L11a263

L11a262

L11a264

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Link Presentations

[edit Notes on L11a263's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,4,13,3 X22,12,9,11 X14,6,15,5 X18,8,19,7 X20,18,21,17 X16,22,17,21 X2,9,3,10 X4,14,5,13 X6,16,7,15 X8,20,1,19
Gauss code	{1, -8, 2, -9, 4, -10, 5, -11}, {8, -1, 3, -2, 9, -4, 10, -7, 6, -5, 11, -6, 7, -3}

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{2 u^3 v^3-3 u^3 v^2+u^3 v-3 u^2 v^3+5 u^2 v^2-4 u^2 v+u^2+u v^3-4 u v^2+5 u v-3 u+v^2-3 v+2}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{23/2}-3 q^{21/2}+5 q^{19/2}-8 q^{17/2}+11 q^{15/2}-12 q^{13/2}+11 q^{11/2}-10 q^{9/2}+7 q^{7/2}-5 q^{5/2}+2 q^{3/2}-\sqrt{q} }[/math] (db)
Signature	5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-5} -z^7 a^{-7} +z^5 a^{-3} -4 z^5 a^{-5} -4 z^5 a^{-7} +z^5 a^{-9} +4 z^3 a^{-3} -4 z^3 a^{-5} -4 z^3 a^{-7} +3 z^3 a^{-9} +4 z a^{-3} -2 z a^{-5} -z a^{-7} +z a^{-9} + a^{-3} z^{-1} - a^{-5} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-14} -z^2 a^{-14} +3 z^5 a^{-13} -4 z^3 a^{-13} +z a^{-13} +4 z^6 a^{-12} -4 z^4 a^{-12} +z^2 a^{-12} +4 z^7 a^{-11} -2 z^5 a^{-11} -3 z^3 a^{-11} +2 z a^{-11} +4 z^8 a^{-10} -5 z^6 a^{-10} +2 z^4 a^{-10} +z^2 a^{-10} +3 z^9 a^{-9} -5 z^7 a^{-9} +5 z^5 a^{-9} -5 z^3 a^{-9} +2 z a^{-9} +z^{10} a^{-8} +3 z^8 a^{-8} -13 z^6 a^{-8} +11 z^4 a^{-8} -3 z^2 a^{-8} +5 z^9 a^{-7} -15 z^7 a^{-7} +12 z^5 a^{-7} -4 z^3 a^{-7} +z a^{-7} +z^{10} a^{-6} +z^8 a^{-6} -12 z^6 a^{-6} +12 z^4 a^{-6} -3 z^2 a^{-6} +2 z^9 a^{-5} -5 z^7 a^{-5} -3 z^5 a^{-5} +10 z^3 a^{-5} -5 z a^{-5} + a^{-5} z^{-1} +2 z^8 a^{-4} -8 z^6 a^{-4} +8 z^4 a^{-4} -z^2 a^{-4} - a^{-4} +z^7 a^{-3} -5 z^5 a^{-3} +8 z^3 a^{-3} -5 z a^{-3} + 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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 24                       1 -1 22                     2   2 20                   3 1   -2 18                 5 2     3 16               6 3       -3 14             6 5         1 12           6 7           1 10         4 5             -1 8       3 6               3 6     2 4                 -2 4   1 4                   3 2   1                     -1 0 1                       1

Integral Khovanov Homology (db, data source)

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