L10n52

L10n51

L10n53

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Link Presentations

[edit Notes on L10n52's Link Presentations]

Planar diagram presentation	X8192 X9,19,10,18 X14,6,15,5 X16,12,17,11 X3,10,4,11 X12,7,13,8 X20,15,7,16 X6,14,1,13 X4,19,5,20 X17,2,18,3
Gauss code	{1, 10, -5, -9, 3, -8}, {6, -1, -2, 5, 4, -6, 8, -3, 7, -4, -10, 2, 9, -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^2 v^2+2 u^2 v-u^2+u v^4-3 u v^3+5 u v^2-3 u v+u-v^4+2 v^3-v^2}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -2 q^{3/2}+4 \sqrt{q}-\frac{6}{\sqrt{q}}+\frac{7}{q^{3/2}}-\frac{8}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^5+z a^5+a^5 z^{-1} -z^5 a^3-3 z^3 a^3-5 z a^3-2 a^3 z^{-1} +3 z^3 a+5 z a+2 a z^{-1} -2 z a^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 a^4 z^8-2 a^2 z^8-4 a^5 z^7-7 a^3 z^7-3 a z^7-3 a^6 z^6+2 a^4 z^6+4 a^2 z^6-z^6-a^7 z^5+11 a^5 z^5+21 a^3 z^5+9 a z^5+7 a^6 z^4+2 a^4 z^4-7 a^2 z^4-2 z^4+2 a^7 z^3-9 a^5 z^3-26 a^3 z^3-18 a z^3-3 z^3 a^{-1} -2 a^6 z^2-a^4 z^2+4 a^2 z^2+3 z^2+4 a^5 z+12 a^3 z+12 a z+4 z a^{-1} -a^2-a^5 z^{-1} -2 a^3 z^{-1} -2 a z^{-1} - a^{-1} 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 4                 2 2 2               2   -2 0             4 2   2 -2           4 3     -1 -4         4 3       1 -6       3 5         2 -8     2 3           -1 -10   1 3             2 -12   2               -2 -14 1                 1

Integral Khovanov Homology (db, data source)

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