L11n279

L11n278

L11n280

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Link Presentations

[edit Notes on L11n279's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X2,14,3,13 X14,7,15,8 X18,10,19,9 X17,11,18,22 X11,21,12,20 X21,17,22,16 X15,1,16,4 X10,20,5,19
Gauss code	{1, -4, -3, 10}, {-2, -1, 5, 3, 6, -11}, {-8, 2, 4, -5, -10, 9, -7, -6, 11, 8, -9, 7}

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{2 u v w^2-2 u v w+u v-u w^2+u w+v^2 (-w)+v^2-v w^2+2 v w-2 v}{\sqrt{u} v w} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -2 q^6+3 q^5-4 q^4+6 q^3-4 q^2+5 q-2+2 q^{-1} }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -2 z^4 a^{-2} -7 z^2 a^{-2} +4 z^2 a^{-4} +2 z^2-10 a^{-2} +8 a^{-4} -2 a^{-6} +4-5 a^{-2} z^{-2} +4 a^{-4} z^{-2} - a^{-6} z^{-2} +2 z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 3 z^3 a^{-7} -5 z a^{-7} + a^{-7} z^{-1} +z^6 a^{-6} -2 z^2 a^{-6} - a^{-6} z^{-2} +3 a^{-6} +2 z^7 a^{-5} -8 z^5 a^{-5} +20 z^3 a^{-5} -18 z a^{-5} +5 a^{-5} z^{-1} +z^8 a^{-4} -3 z^6 a^{-4} +10 z^4 a^{-4} -14 z^2 a^{-4} -4 a^{-4} z^{-2} +12 a^{-4} +3 z^7 a^{-3} -11 z^5 a^{-3} +24 z^3 a^{-3} -24 z a^{-3} +9 a^{-3} z^{-1} +z^8 a^{-2} -4 z^6 a^{-2} +13 z^4 a^{-2} -21 z^2 a^{-2} -5 a^{-2} z^{-2} +15 a^{-2} +z^7 a^{-1} -3 z^5 a^{-1} +7 z^3 a^{-1} -11 z a^{-1} +5 a^{-1} z^{-1} +3 z^4-9 z^2-2 z^{-2} +7 }[/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   \ -2 -1 0 1 2 3 4 5 χ 13               2 -2 11             1   1 9           3 2   -1 7         3 1     2 5       2 4       2 3     3 2         1 1   1 4           3 -1 1 1             0 -3 2               2

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=3 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/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}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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