3}, {3, -1, 2, 6, -8, -5, 4, -9, 7, 8, -10, -2, 9, -4}

A Braid Representative	{{{braid_table}}}
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^2 v^4-2 u^2 v^3-u v^2-2 v+1}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{2}{q^{9/2}}+\frac{2}{q^{7/2}}-\frac{2}{q^{5/2}}+\frac{1}{q^{3/2}}-\frac{1}{q^{17/2}}+\frac{1}{q^{15/2}}-\frac{2}{q^{13/2}}+\frac{2}{q^{11/2}}-\frac{1}{\sqrt{q}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z a^9+a^9 z^{-1} -z^5 a^7-5 z^3 a^7-6 z a^7-2 a^7 z^{-1} +z^7 a^5+6 z^5 a^5+11 z^3 a^5+8 z a^5+2 a^5 z^{-1} -z^5 a^3-5 z^3 a^3-6 z a^3-a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{11} z+a^{10} z^2+2 a^9 z^3-4 a^9 z+a^9 z^{-1} +a^8 z^6-3 a^8 z^4+a^8 z^2+2 a^7 z^7-10 a^7 z^5+15 a^7 z^3-10 a^7 z+2 a^7 z^{-1} +a^6 z^8-4 a^6 z^6+3 a^6 z^4-a^6 z^2+a^6+3 a^5 z^7-16 a^5 z^5+24 a^5 z^3-12 a^5 z+2 a^5 z^{-1} +a^4 z^8-5 a^4 z^6+6 a^4 z^4-a^4 z^2+a^3 z^7-6 a^3 z^5+11 a^3 z^3-7 a^3 z+a^3 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 0                 1 1 -2                   0 -4             2 1   1 -6           1 1     0 -8         1 1       0 -10       1 1         0 -12     1 1           0 -14     1             1 -16 1 1               0 -18 1                 1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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