L11n297

L11n296

L11n298

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Link Presentations

[edit Notes on L11n297's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X15,20,16,21 X14,8,15,7 X21,10,22,5 X18,11,19,12 X9,17,10,16 X22,17,11,18 X8,19,9,20 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -9, -7, 5}, {6, -2, 11, -4, -3, 7, 8, -6, 9, 3, -5, -8}

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-t(3)^2 t(2)^2-t(1) t(3) t(2)^2+2 t(3) t(2)^2-t(2)^2-t(1) t(3)^2 t(2)+2 t(1) t(3) t(2)-2 t(3) t(2)+t(2)+t(1) t(3)^2+t(1)-2 t(1) t(3)+t(3)-1}{\sqrt{t(1)} t(2) t(3)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^2-2 q+5-4 q^{-1} +7 q^{-2} -5 q^{-3} +5 q^{-4} -4 q^{-5} +2 q^{-6} - q^{-7} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^2 a^6-a^6 z^{-2} -2 a^6+2 z^4 a^4+7 z^2 a^4+4 a^4 z^{-2} +8 a^4-z^6 a^2-5 z^4 a^2-10 z^2 a^2-5 a^2 z^{-2} -10 a^2+z^4+3 z^2+2 z^{-2} +4 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^5 z^9+a^3 z^9+2 a^6 z^8+5 a^4 z^8+3 a^2 z^8+a^7 z^7+a^3 z^7+2 a z^7-9 a^6 z^6-22 a^4 z^6-13 a^2 z^6-5 a^7 z^5-16 a^5 z^5-17 a^3 z^5-6 a z^5+11 a^6 z^4+30 a^4 z^4+24 a^2 z^4+5 z^4+8 a^7 z^3+29 a^5 z^3+29 a^3 z^3+10 a z^3+2 z^3 a^{-1} -5 a^6 z^2-22 a^4 z^2-27 a^2 z^2+z^2 a^{-2} -9 z^2-5 a^7 z-18 a^5 z-24 a^3 z-11 a z+3 a^6+12 a^4+15 a^2+7+a^7 z^{-1} +5 a^5 z^{-1} +9 a^3 z^{-1} +5 a z^{-1} -a^6 z^{-2} -4 a^4 z^{-2} -5 a^2 z^{-2} -2 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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 5                   1 1 3                 2 1 -1 1               3     3 -1             3 4     1 -3           4 1       3 -5         2 4         2 -7       3 3           0 -9     1 2             1 -11   1 3               -2 -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=-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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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

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