L11a468

L11a467

L11a469

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Link Presentations

[edit Notes on L11a468's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X18,11,19,12 X16,8,17,7 X8,16,9,15 X22,17,15,18 X20,13,21,14 X12,19,13,20 X14,21,5,22 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {5, -4, 6, -3, 8, -7, 9, -6}, {10, -1, 4, -5, 11, -2, 3, -8, 7, -9}

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(3)-1) \left(t(2)^2 t(3)^3-t(2) t(3)^3+2 t(1) t(2)^2 t(3)^2-2 t(2)^2 t(3)^2-2 t(1) t(2) t(3)^2+2 t(2) t(3)^2-t(3)^2-t(1) t(2)^2 t(3)-2 t(1) t(3)+2 t(1) t(2) t(3)-2 t(2) t(3)+2 t(3)+t(1)-t(1) t(2)\right)}{\sqrt{t(1)} t(2) t(3)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 1-2 q^{-1} +5 q^{-2} -8 q^{-3} +12 q^{-4} -13 q^{-5} +15 q^{-6} -12 q^{-7} +10 q^{-8} -6 q^{-9} +3 q^{-10} - q^{-11} }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{10} \left(-z^2\right)-2 a^{10}+3 a^8 z^4+9 a^8 z^2+a^8 z^{-2} +6 a^8-2 a^6 z^6-8 a^6 z^4-11 a^6 z^2-2 a^6 z^{-2} -8 a^6-a^4 z^6-2 a^4 z^4+2 a^4 z^2+a^4 z^{-2} +3 a^4+a^2 z^4+3 a^2 z^2+a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{13}-2 z^3 a^{13}+z a^{13}+3 z^6 a^{12}-6 z^4 a^{12}+3 z^2 a^{12}-a^{12}+4 z^7 a^{11}-5 z^5 a^{11}-2 z^3 a^{11}+z a^{11}+4 z^8 a^{10}-4 z^6 a^{10}-2 z^4 a^{10}+z^2 a^{10}+3 z^9 a^9-2 z^7 a^9-4 z^5 a^9+8 z^3 a^9-3 z a^9+z^{10} a^8+6 z^8 a^8-25 z^6 a^8+42 z^4 a^8-29 z^2 a^8-a^8 z^{-2} +10 a^8+6 z^9 a^7-16 z^7 a^7+14 z^5 a^7+7 z^3 a^7-10 z a^7+2 a^7 z^{-1} +z^{10} a^6+5 z^8 a^6-28 z^6 a^6+51 z^4 a^6-40 z^2 a^6-2 a^6 z^{-2} +14 a^6+3 z^9 a^5-8 z^7 a^5+6 z^5 a^5+2 z^3 a^5-7 z a^5+2 a^5 z^{-1} +3 z^8 a^4-9 z^6 a^4+9 z^4 a^4-9 z^2 a^4-a^4 z^{-2} +5 a^4+2 z^7 a^3-6 z^5 a^3+3 z^3 a^3+z^6 a^2-4 z^4 a^2+4 z^2 a^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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                       1 1 -1                     1   -1 -3                   4 1   3 -5                 5 2     -3 -7               7 3       4 -9             7 6         -1 -11           8 6           2 -13         5 8             3 -15       5 7               -2 -17     2 6                 4 -19   1 4                   -3 -21   2                     2 -23 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=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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