L10a43

L10a42

L10a44

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Link Presentations

[edit Notes on L10a43's Link Presentations]

Planar diagram presentation	X6172 X2,9,3,10 X12,3,13,4 X20,12,5,11 X18,6,19,5 X10,20,11,19 X4,17,1,18 X16,14,17,13 X14,8,15,7 X8,16,9,15
Gauss code	{1, -2, 3, -7}, {5, -1, 9, -10, 2, -6, 4, -3, 8, -9, 10, -8, 7, -5, 6, -4}

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^5-4 u v^4+8 u v^3-8 u v^2+4 u v+4 v^4-8 v^3+8 v^2-4 v+1}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -9 q^{9/2}+13 q^{7/2}-\frac{1}{q^{7/2}}-16 q^{5/2}+\frac{3}{q^{5/2}}+17 q^{3/2}-\frac{8}{q^{3/2}}-q^{13/2}+5 q^{11/2}-16 \sqrt{q}+\frac{11}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-1} +a z^5-5 z^5 a^{-1} +2 z^5 a^{-3} +3 a z^3-12 z^3 a^{-1} +5 z^3 a^{-3} -z^3 a^{-5} +5 a z-12 z a^{-1} +5 z a^{-3} +3 a z^{-1} -5 a^{-1} z^{-1} +2 a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-7} +5 z^6 a^{-6} -6 z^4 a^{-6} + a^{-6} +9 z^7 a^{-5} -15 z^5 a^{-5} +5 z^3 a^{-5} +7 z^8 a^{-4} -4 z^6 a^{-4} -9 z^4 a^{-4} +4 z^2 a^{-4} +2 z^9 a^{-3} +15 z^7 a^{-3} +a^3 z^5-34 z^5 a^{-3} -2 a^3 z^3+18 z^3 a^{-3} +a^3 z-5 z a^{-3} +2 a^{-3} z^{-1} +12 z^8 a^{-2} +3 a^2 z^6-13 z^6 a^{-2} -4 a^2 z^4-10 z^4 a^{-2} +a^2 z^2+17 z^2 a^{-2} -5 a^{-2} +2 z^9 a^{-1} +6 a z^7+12 z^7 a^{-1} -12 a z^5-31 z^5 a^{-1} +14 a z^3+29 z^3 a^{-1} -10 a z-16 z a^{-1} +3 a z^{-1} +5 a^{-1} z^{-1} +5 z^8-z^6-11 z^4+14 z^2-5 }[/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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                     1 1 12                   4   -4 10                 5 1   4 8               8 4     -4 6             8 5       3 4           9 8         -1 2         7 8           -1 0       5 10             5 -2     3 6               -3 -4     5                 5 -6 1 3                   -2 -8 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=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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