L10a132

L10a131

L10a133

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Link Presentations

[edit Notes on L10a132's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X20,14,11,13 X10,15,5,16 X8,17,9,18 X16,7,17,8 X18,9,19,10 X14,20,15,19 X2536 X4,11,1,12
Gauss code	{1, -9, 2, -10}, {9, -1, 6, -5, 7, -4}, {10, -2, 3, -8, 4, -6, 5, -7, 8, -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) t(3)^2 t(2)^2-2 t(3)^2 t(2)^2-2 t(1) t(3) t(2)^2+2 t(3) t(2)^2-2 t(1) t(3)^2 t(2)+2 t(3)^2 t(2)-2 t(1) t(2)+4 t(1) t(3) t(2)-4 t(3) t(2)+2 t(2)+2 t(1)-2 t(1) t(3)+2 t(3)-1}{\sqrt{t(1)} t(2) t(3)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-10} -2 q^{-9} +5 q^{-8} -7 q^{-7} +10 q^{-6} -9 q^{-5} +10 q^{-4} -7 q^{-3} +5 q^{-2} -3 q^{-1} +1 }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{10} z^{-2} +a^{10}-3 a^8 z^2-2 a^8 z^{-2} -6 a^8+3 a^6 z^4+8 a^6 z^2+a^6 z^{-2} +5 a^6-a^4 z^6-3 a^4 z^4-2 a^4 z^2+a^2 z^4+2 a^2 z^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{12}-2 z^2 a^{12}+a^{12}+2 z^5 a^{11}-2 z^3 a^{11}+3 z^6 a^{10}-3 z^4 a^{10}+3 z^2 a^{10}+a^{10} z^{-2} -3 a^{10}+3 z^7 a^9-z^5 a^9-3 z^3 a^9+6 z a^9-2 a^9 z^{-1} +2 z^8 a^8+3 z^6 a^8-12 z^4 a^8+15 z^2 a^8+2 a^8 z^{-2} -8 a^8+z^9 a^7+3 z^7 a^7-5 z^5 a^7-3 z^3 a^7+6 z a^7-2 a^7 z^{-1} +5 z^8 a^6-9 z^6 a^6+6 z^2 a^6+a^6 z^{-2} -5 a^6+z^9 a^5+3 z^7 a^5-12 z^5 a^5+6 z^3 a^5+3 z^8 a^4-8 z^6 a^4+5 z^4 a^4-2 z^2 a^4+3 z^7 a^3-10 z^5 a^3+8 z^3 a^3+z^6 a^2-3 z^4 a^2+2 z^2 a^2 }[/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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                     1 1 -1                   2   -2 -3                 3 1   2 -5               5 3     -2 -7             5 2       3 -9           4 5         1 -11         6 5           1 -13       2 5             3 -15     3 5               -2 -17   1 4                 3 -19   1                   -1 -21 1                     1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/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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