L11n302

L11n301

L11n303

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Link Presentations

[edit Notes on L11n302's Link Presentations]

Planar diagram presentation	X6172 X3,13,4,12 X20,13,21,14 X22,19,11,20 X15,5,16,10 X17,9,18,8 X7,17,8,16 X9,19,10,18 X14,21,15,22 X2536 X11,1,12,4
Gauss code	{1, -10, -2, 11}, {10, -1, -7, 6, -8, 5}, {-11, 2, 3, -9, -5, 7, -6, 8, 4, -3, 9, -4}

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^2 w+u v w-u v-u w^2+u w-v^2 w^3+v^2 w^2+v w^4-v w^3+w^3}{\sqrt{u} v w^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^6-q^5+2 q^4-q^3+2 q^2+1+ q^{-1} - q^{-2} + q^{-3} - q^{-4} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^6 a^{-2} +z^6-a^2 z^4-7 z^4 a^{-2} +z^4 a^{-4} +7 z^4-4 a^2 z^2-17 z^2 a^{-2} +4 z^2 a^{-4} +16 z^2-4 a^2-16 a^{-2} +5 a^{-4} +15-a^2 z^{-2} -5 a^{-2} z^{-2} +2 a^{-4} z^{-2} +4 z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^2 z^8+z^8 a^{-2} +z^8 a^{-4} +z^8+a^3 z^7+2 a z^7+2 z^7 a^{-1} +2 z^7 a^{-3} +z^7 a^{-5} -6 a^2 z^6-8 z^6 a^{-2} -5 z^6 a^{-4} +z^6 a^{-6} -8 z^6-6 a^3 z^5-14 a z^5-16 z^5 a^{-1} -12 z^5 a^{-3} -4 z^5 a^{-5} +10 a^2 z^4+22 z^4 a^{-2} +7 z^4 a^{-4} -5 z^4 a^{-6} +20 z^4+10 a^3 z^3+28 a z^3+37 z^3 a^{-1} +21 z^3 a^{-3} +2 z^3 a^{-5} -7 a^2 z^2-31 z^2 a^{-2} -7 z^2 a^{-4} +6 z^2 a^{-6} -25 z^2-5 a^3 z-21 a z-33 z a^{-1} -16 z a^{-3} +z a^{-5} +4 a^2+20 a^{-2} +6 a^{-4} -2 a^{-6} +17+a^3 z^{-1} +5 a z^{-1} +9 a^{-1} z^{-1} +5 a^{-3} z^{-1} -a^2 z^{-2} -5 a^{-2} z^{-2} -2 a^{-4} z^{-2} -4 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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                       1 1 11                     1 1 0 9                   1     1 7               1 1 1     1 5             1 3 1       1 3           1   1         2 1         1 4 2           1 -1       1   1             2 -3       1 1               0 -5   1 1                   0 -7                         0 -9 1                       -1

Integral Khovanov Homology (db, data source)

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