L10n66

L10n65

L10n67

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Link Presentations

[edit Notes on L10n66's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X16,12,17,11 X18,13,19,14 X20,18,9,17 X12,19,13,20 X15,8,16,5 X7,14,8,15 X2536 X9,1,10,4
Gauss code	{1, -9, -2, 10}, {9, -1, -8, 7}, {-10, 2, 3, -6, 4, 8, -7, -3, 5, -4, 6, -5}

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 w^2+u v w+u w^3-2 u w^2+2 v w-v-w^2+w}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-6} - q^{-5} +2 q^{-4} +q^3- q^{-3} -q^2+ q^{-2} +q+1 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^6 z^{-2} +a^6-2 a^4 z^2-3 a^4 z^{-2} -5 a^4+a^2 z^4+5 a^2 z^2+4 a^2 z^{-2} +z^2 a^{-2} + a^{-2} z^{-2} +8 a^2+2 a^{-2} -z^4-5 z^2-3 z^{-2} -6 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^5 z^7+a^3 z^7+a z^7+z^7 a^{-1} +a^6 z^6+3 a^4 z^6+3 a^2 z^6+z^6 a^{-2} +2 z^6-4 a^5 z^5-5 a^3 z^5-6 a z^5-5 z^5 a^{-1} -5 a^6 z^4-16 a^4 z^4-20 a^2 z^4-5 z^4 a^{-2} -14 z^4+2 a^5 z^3+4 a^3 z^3+6 a z^3+4 z^3 a^{-1} +7 a^6 z^2+23 a^4 z^2+35 a^2 z^2+6 z^2 a^{-2} +25 z^2+a^5 z+a^3 z+a z+z a^{-1} -4 a^6-14 a^4-21 a^2-4 a^{-2} -14-a^5 z^{-1} -a^3 z^{-1} -a z^{-1} - a^{-1} z^{-1} +a^6 z^{-2} +3 a^4 z^{-2} +4 a^2 z^{-2} + a^{-2} z^{-2} +3 z^{-2} }[/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 3 4 χ 7                     1 1 5                       0 3                 1 1   0 1             3 1       2 -1           1 3 1       1 -3         1 2 2         1 -5       1 1             0 -7     1 1 1             1 -9   1 2                 1 -11                       0 -13 1                     1

Integral Khovanov Homology (db, data source)

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