L11n458

L11n457

L11n459

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Link L11n458.

A graph, L11n458.

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

Link Presentations

[edit Notes on L11n458's Link Presentations]

Planar diagram presentation	X6172 X3,13,4,12 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 X11,1,12,4
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{t(1) t(4)^2 t(3)^2-t(4)^2 t(3)^2+t(1) t(3)^2-t(1) t(2) t(3)^2-2 t(1) t(4) t(3)^2+2 t(1) t(2) t(4) t(3)^2+t(4) t(3)^2-t(1) t(4)^2 t(3)-2 t(2) t(4)^2 t(3)+2 t(4)^2 t(3)-2 t(1) t(3)+2 t(1) t(2) t(3)-t(2) t(3)+3 t(1) t(4) t(3)-3 t(1) t(2) t(4) t(3)+3 t(2) t(4) t(3)-3 t(4) t(3)+t(2) t(4)^2-t(4)^2-t(1) t(2)+t(2)+t(1) t(2) t(4)-2 t(2) t(4)+2 t(4)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 11 q^{9/2}-11 q^{7/2}+5 q^{5/2}-3 q^{3/2}+q^{21/2}-4 q^{19/2}+7 q^{17/2}-11 q^{15/2}+12 q^{13/2}-15 q^{11/2} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^{-9} - a^{-9} z^{-3} - a^{-9} z^{-1} -z^5 a^{-7} +3 a^{-7} z^{-3} +4 z a^{-7} +6 a^{-7} z^{-1} -2 z^5 a^{-5} -6 z^3 a^{-5} -3 a^{-5} z^{-3} -10 z a^{-5} -9 a^{-5} z^{-1} +3 z^3 a^{-3} + a^{-3} z^{-3} +6 z a^{-3} +4 a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{-12} -2 z^4 a^{-12} +z^2 a^{-12} +4 z^7 a^{-11} -11 z^5 a^{-11} +8 z^3 a^{-11} -2 z a^{-11} +5 z^8 a^{-10} -11 z^6 a^{-10} +2 z^4 a^{-10} +2 z^2 a^{-10} +2 z^9 a^{-9} +7 z^7 a^{-9} -33 z^5 a^{-9} +32 z^3 a^{-9} - a^{-9} z^{-3} -16 z a^{-9} +5 a^{-9} z^{-1} +11 z^8 a^{-8} -25 z^6 a^{-8} +9 z^4 a^{-8} +10 z^2 a^{-8} +3 a^{-8} z^{-2} -10 a^{-8} +2 z^9 a^{-7} +10 z^7 a^{-7} -41 z^5 a^{-7} +52 z^3 a^{-7} -3 a^{-7} z^{-3} -31 z a^{-7} +12 a^{-7} z^{-1} +6 z^8 a^{-6} -10 z^6 a^{-6} +2 z^4 a^{-6} +18 z^2 a^{-6} +6 a^{-6} z^{-2} -19 a^{-6} +7 z^7 a^{-5} -19 z^5 a^{-5} +34 z^3 a^{-5} -3 a^{-5} z^{-3} -27 z a^{-5} +12 a^{-5} z^{-1} +3 z^6 a^{-4} -3 z^4 a^{-4} +9 z^2 a^{-4} +3 a^{-4} z^{-2} -10 a^{-4} +6 z^3 a^{-3} - a^{-3} z^{-3} -10 z a^{-3} +5 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   \ 0 1 2 3 4 5 6 7 8 9 χ 22                   1 -1 20                 3   3 18               4 1   -3 16             7 3     4 14           7 6       -1 12         8 5         3 10       5 9           4 8     6 6             0 6   1 7               6 4 2 4                 -2 2 3                   3

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ i=4 }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=9 }[/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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