L10a90

L10a89

L10a91

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Link Presentations

[edit Notes on L10a90's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,4,13,3 X20,12,9,11 X14,6,15,5 X2,9,3,10 X4,14,5,13 X6,20,7,19 X16,7,17,8 X18,15,19,16 X8,17,1,18
Gauss code	{1, -5, 2, -6, 4, -7, 8, -10}, {5, -1, 3, -2, 6, -4, 9, -8, 10, -9, 7, -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{(t(1)-1)^2 (t(2)-1)^2 (t(1) t(2)+1)}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -3 q^{9/2}+\frac{1}{q^{9/2}}+5 q^{7/2}-\frac{3}{q^{7/2}}-8 q^{5/2}+\frac{5}{q^{5/2}}+10 q^{3/2}-\frac{8}{q^{3/2}}+q^{11/2}-11 \sqrt{q}+\frac{9}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^5 a^{-3} -a^3 z^3+3 z^3 a^{-3} -2 a^3 z+2 z a^{-3} -z^7 a^{-1} +2 a z^5-5 z^5 a^{-1} +7 a z^3-9 z^3 a^{-1} +6 a z-6 z a^{-1} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a z^9-z^9 a^{-1} -3 a^2 z^8-3 z^8 a^{-2} -6 z^8-3 a^3 z^7-5 a z^7-6 z^7 a^{-1} -4 z^7 a^{-3} -a^4 z^6+7 a^2 z^6+z^6 a^{-2} -4 z^6 a^{-4} +13 z^6+10 a^3 z^5+22 a z^5+18 z^5 a^{-1} +3 z^5 a^{-3} -3 z^5 a^{-5} +3 a^4 z^4-a^2 z^4+3 z^4 a^{-2} +4 z^4 a^{-4} -z^4 a^{-6} -6 z^4-9 a^3 z^3-22 a z^3-17 z^3 a^{-1} +4 z^3 a^{-5} -2 a^4 z^2-2 a^2 z^2-3 z^2 a^{-2} -z^2 a^{-4} +z^2 a^{-6} -z^2+3 a^3 z+8 a z+6 z a^{-1} -z a^{-5} +1-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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                     1 -1 10                   2   2 8                 3 1   -2 6               5 2     3 4             5 3       -2 2           6 5         1 0         5 7           2 -2       3 4             -1 -4     2 5               3 -6   1 3                 -2 -8   2                   2 -10 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=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/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}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=5 }[/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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