L10n62

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Link Presentations

[edit Notes on L10n62's Link Presentations]

Planar diagram presentation	X12,1,13,2 X16,8,17,7 X5,1,6,10 X3746 X9,5,10,4 X18,14,19,13 X20,16,11,15 X14,20,15,19 X2,11,3,12 X8,18,9,17
Gauss code	{1, -9, -4, 5, -3, 4, 2, -10, -5, 3}, {9, -1, 6, -8, 7, -2, 10, -6, 8, -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) \left(t(2)^2 t(1)^2+t(1)^2-t(2) t(1)+t(2)^2+1\right)}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -4 q^{9/2}+3 q^{7/2}-4 q^{5/2}+q^{3/2}-q^{17/2}+3 q^{15/2}-3 q^{13/2}+4 q^{11/2}-\sqrt{q} }[/math] (db)
Signature	5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^{-9} - a^{-9} z^{-1} +z^5 a^{-7} +5 z^3 a^{-7} +8 z a^{-7} +5 a^{-7} z^{-1} -z^7 a^{-5} -6 z^5 a^{-5} -13 z^3 a^{-5} -15 z a^{-5} -8 a^{-5} z^{-1} +z^5 a^{-3} +5 z^3 a^{-3} +8 z a^{-3} +4 a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^8 a^{-4} -z^8 a^{-6} -z^7 a^{-3} -5 z^7 a^{-5} -4 z^7 a^{-7} +3 z^6 a^{-4} -2 z^6 a^{-6} -5 z^6 a^{-8} +6 z^5 a^{-3} +24 z^5 a^{-5} +16 z^5 a^{-7} -2 z^5 a^{-9} +3 z^4 a^{-4} +23 z^4 a^{-6} +20 z^4 a^{-8} -13 z^3 a^{-3} -35 z^3 a^{-5} -18 z^3 a^{-7} +4 z^3 a^{-9} -13 z^2 a^{-4} -33 z^2 a^{-6} -23 z^2 a^{-8} -3 z^2 a^{-10} +12 z a^{-3} +23 z a^{-5} +11 z a^{-7} -z a^{-9} -z a^{-11} +8 a^{-4} +14 a^{-6} +9 a^{-8} +2 a^{-10} -4 a^{-3} z^{-1} -8 a^{-5} z^{-1} -5 a^{-7} z^{-1} - a^{-9} z^{-1} }[/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 6 χ 18                 1 1 16               2   -2 14             2 2   0 12           3 2     -1 10         1 2 1     0 8       2 3         1 6     2 1           1 4   1 4             3 2                   0 0 1                 1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ i=4 }[/math] [math]\displaystyle{ i=6 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/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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