L11a218

L11a217

L11a219

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Link Presentations

[edit Notes on L11a218's Link Presentations]

Planar diagram presentation	X8192 X12,4,13,3 X20,10,21,9 X22,12,7,11 X10,22,11,21 X16,6,17,5 X18,16,19,15 X14,20,15,19 X2738 X4,14,5,13 X6,18,1,17
Gauss code	{1, -9, 2, -10, 6, -11}, {9, -1, 3, -5, 4, -2, 10, -8, 7, -6, 11, -7, 8, -3, 5, -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{2 u^2 v^4-3 u^2 v^3+3 u^2 v^2-3 u^2 v+u^2-3 u v^4+5 u v^3-5 u v^2+5 u v-3 u+v^4-3 v^3+3 v^2-3 v+2}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{23/2}-3 q^{21/2}+6 q^{19/2}-10 q^{17/2}+13 q^{15/2}-14 q^{13/2}+14 q^{11/2}-12 q^{9/2}+8 q^{7/2}-6 q^{5/2}+2 q^{3/2}-\sqrt{q} }[/math] (db)
Signature	5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^5 a^{-9} +3 z^3 a^{-9} +2 z a^{-9} -z^7 a^{-7} -4 z^5 a^{-7} -5 z^3 a^{-7} -2 z a^{-7} + a^{-7} z^{-1} -z^7 a^{-5} -4 z^5 a^{-5} -5 z^3 a^{-5} -4 z a^{-5} -3 a^{-5} z^{-1} +z^5 a^{-3} +4 z^3 a^{-3} +5 z a^{-3} +2 a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^{10} a^{-6} -z^{10} a^{-8} -2 z^9 a^{-5} -6 z^9 a^{-7} -4 z^9 a^{-9} -2 z^8 a^{-4} -3 z^8 a^{-6} -8 z^8 a^{-8} -7 z^8 a^{-10} -z^7 a^{-3} +3 z^7 a^{-5} +14 z^7 a^{-7} +3 z^7 a^{-9} -7 z^7 a^{-11} +7 z^6 a^{-4} +16 z^6 a^{-6} +29 z^6 a^{-8} +15 z^6 a^{-10} -5 z^6 a^{-12} +5 z^5 a^{-3} +9 z^5 a^{-5} -3 z^5 a^{-7} +8 z^5 a^{-9} +12 z^5 a^{-11} -3 z^5 a^{-13} -5 z^4 a^{-4} -11 z^4 a^{-6} -28 z^4 a^{-8} -16 z^4 a^{-10} +5 z^4 a^{-12} -z^4 a^{-14} -9 z^3 a^{-3} -17 z^3 a^{-5} -5 z^3 a^{-7} -10 z^3 a^{-9} -10 z^3 a^{-11} +3 z^3 a^{-13} -3 z^2 a^{-4} -2 z^2 a^{-6} +9 z^2 a^{-8} +6 z^2 a^{-10} -z^2 a^{-12} +z^2 a^{-14} +7 z a^{-3} +10 z a^{-5} +3 z a^{-7} +2 z a^{-9} +2 z a^{-11} +3 a^{-4} +3 a^{-6} + a^{-8} -2 a^{-3} z^{-1} -3 a^{-5} z^{-1} - a^{-7} 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                   4 1   -3 18                 6 2     4 16               7 4       -3 14             7 6         1 12           7 7           0 10         5 7             -2 8       4 8               4 6     2 4                 -2 4   1 5                   4 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}^{5}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\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^{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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