L11a194

L11a193

L11a195

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Link Presentations

[edit Notes on L11a194's Link Presentations]

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

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