L10a64

L10a63

L10a65

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Link Presentations

[edit Notes on L10a64's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X20,10,7,9 X2738 X16,14,17,13 X14,5,15,6 X4,15,5,16 X18,12,19,11 X12,18,13,17 X6,20,1,19
Gauss code	{1, -4, 2, -7, 6, -10}, {4, -1, 3, -2, 8, -9, 5, -6, 7, -5, 9, -8, 10, -3}

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