L11n362

L11n361

L11n363

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Link Presentations

[edit Notes on L11n362's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X18,6,19,5 X8493 X9,21,10,20 X19,11,20,10 X14,18,15,17 X22,16,17,15 X16,22,5,21 X2,12,3,11
Gauss code	{1, -11, 5, -3}, {8, -4, -7, 6, 10, -9}, {4, -1, 2, -5, -6, 7, 11, -2, 3, -8, 9, -10}

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)-1) (t(3)-1)^2 \left(t(3) t(2)^2+t(3)^2 t(2)-t(3) t(2)+t(2)+t(3)\right)}{\sqrt{t(1)} t(2) t(3)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 3 q^7-5 q^6+10 q^5-12 q^4+14 q^3-13 q^2- q^{-2} +11 q+4 q^{-1} -7 }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^6 a^{-2} +z^6 a^{-4} +2 z^4 a^{-2} +3 z^4 a^{-4} -z^4 a^{-6} -z^4+4 z^2 a^{-4} -3 z^2 a^{-6} -z^2- a^{-2} +3 a^{-4} -4 a^{-6} + a^{-8} +1+ a^{-4} z^{-2} -2 a^{-6} z^{-2} + a^{-8} z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 6 z^4 a^{-8} -14 z^2 a^{-8} - a^{-8} z^{-2} +8 a^{-8} +3 z^7 a^{-7} -6 z^5 a^{-7} +10 z^3 a^{-7} -10 z a^{-7} +2 a^{-7} z^{-1} +5 z^8 a^{-6} -16 z^6 a^{-6} +33 z^4 a^{-6} -33 z^2 a^{-6} -2 a^{-6} z^{-2} +15 a^{-6} +2 z^9 a^{-5} +3 z^7 a^{-5} -13 z^5 a^{-5} +20 z^3 a^{-5} -12 z a^{-5} +2 a^{-5} z^{-1} +10 z^8 a^{-4} -25 z^6 a^{-4} +33 z^4 a^{-4} -24 z^2 a^{-4} - a^{-4} z^{-2} +9 a^{-4} +2 z^9 a^{-3} +6 z^7 a^{-3} -18 z^5 a^{-3} +14 z^3 a^{-3} -3 z a^{-3} +5 z^8 a^{-2} -5 z^6 a^{-2} -z^4 a^{-2} -3 z^2 a^{-2} +2 a^{-2} +6 z^7 a^{-1} +a z^5-10 z^5 a^{-1} -a z^3+3 z^3 a^{-1} -z a^{-1} +4 z^6-7 z^4+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   \ -3 -2 -1 0 1 2 3 4 5 6 χ 15                   3 3 13                 4 2 -2 11               6 1   5 9             6 4     -2 7           8 6       2 5         5 6         1 3       6 8           -2 1     3 7             4 -1   1 4               -3 -3   3                 3 -5 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=-3 }[/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} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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