L10a80

L10a79

L10a81

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Link Presentations

[edit Notes on L10a80's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X18,10,19,9 X20,13,7,14 X14,19,15,20 X10,16,11,15 X16,6,17,5 X2738 X4,11,5,12 X6,18,1,17
Gauss code	{1, -8, 2, -9, 7, -10}, {8, -1, 3, -6, 9, -2, 4, -5, 6, -7, 10, -3, 5, -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{2 u^2 v^3-4 u^2 v^2+3 u^2 v-u^2+u v^4-4 u v^3+5 u v^2-4 u v+u-v^4+3 v^3-4 v^2+2 v}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{9/2}+3 q^{7/2}-6 q^{5/2}+9 q^{3/2}-11 \sqrt{q}+\frac{11}{\sqrt{q}}-\frac{11}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{1}{q^{11/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z+a^5 z^{-1} -3 a^3 z^3-z^3 a^{-3} -6 a^3 z-2 a^3 z^{-1} -z a^{-3} +2 a z^5+z^5 a^{-1} +6 a z^3+z^3 a^{-1} +6 a z+2 a z^{-1} -z a^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 a^3 z^9-2 a z^9-3 a^4 z^8-10 a^2 z^8-7 z^8-a^5 z^7+a^3 z^7-8 a z^7-10 z^7 a^{-1} +12 a^4 z^6+34 a^2 z^6-9 z^6 a^{-2} +13 z^6+4 a^5 z^5+19 a^3 z^5+42 a z^5+21 z^5 a^{-1} -6 z^5 a^{-3} -14 a^4 z^4-30 a^2 z^4+13 z^4 a^{-2} -3 z^4 a^{-4} -6 a^5 z^3-32 a^3 z^3-44 a z^3-13 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} +4 a^4 z^2+7 a^2 z^2-5 z^2 a^{-2} -2 z^2+4 a^5 z+15 a^3 z+17 a z+5 z a^{-1} -z a^{-3} -a^2-a^5 z^{-1} -2 a^3 z^{-1} -2 a z^{-1} - a^{-1} 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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 10                     1 1 8                   2   -2 6                 4 1   3 4               5 2     -3 2             6 4       2 0           6 6         0 -2         5 5           0 -4       4 7             3 -6     2 4               -2 -8   1 4                 3 -10   2                   -2 -12 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=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\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^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/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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