L11n459

L11n458

L11n459

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Link L11n459.

A graph, L11n459.

A part of a knot and a part of a graph.

Link Presentations

[edit Notes on L11n459's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X7,21,8,20 X19,5,20,10 X13,19,14,22 X21,11,22,18 X17,15,18,14 X9,17,10,16 X15,9,16,8 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {-4, 3, -6, 5}, {10, -1, -3, 9, -8, 4}, {11, -2, -5, 7, -9, 8, -7, 6}

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{(w x-1) (u w x+u (-w)-u x+2 u-2 v w x+v w+v x-v)}{\sqrt{u} \sqrt{v} w x} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -2 q^{9/2}-q^{5/2}-\frac{1}{q^{5/2}}-2 q^{3/2}+\frac{1}{q^{3/2}}+q^{15/2}-2 q^{13/2}+q^{11/2}-\frac{3}{\sqrt{q}} }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z a^{-7} -z^3 a^{-5} - a^{-5} z^{-3} -2 z a^{-5} -2 a^{-5} z^{-1} +z^3 a^{-3} +3 a^{-3} z^{-3} +6 z a^{-3} +7 a^{-3} z^{-1} -z^5 a^{-1} +a z^3-5 z^3 a^{-1} +a z^{-3} -3 a^{-1} z^{-3} +3 a z-8 z a^{-1} +3 a z^{-1} -8 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^9 a^{-1} -z^9 a^{-3} -3 z^8 a^{-2} -3 z^8 a^{-4} -z^8 a^{-6} -z^8-a z^7+4 z^7 a^{-1} +3 z^7 a^{-3} -4 z^7 a^{-5} -2 z^7 a^{-7} +17 z^6 a^{-2} +18 z^6 a^{-4} +4 z^6 a^{-6} -z^6 a^{-8} +4 z^6+6 a z^5+3 z^5 a^{-1} +12 z^5 a^{-3} +25 z^5 a^{-5} +10 z^5 a^{-7} -18 z^4 a^{-2} -23 z^4 a^{-4} +4 z^4 a^{-8} +z^4-11 a z^3-19 z^3 a^{-1} -35 z^3 a^{-3} -39 z^3 a^{-5} -12 z^3 a^{-7} -10 z^2 a^{-2} -2 z^2 a^{-6} -2 z^2 a^{-8} -10 z^2+10 a z+23 z a^{-1} +27 z a^{-3} +20 z a^{-5} +6 z a^{-7} +19 a^{-2} +10 a^{-4} +10-5 a z^{-1} -12 a^{-1} z^{-1} -12 a^{-3} z^{-1} -5 a^{-5} z^{-1} -6 a^{-2} z^{-2} -3 a^{-4} z^{-2} -3 z^{-2} +a z^{-3} +3 a^{-1} z^{-3} +3 a^{-3} z^{-3} + a^{-5} z^{-3} }[/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 6 7 χ 16                       1 -1 14                     1   1 12                 1 1 1   1 10               1 3 1     1 8             2 2 2       2 6           3 6 2         1 4         1 2 4           3 2       2 5 1             2 0     1 1 3               3 -2     2                   2 -4 1 1                     0 -6 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{ i=4 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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} }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=7 }[/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

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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