L11n421

L11n420

L11n422

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Link Presentations

[edit Notes on L11n421's Link Presentations]

Planar diagram presentation	X8192 X16,8,17,7 X3,10,4,11 X17,2,18,3 X9,19,10,18 X20,12,21,11 X14,6,15,5 X22,15,13,16 X6,14,1,13 X4,19,5,20 X12,22,7,21
Gauss code	{1, 4, -3, -10, 7, -9}, {2, -1, -5, 3, 6, -11}, {9, -7, 8, -2, -4, 5, 10, -6, 11, -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(2)^2 t(3)^3-t(2)^2 t(3)^3-2 t(1) t(2)^2 t(3)^2+t(2)^2 t(3)^2+t(1) t(2) t(3)^2-t(1)^2 t(3)+2 t(1) t(3)-t(1) t(2) t(3)+t(1)^2-t(1)}{t(1) t(2) t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^5-q^4+3 q^3-3 q^2+4 q-3+4 q^{-1} -2 q^{-2} +2 q^{-3} - q^{-4} }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^2 a^{-4} + a^{-4} z^{-2} +3 a^{-4} -a^2 z^4-2 z^4 a^{-2} -3 a^2 z^2-8 z^2 a^{-2} -2 a^{-2} z^{-2} -a^2-8 a^{-2} +z^6+5 z^4+8 z^2+ z^{-2} +6 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^2 a^{-6} - a^{-6} +z^3 a^{-5} +2 z^4 a^{-4} -4 z^2 a^{-4} - a^{-4} z^{-2} +4 a^{-4} +a^3 z^7+z^7 a^{-3} -5 a^3 z^5-4 z^5 a^{-3} +6 a^3 z^3+7 z^3 a^{-3} -a^3 z-6 z a^{-3} +2 a^{-3} z^{-1} +2 a^2 z^8+2 z^8 a^{-2} -11 a^2 z^6-11 z^6 a^{-2} +18 a^2 z^4+23 z^4 a^{-2} -11 a^2 z^2-25 z^2 a^{-2} -2 a^{-2} z^{-2} +2 a^2+12 a^{-2} +a z^9+z^9 a^{-1} -3 a z^7-3 z^7 a^{-1} -3 a z^5-2 z^5 a^{-1} +9 a z^3+9 z^3 a^{-1} -3 a z-8 z a^{-1} +2 a^{-1} z^{-1} +4 z^8-22 z^6+39 z^4-31 z^2- z^{-2} +10 }[/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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 11                   1 1 9                 1 1 0 7               2     2 5             2 2     0 3           2 1       1 1         2 3         1 -1       2 1           1 -3     1 3             2 -5   1 1               0 -7   1                 1 -9 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=3 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/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=-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=0 }[/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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=4 }[/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

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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