L10n77

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Link Presentations

[edit Notes on L10n77's Link Presentations]

Planar diagram presentation	X6172 X5,14,6,15 X3849 X15,2,16,3 X16,7,17,8 X9,18,10,19 X11,20,12,13 X13,12,14,5 X4,17,1,18 X19,10,20,11
Gauss code	{1, 4, -3, -9}, {-2, -1, 5, 3, -6, 10, -7, 8}, {-8, 2, -4, -5, 9, 6, -10, 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{u v^3 w^3-1}{\sqrt{u} v^{3/2} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-12} + q^{-8} + q^{-6} + q^{-4} }[/math] (db)
Signature	-8 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{12} z^2+a^{12} z^{-2} +3 a^{12}-a^{10} z^6-7 a^{10} z^4-15 a^{10} z^2-2 a^{10} z^{-2} -11 a^{10}+a^8 z^8+8 a^8 z^6+21 a^8 z^4+21 a^8 z^2+a^8 z^{-2} +8 a^8 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{16}+a^{12} z^4-5 a^{12} z^2-a^{12} z^{-2} +5 a^{12}+a^{11} z^7-7 a^{11} z^5+15 a^{11} z^3-11 a^{11} z+2 a^{11} z^{-1} +a^{10} z^8-8 a^{10} z^6+22 a^{10} z^4-26 a^{10} z^2-2 a^{10} z^{-2} +13 a^{10}+a^9 z^7-7 a^9 z^5+15 a^9 z^3-11 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} +8 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         2 1       1 -17     1             1 -19     1 1           0 -21 1 1               0 -23 2 1               1 -25 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}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/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}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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