L11n425

L11n424

L11n426

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Link Presentations

[edit Notes on L11n425's Link Presentations]

Planar diagram presentation	X8192 X16,8,17,7 X5,14,6,15 X3,10,4,11 X13,4,14,5 X17,2,18,3 X9,19,10,18 X21,7,22,12 X11,13,12,22 X20,16,21,15 X6,19,1,20
Gauss code	{1, 6, -4, 5, -3, -11}, {2, -1, -7, 4, -9, 8}, {-5, 3, 10, -2, -6, 7, 11, -10, -8, 9}

A Braid Representative	{{{braid_table}}}
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(2)^2 t(3)^3+t(1) t(2) t(3)^3-t(2) t(3)^3+t(1) t(2)^2 t(3)^2-t(2)^2 t(3)^2-2 t(1) t(2) t(3)^2+t(2) t(3)^2+t(1)^2 t(3)-t(1) t(3)-t(1)^2 t(2) t(3)+2 t(1) t(2) t(3)+t(1)+t(1)^2 t(2)-t(1) t(2)}{t(1) t(2) t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 2 q^2-3 q+5-5 q^{-1} +6 q^{-2} -4 q^{-3} +4 q^{-4} -2 q^{-5} + q^{-6} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^2 z^6+a^4 z^4-5 a^2 z^4+z^4+3 a^4 z^2-9 a^2 z^2+2 z^2+3 a^4-6 a^2+ a^{-2} +2+a^4 z^{-2} -2 a^2 z^{-2} + z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^3 z^9+a z^9+3 a^4 z^8+4 a^2 z^8+z^8+2 a^5 z^7-a^3 z^7-3 a z^7+a^6 z^6-14 a^4 z^6-19 a^2 z^6-4 z^6-7 a^5 z^5-8 a^3 z^5+z^5 a^{-1} -4 a^6 z^4+22 a^4 z^4+33 a^2 z^4+7 z^4+4 a^5 z^3+14 a^3 z^3+10 a z^3+3 a^6 z^2-19 a^4 z^2-27 a^2 z^2+2 z^2 a^{-2} -3 z^2-9 a^3 z-9 a z+7 a^4+11 a^2-2 a^{-2} +3+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-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   \ -6 -5 -4 -3 -2 -1 0 1 2 χ 5                 2 2 3               2 1 -1 1             3 1   2 -1           4 4     0 -3         2 1       1 -5       2 4         2 -7     2 2           0 -9     2             2 -11 1 2               -1 -13 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=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/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=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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}\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}^{2} }[/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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