L11n240

L11n239

L11n241

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Link Presentations

[edit Notes on L11n240's Link Presentations]

Planar diagram presentation	X12,1,13,2 X16,8,17,7 X5,1,6,10 X3746 X9,5,10,4 X13,18,14,19 X19,22,20,11 X15,21,16,20 X21,15,22,14 X2,11,3,12 X8,18,9,17
Gauss code	{1, -10, -4, 5, -3, 4, 2, -11, -5, 3}, {10, -1, -6, 9, -8, -2, 11, 6, -7, 8, -9, 7}

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) (t(2)-1) \left(t(2) t(1)^2-t(2)^2 t(1)+2 t(2) t(1)-t(1)+t(2)\right)}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{15/2}-2 q^{13/2}+3 q^{11/2}-4 q^{9/2}+3 q^{7/2}-3 q^{5/2}+2 q^{3/2}-\sqrt{q}-\frac{1}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{1}{q^{5/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z a^{-7} + a^{-7} z^{-1} -2 z^3 a^{-5} -5 z a^{-5} -4 a^{-5} z^{-1} +z^5 a^{-3} +5 z^3 a^{-3} +10 z a^{-3} +6 a^{-3} z^{-1} -z^5 a^{-1} +a z^3-6 z^3 a^{-1} +3 a z-9 z a^{-1} +2 a z^{-1} -5 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{-8} -4 z^4 a^{-8} +4 z^2 a^{-8} - a^{-8} +2 z^7 a^{-7} -8 z^5 a^{-7} +8 z^3 a^{-7} -4 z a^{-7} + a^{-7} z^{-1} +z^8 a^{-6} -z^6 a^{-6} -8 z^4 a^{-6} +9 z^2 a^{-6} -3 a^{-6} +4 z^7 a^{-5} -18 z^5 a^{-5} +24 z^3 a^{-5} -16 z a^{-5} +4 a^{-5} z^{-1} +z^8 a^{-4} -3 z^6 a^{-4} -2 z^4 a^{-4} +7 z^2 a^{-4} -3 a^{-4} +3 z^7 a^{-3} -20 z^5 a^{-3} +39 z^3 a^{-3} -27 z a^{-3} +6 a^{-3} z^{-1} +z^8 a^{-2} -7 z^6 a^{-2} +10 z^4 a^{-2} - a^{-2} +a z^7+2 z^7 a^{-1} -6 a z^5-16 z^5 a^{-1} +10 a z^3+33 z^3 a^{-1} -7 a z-22 z a^{-1} +2 a z^{-1} +5 a^{-1} z^{-1} +z^8-6 z^6+8 z^4-2 z^2-1 }[/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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 16                       1 -1 14                     1   1 12                   2 1   -1 10                 2 1     1 8             1 2 2       1 6             2 2         0 4         1 2 2           1 2       1 2 2             -1 0       1 3               2 -2   1 1                   0 -4                         0 -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=0 }[/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 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/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}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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