L11n39

L11n38

L11n40

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Link Presentations

[edit Notes on L11n39's Link Presentations]

Planar diagram presentation	X6172 X20,7,21,8 X4,21,1,22 X5,12,6,13 X8493 X9,16,10,17 X13,19,14,18 X17,15,18,14 X15,10,16,11 X11,22,12,5 X2,20,3,19
Gauss code	{1, -11, 5, -3}, {-4, -1, 2, -5, -6, 9, -10, 4, -7, 8, -9, 6, -8, 7, 11, -2, 3, 10}

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-1) (v-1)}{\sqrt{u} \sqrt{v}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{3}{q^{9/2}}+\frac{2}{q^{7/2}}-q^{5/2}-\frac{2}{q^{5/2}}+2 q^{3/2}+\frac{1}{q^{15/2}}-\frac{2}{q^{13/2}}+\frac{3}{q^{11/2}}-2 \sqrt{q} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 (-z)-a^7 z^{-1} +2 a^5 z^3+5 a^5 z+3 a^5 z^{-1} -a^3 z^5-5 a^3 z^3-8 a^3 z-3 a^3 z^{-1} +a z^5+5 a z^3-z^3 a^{-1} +6 a z+2 a z^{-1} -2 z a^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^8 z^6-4 a^8 z^4+4 a^8 z^2-a^8+2 a^7 z^7-8 a^7 z^5+7 a^7 z^3-3 a^7 z+a^7 z^{-1} +a^6 z^8-2 a^6 z^6-5 a^6 z^4+5 a^6 z^2-2 a^6+3 a^5 z^7-15 a^5 z^5+19 a^5 z^3-12 a^5 z+3 a^5 z^{-1} +2 a^4 z^8-12 a^4 z^6+17 a^4 z^4-7 a^4 z^2+a^3 z^9-4 a^3 z^7-7 a^3 z^5+27 a^3 z^3-18 a^3 z+3 a^3 z^{-1} +3 a^2 z^8-20 a^2 z^6+32 a^2 z^4-13 a^2 z^2+2 a^2+a z^9-4 a z^7+z^7 a^{-1} -5 a z^5-5 z^5 a^{-1} +21 a z^3+6 z^3 a^{-1} -13 a z-4 z a^{-1} +2 a z^{-1} + a^{-1} z^{-1} +2 z^8-11 z^6+14 z^4-5 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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                       1 1 4                     1   -1 2                   1 1   0 0               3 2 1     2 -2             2 3 1       0 -4           2 2 2         2 -6         2 3 1           0 -8       2 2 1             1 -10     1 2 1               0 -12   1 2                   -1 -14   1                     1 -16 1                       -1

Integral Khovanov Homology (db, data source)

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