L11a238

L11a237

L11a239

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Link Presentations

[edit Notes on L11a238's Link Presentations]

Planar diagram presentation	X8192 X12,4,13,3 X22,12,7,11 X16,9,17,10 X14,22,15,21 X10,15,11,16 X18,6,19,5 X20,18,21,17 X2738 X4,14,5,13 X6,20,1,19
Gauss code	{1, -9, 2, -10, 7, -11}, {9, -1, 4, -6, 3, -2, 10, -5, 6, -4, 8, -7, 11, -8, 5, -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)^2 t(2)^4-2 t(1) t(2)^4+t(2)^4-5 t(1)^2 t(2)^3+8 t(1) t(2)^3-4 t(2)^3+7 t(1)^2 t(2)^2-11 t(1) t(2)^2+7 t(2)^2-4 t(1)^2 t(2)+8 t(1) t(2)-5 t(2)+t(1)^2-2 t(1)+1}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{17/2}-4 q^{15/2}+9 q^{13/2}-15 q^{11/2}+19 q^{9/2}-22 q^{7/2}+21 q^{5/2}-18 q^{3/2}+13 \sqrt{q}-\frac{8}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{1}{q^{5/2}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^{-7} +z a^{-7} -2 z^5 a^{-5} -4 z^3 a^{-5} -2 z a^{-5} - a^{-5} z^{-1} +z^7 a^{-3} +3 z^5 a^{-3} +4 z^3 a^{-3} +4 z a^{-3} +2 a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-5 z^3 a^{-1} +2 a z-4 z a^{-1} +a z^{-1} -2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-10} +4 z^5 a^{-9} -z^3 a^{-9} +9 z^6 a^{-8} -7 z^4 a^{-8} +2 z^2 a^{-8} +14 z^7 a^{-7} -20 z^5 a^{-7} +12 z^3 a^{-7} -3 z a^{-7} +14 z^8 a^{-6} -21 z^6 a^{-6} +8 z^4 a^{-6} -z^2 a^{-6} +8 z^9 a^{-5} +z^7 a^{-5} -33 z^5 a^{-5} +29 z^3 a^{-5} -9 z a^{-5} + a^{-5} z^{-1} +2 z^{10} a^{-4} +18 z^8 a^{-4} -55 z^6 a^{-4} +39 z^4 a^{-4} -8 z^2 a^{-4} +12 z^9 a^{-3} -22 z^7 a^{-3} -10 z^5 a^{-3} +28 z^3 a^{-3} -13 z a^{-3} +2 a^{-3} z^{-1} +2 z^{10} a^{-2} +7 z^8 a^{-2} -35 z^6 a^{-2} +34 z^4 a^{-2} -9 z^2 a^{-2} + a^{-2} +4 z^9 a^{-1} +a z^7-8 z^7 a^{-1} -4 a z^5-5 z^5 a^{-1} +6 a z^3+18 z^3 a^{-1} -4 a z-11 z a^{-1} +a z^{-1} +2 a^{-1} z^{-1} +3 z^8-10 z^6+11 z^4-4 z^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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 18                       1 -1 16                     3   3 14                   6 1   -5 12                 9 3     6 10               10 6       -4 8             12 9         3 6           10 11           1 4         8 11             -3 2       6 11               5 0     2 7                 -5 -2   1 6                   5 -4   2                     -2 -6 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ i=4 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=7 }[/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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