L10n37

L10n36

L10n38

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Link Presentations

[edit Notes on L10n37's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X13,17,14,16 X9,15,10,14 X15,11,16,10 X17,5,18,20 X7,19,8,18 X19,9,20,8 X2536 X4,11,1,12
Gauss code	{1, -9, 2, -10}, {9, -1, -7, 8, -4, 5, 10, -2, -3, 4, -5, 3, -6, 7, -8, 6}

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{\left(v^2-v+1\right) \left(-u v^2+u v+u+v^3+v^2-v\right)}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -2 q^{9/2}+3 q^{7/2}-2 q^{5/2}-\frac{1}{q^{5/2}}+2 q^{3/2}-q^{13/2}+2 q^{11/2}-2 \sqrt{q}-\frac{1}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^5 a^{-1} +z^5 a^{-3} +a z^3-7 z^3 a^{-1} +5 z^3 a^{-3} -z^3 a^{-5} +4 a z-13 z a^{-1} +9 z a^{-3} -2 z a^{-5} +4 a z^{-1} -8 a^{-1} z^{-1} +5 a^{-3} z^{-1} - a^{-5} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-7} -3 z^3 a^{-7} +z a^{-7} +2 z^6 a^{-6} -7 z^4 a^{-6} +5 z^2 a^{-6} -2 a^{-6} +z^7 a^{-5} -2 z^5 a^{-5} -2 z^3 a^{-5} +z a^{-5} + a^{-5} z^{-1} +3 z^6 a^{-4} -13 z^4 a^{-4} +18 z^2 a^{-4} -9 a^{-4} +z^7 a^{-3} -5 z^5 a^{-3} +9 z^3 a^{-3} -8 z a^{-3} +5 a^{-3} z^{-1} +2 z^6 a^{-2} -14 z^4 a^{-2} +28 z^2 a^{-2} -14 a^{-2} +a z^7+z^7 a^{-1} -7 a z^5-9 z^5 a^{-1} +15 a z^3+23 z^3 a^{-1} -13 a z-21 z a^{-1} +4 a z^{-1} +8 a^{-1} z^{-1} +z^6-8 z^4+15 z^2-8 }[/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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                     1 1 12                   1   -1 10                 1 1   0 8               2 1     -1 6           1 1 1       -1 4           2 2         0 2       1 2 1           0 0         3             3 -2     1                 1 -4 1                     1 -6 1                     1

Integral Khovanov Homology (db, data source)

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