L11n334

L11n333

L11n335

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Link Presentations

[edit Notes on L11n334's Link Presentations]

Planar diagram presentation	X6172 X11,16,12,17 X8493 X2,18,3,17 X5,14,6,15 X18,7,19,8 X15,12,16,5 X13,20,14,21 X9,13,10,22 X21,11,22,10 X4,19,1,20
Gauss code	{1, -4, 3, -11}, {-5, -1, 6, -3, -9, 10, -2, 7}, {-8, 5, -7, 2, 4, -6, 11, 8, -10, 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{ 0 }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^3-2 q^2+2 q-1+2 q^{-1} + q^{-2} +2 q^{-4} -2 q^{-5} +2 q^{-6} - q^{-7} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^2 a^6-a^6+z^4 a^4+3 z^2 a^4+a^4 z^{-2} +2 a^4-2 a^2 z^{-2} -a^2-z^4-3 z^2+ z^{-2} -1+z^2 a^{-2} + a^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^7-5 a^7 z^5+6 a^7 z^3-2 a^7 z+2 a^6 z^8-11 a^6 z^6+16 a^6 z^4-9 a^6 z^2+3 a^6+a^5 z^9-4 a^5 z^7-2 a^5 z^5+12 a^5 z^3-7 a^5 z+3 a^4 z^8-20 a^4 z^6+37 a^4 z^4-27 a^4 z^2-a^4 z^{-2} +9 a^4+a^3 z^9-5 a^3 z^7+13 a^3 z^3-10 a^3 z+2 a^3 z^{-1} +2 a^2 z^8-14 a^2 z^6+z^6 a^{-2} +26 a^2 z^4-4 z^4 a^{-2} -20 a^2 z^2+3 z^2 a^{-2} -2 a^2 z^{-2} +7 a^2- a^{-2} +2 a z^7+2 z^7 a^{-1} -12 a z^5-9 z^5 a^{-1} +15 a z^3+8 z^3 a^{-1} -6 a z-z a^{-1} +2 a z^{-1} +z^8-4 z^6+z^4+z^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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                     1   -1 3                   1 1   0 1               2 2 1     1 -1             1 3 1       1 -3           2 2 3         3 -5         2 4 1           1 -7       1 1 2             2 -9     1 2 1               0 -11   1 1                   0 -13   1                     1 -15 1                       -1

Integral Khovanov Homology (db, data source)

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