L10n93

L10n92

L10n94

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Link Presentations

[edit Notes on L10n93's Link Presentations]

Planar diagram presentation	X8192 X9,18,10,19 X17,6,18,1 X16,7,17,8 X3,10,4,11 X5,14,6,15 X13,4,14,5 X20,11,13,12 X12,15,7,16 X2,19,3,20
Gauss code	{1, -10, -5, 7, -6, 3}, {4, -1, -2, 5, 8, -9}, {-7, 6, 9, -4, -3, 2, 10, -8}

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 v w+1) \left(u v w^2-1\right)}{u v w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-11} + q^{-9} + q^{-6} + q^{-4} }[/math] (db)
Signature	-7 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^2 a^{12}+a^{12} z^{-2} +2 a^{12}-z^6 a^{10}-7 z^4 a^{10}-14 z^2 a^{10}-2 a^{10} z^{-2} -9 a^{10}+z^8 a^8+8 z^6 a^8+21 z^4 a^8+21 z^2 a^8+a^8 z^{-2} +7 a^8 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{14} z^2-2 a^{14}-a^{12} z^2-a^{12} z^{-2} +3 a^{12}+a^{11} z^7-7 a^{11} z^5+14 a^{11} z^3-9 a^{11} z+2 a^{11} z^{-1} +a^{10} z^8-8 a^{10} z^6+21 a^{10} z^4-23 a^{10} z^2-2 a^{10} z^{-2} +11 a^{10}+a^9 z^7-7 a^9 z^5+14 a^9 z^3-9 a^9 z+2 a^9 z^{-1} +a^8 z^8-8 a^8 z^6+21 a^8 z^4-21 a^8 z^2-a^8 z^{-2} +7 a^8 }[/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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -7                 1 1 -9                 1 1 -11             1     1 -13         1         1 -15         1 1       0 -17     2 1           1 -19     2 1           1 -21 1                 1 -23 1                 1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-9 }[/math] [math]\displaystyle{ i=-7 }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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