L10n21

L10n20

L10n22

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Link Presentations

[edit Notes on L10n21's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X9,16,10,17 X17,5,18,20 X13,19,14,18 X19,15,20,14 X15,8,16,9 X2536 X4,12,1,11
Gauss code	{1, -9, 2, -10}, {9, -1, 3, 8, -4, -2, 10, -3, -6, 7, -8, 4, -5, 6, -7, 5}

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

Integral Khovanov Homology (db, data source)

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