L11n284

L11n283

L11n285

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Link Presentations

[edit Notes on L11n284's Link Presentations]

Planar diagram presentation	X6172 X12,6,13,5 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(2) t(3)^3+1\right)}{\sqrt{t(1)} t(2) t(3)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -2 q^7+3 q^6-4 q^5+6 q^4-4 q^3+6 q^2-3 q- q^{-1} +3 }[/math] (db)
Signature	4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^2 a^{-8} - a^{-8} z^{-2} -z^6 a^{-6} -5 z^4 a^{-6} -5 z^2 a^{-6} +4 a^{-6} z^{-2} +2 a^{-6} +z^8 a^{-4} +6 z^6 a^{-4} +11 z^4 a^{-4} +6 z^2 a^{-4} -5 a^{-4} z^{-2} -5 a^{-4} -z^6 a^{-2} -4 z^4 a^{-2} -2 z^2 a^{-2} +2 a^{-2} z^{-2} +3 a^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 2 z^9 a^{-3} +2 z^9 a^{-5} +3 z^8 a^{-2} +7 z^8 a^{-4} +4 z^8 a^{-6} +z^7 a^{-1} -7 z^7 a^{-3} -6 z^7 a^{-5} +2 z^7 a^{-7} -15 z^6 a^{-2} -35 z^6 a^{-4} -20 z^6 a^{-6} -4 z^5 a^{-1} +z^5 a^{-3} -3 z^5 a^{-5} -8 z^5 a^{-7} +21 z^4 a^{-2} +49 z^4 a^{-4} +29 z^4 a^{-6} +z^4 a^{-8} +3 z^3 a^{-1} +4 z^3 a^{-3} +7 z^3 a^{-5} +6 z^3 a^{-7} -9 z^2 a^{-2} -21 z^2 a^{-4} -14 z^2 a^{-6} -2 z^2 a^{-8} +5 z a^{-3} +9 z a^{-5} +5 z a^{-7} +z a^{-9} -3 a^{-2} -4 a^{-4} -2 a^{-6} -5 a^{-3} z^{-1} -9 a^{-5} z^{-1} -5 a^{-7} z^{-1} - a^{-9} z^{-1} +2 a^{-2} z^{-2} +5 a^{-4} z^{-2} +4 a^{-6} z^{-2} + a^{-8} 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 15                 2 -2 13               2 1 1 11             3 2   -1 9           3 2 1   2 7         3 5       2 5       3 2 1       2 3     2 5           3 1   1 1             0 -1   2               2 -3 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{ i=5 }[/math] [math]\displaystyle{ r=-3 }[/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=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}\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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