L11n327

L11n326

L11n328

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Link Presentations

[edit Notes on L11n327's Link Presentations]

Planar diagram presentation	X6172 X5,14,6,15 X3849 X2,16,3,15 X16,7,17,8 X13,18,14,19 X9,21,10,20 X19,5,20,12 X11,13,12,22 X21,11,22,10 X17,1,18,4
Gauss code	{1, -4, -3, 11}, {-2, -1, 5, 3, -7, 10, -9, 8}, {-6, 2, 4, -5, -11, 6, -8, 7, -10, 9}

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^3-2 u v^2 w^2-u v w^3+3 u v w^2-2 u v w-u w^2+u w+v^3 \left(-w^2\right)+v^3 w+2 v^2 w^2-3 v^2 w+v^2+2 v w-v}{\sqrt{u} v^{3/2} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 2 q^5-4 q^4+6 q^3-6 q^2+8 q-6+6 q^{-1} -3 q^{-2} +2 q^{-3} - q^{-4} }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ 2 z^2 a^{-4} + a^{-4} z^{-2} +3 a^{-4} -a^2 z^4-3 z^4 a^{-2} -3 a^2 z^2-10 z^2 a^{-2} -2 a^{-2} z^{-2} -2 a^2-9 a^{-2} +z^6+5 z^4+10 z^2+ z^{-2} +8 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 3 z^2 a^{-6} - a^{-6} +z^5 a^{-5} +3 z^3 a^{-5} -z a^{-5} +3 z^6 a^{-4} -3 z^4 a^{-4} - a^{-4} z^{-2} +3 a^{-4} +a^3 z^7+4 z^7 a^{-3} -5 a^3 z^5-10 z^5 a^{-3} +7 a^3 z^3+12 z^3 a^{-3} -2 a^3 z-8 z a^{-3} +2 a^{-3} z^{-1} +2 a^2 z^8+3 z^8 a^{-2} -10 a^2 z^6-8 z^6 a^{-2} +16 a^2 z^4+12 z^4 a^{-2} -10 a^2 z^2-19 z^2 a^{-2} -2 a^{-2} z^{-2} +3 a^2+11 a^{-2} +a z^9+z^9 a^{-1} -a z^7+2 z^7 a^{-1} -9 a z^5-15 z^5 a^{-1} +16 a z^3+18 z^3 a^{-1} -7 a z-12 z a^{-1} +2 a^{-1} z^{-1} +5 z^8-21 z^6+31 z^4-26 z^2- z^{-2} +11 }[/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 χ 11                   2 2 9                 3 1 -2 7               3 1   2 5             3 3     0 3           5 3       2 1         4 6         2 -1       2 2           0 -3     1 4             3 -5   1 2               -1 -7   1                 1 -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=3 }[/math] [math]\displaystyle{ r=-5 }[/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=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/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}\oplus{\mathbb Z}_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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