L11n173

L11n172

L11n174

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Link Presentations

[edit Notes on L11n173's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X12,7,13,8 X15,7,16,22 X14,6,15,5 X6,14,1,13 X21,17,22,16 X18,10,19,9 X20,11,21,12 X4,18,5,17 X2,19,3,20
Gauss code	{1, -11, 2, -10, 5, -6}, {3, -1, 8, -2, 9, -3, 6, -5, -4, 7, 10, -8, 11, -9, -7, 4}

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)^2 t(2)^4-t(1) t(2)^4-3 t(1)^2 t(2)^3+5 t(1) t(2)^3-2 t(2)^3+4 t(1)^2 t(2)^2-9 t(1) t(2)^2+4 t(2)^2-2 t(1)^2 t(2)+5 t(1) t(2)-3 t(2)-t(1)+1}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -6 q^{9/2}+9 q^{7/2}-\frac{1}{q^{7/2}}-13 q^{5/2}+\frac{4}{q^{5/2}}+14 q^{3/2}-\frac{8}{q^{3/2}}+2 q^{11/2}-14 \sqrt{q}+\frac{11}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z a^{-5} +z^5 a^{-3} +z^3 a^{-3} -z a^{-3} - a^{-3} z^{-1} -z^7 a^{-1} +a z^5-4 z^5 a^{-1} +2 a z^3-5 z^3 a^{-1} +a z+ a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 z^9 a^{-1} -2 z^9 a^{-3} -9 z^8 a^{-2} -3 z^8 a^{-4} -6 z^8-7 a z^7-7 z^7 a^{-1} -z^7 a^{-3} -z^7 a^{-5} -4 a^2 z^6+15 z^6 a^{-2} +3 z^6 a^{-4} +8 z^6-a^3 z^5+13 a z^5+17 z^5 a^{-1} -2 z^5 a^{-3} -5 z^5 a^{-5} +6 a^2 z^4-9 z^4 a^{-2} -5 z^4 a^{-4} -3 z^4 a^{-6} -z^4+a^3 z^3-6 a z^3-7 z^3 a^{-1} +9 z^3 a^{-3} +9 z^3 a^{-5} -a^2 z^2+2 z^2 a^{-2} +4 z^2 a^{-4} +3 z^2 a^{-6} +a z-2 z a^{-1} -6 z a^{-3} -3 z a^{-5} - a^{-2} + a^{-1} z^{-1} + a^{-3} z^{-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 χ 12                   2 -2 10                 4   4 8               5 2   -3 6             8 4     4 4           7 6       -1 2         7 7         0 0       5 8           3 -2     3 6             -3 -4   1 5               4 -6   3                 -3 -8 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{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{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

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