L11n298

L11n297

L11n299

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Link Presentations

[edit Notes on L11n298's Link Presentations]

Planar diagram presentation	X6172 X3,13,4,12 X13,19,14,18 X17,11,18,22 X7,17,8,16 X21,8,22,9 X9,14,10,15 X15,20,16,21 X19,5,20,10 X2536 X11,1,12,4
Gauss code	{1, -10, -2, 11}, {10, -1, -5, 6, -7, 9}, {-11, 2, -3, 7, -8, 5, -4, 3, -9, 8, -6, 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{-u v^2 w^3+3 u v^2 w^2-3 u v^2 w+u v^2+2 u v w^3-5 u v w^2+3 u v w-u v-u w^3+u w^2-v^2 w^2+v^2 w+v w^4-3 v w^3+5 v w^2-2 v w-w^4+3 w^3-3 w^2+w}{\sqrt{u} v w^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^6-4 q^5+8 q^4-11 q^3-2 q^{-3} +14 q^2+7 q^{-2} -14 q-9 q^{-1} +14 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^4 a^{-4} +z^2 a^{-4} + a^{-4} -z^6 a^{-2} -3 z^4 a^{-2} -2 a^2 z^2-5 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -2 a^{-2} +3 z^4+5 z^2-2 z^{-2} +1 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 3 z^9 a^{-1} +3 z^9 a^{-3} +14 z^8 a^{-2} +6 z^8 a^{-4} +8 z^8+6 a z^7+5 z^7 a^{-1} +3 z^7 a^{-3} +4 z^7 a^{-5} +a^2 z^6-40 z^6 a^{-2} -15 z^6 a^{-4} +z^6 a^{-6} -23 z^6-11 a z^5-28 z^5 a^{-1} -27 z^5 a^{-3} -10 z^5 a^{-5} +8 a^2 z^4+39 z^4 a^{-2} +8 z^4 a^{-4} -2 z^4 a^{-6} +37 z^4+3 a^3 z^3+14 a z^3+30 z^3 a^{-1} +25 z^3 a^{-3} +6 z^3 a^{-5} -9 a^2 z^2-20 z^2 a^{-2} -3 z^2 a^{-4} +z^2 a^{-6} -25 z^2-a^3 z-3 a z-7 z a^{-1} -7 z a^{-3} -2 z a^{-5} +4 a^{-2} +2 a^{-4} +3-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + 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   \ -3 -2 -1 0 1 2 3 4 5 6 χ 13                   1 1 11                 3   -3 9               5 1   4 7             6 3     -3 5           8 5       3 3         7 7         0 1       7 7           0 -1     4 9             5 -3   3 5               -2 -5   5                 5 -7 2                   -2

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=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} }[/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}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/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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