L11n282

L11n281

L11n283

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Link Presentations

[edit Notes on L11n282's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X8493 X2,14,3,13 X14,7,15,8 X18,10,19,9 X17,11,18,22 X11,21,12,20 X21,17,22,16 X4,15,1,16 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{(t(1)-1) (t(2)-1) (t(3)-1) \left(t(3)^2-t(3)+1\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -2 q^6+4 q^5-7 q^4+9 q^3-7 q^2- q^{-2} +9 q+4 q^{-1} -5 }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^6 a^{-2} +3 z^4 a^{-2} -z^4 a^{-4} -z^4+z^2 a^{-2} -z^2-5 a^{-2} +4 a^{-4} - a^{-6} +2-5 a^{-2} z^{-2} +4 a^{-4} z^{-2} - a^{-6} z^{-2} +2 z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^8 a^{-2} +z^8 a^{-4} +4 z^7 a^{-1} +6 z^7 a^{-3} +2 z^7 a^{-5} +8 z^6 a^{-2} +5 z^6 a^{-4} +z^6 a^{-6} +4 z^6+a z^5-6 z^5 a^{-1} -8 z^5 a^{-3} -z^5 a^{-5} -20 z^4 a^{-2} -9 z^4 a^{-4} +2 z^4 a^{-6} -9 z^4-a z^3+5 z^3 a^{-3} +7 z^3 a^{-5} +3 z^3 a^{-7} +7 z^2 a^{-2} +3 z^2 a^{-4} +4 z^2-5 z a^{-1} -12 z a^{-3} -10 z a^{-5} -3 z a^{-7} +5 a^{-2} +4 a^{-4} + a^{-6} +3+5 a^{-1} z^{-1} +9 a^{-3} z^{-1} +5 a^{-5} z^{-1} + a^{-7} z^{-1} -5 a^{-2} z^{-2} -4 a^{-4} z^{-2} - a^{-6} z^{-2} -2 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   \ -3 -2 -1 0 1 2 3 4 5 χ 13                 2 -2 11               2   2 9             5 2   -3 7           4 2     2 5         3 5       2 3       6 4         2 1     3 7           4 -1   1 2             -1 -3   3               3 -5 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=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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