L10n72

L10n71

L10n73

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Link Presentations

[edit Notes on L10n72's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X7,17,8,16 X9,11,10,20 X11,18,12,19 X15,9,16,8 X19,5,20,10 X17,14,18,15 X2536 X4,14,1,13
Gauss code	{1, -9, 2, -10}, {9, -1, -3, 6, -4, 7}, {-5, -2, 10, 8, -6, 3, -8, 5, -7, 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+u v w^2-u v w-u w^2+u w-v^2 w+v^2+v w-v-w^2}{\sqrt{u} v w} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^6+2 q^3-q^2+3 q-2+2 q^{-1} -2 q^{-2} + q^{-3} }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{-6} z^{-2} + a^{-6} -z^2 a^{-4} -2 a^{-4} z^{-2} -4 a^{-4} +a^2 z^2+2 z^2 a^{-2} + a^{-2} z^{-2} +a^2+4 a^{-2} -z^4-3 z^2-2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^8 a^{-2} +z^8+2 a z^7+3 z^7 a^{-1} +z^7 a^{-3} +a^2 z^6-5 z^6 a^{-2} +z^6 a^{-6} -3 z^6-9 a z^5-15 z^5 a^{-1} -6 z^5 a^{-3} -4 a^2 z^4+6 z^4 a^{-2} -2 z^4 a^{-4} -6 z^4 a^{-6} -2 z^4+9 a z^3+19 z^3 a^{-1} +8 z^3 a^{-3} -2 z^3 a^{-5} +3 a^2 z^2-z^2 a^{-2} +6 z^2 a^{-4} +9 z^2 a^{-6} +5 z^2-2 a z-6 z a^{-1} +4 z a^{-5} -a^2-3 a^{-2} -6 a^{-4} -4 a^{-6} -1-2 a^{-3} z^{-1} -2 a^{-5} z^{-1} + a^{-2} z^{-2} +2 a^{-4} z^{-2} + a^{-6} 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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                     1 1 11                     1 1 9               1 1     0 7             2         2 5           1 3 1       1 3         3 1           2 1       1 3 1           1 -1     1 2 1             0 -3   1 1                 0 -5   1                   -1 -7 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=-4 }[/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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/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} }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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=5 }[/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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