L11a179

L11a178

L11a180

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Link Presentations

[edit Notes on L11a179's Link Presentations]

Planar diagram presentation	X8192 X20,9,21,10 X6,21,1,22 X16,8,17,7 X14,6,15,5 X4,14,5,13 X12,18,13,17 X10,4,11,3 X18,12,19,11 X22,16,7,15 X2,20,3,19
Gauss code	{1, -11, 8, -6, 5, -3}, {4, -1, 2, -8, 9, -7, 6, -5, 10, -4, 7, -9, 11, -2, 3, -10}

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{\left(v^2-v+1\right) (u v-u-v+2) (2 u v-u-v+1)}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 24 q^{9/2}-25 q^{7/2}+20 q^{5/2}-16 q^{3/2}+\frac{1}{q^{3/2}}-q^{19/2}+4 q^{17/2}-9 q^{15/2}+16 q^{13/2}-21 q^{11/2}+9 \sqrt{q}-\frac{4}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^5 a^{-7} -2 z^3 a^{-7} -z a^{-7} + a^{-7} z^{-1} +z^7 a^{-5} +3 z^5 a^{-5} +3 z^3 a^{-5} -z a^{-5} -3 a^{-5} z^{-1} +z^7 a^{-3} +3 z^5 a^{-3} +4 z^3 a^{-3} +4 z a^{-3} +2 a^{-3} z^{-1} -z^5 a^{-1} -2 z^3 a^{-1} -z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-11} -z^3 a^{-11} +4 z^6 a^{-10} -5 z^4 a^{-10} +z^2 a^{-10} +8 z^7 a^{-9} -11 z^5 a^{-9} +4 z^3 a^{-9} -z a^{-9} +11 z^8 a^{-8} -19 z^6 a^{-8} +14 z^4 a^{-8} -4 z^2 a^{-8} - a^{-8} +9 z^9 a^{-7} -11 z^7 a^{-7} +2 z^5 a^{-7} +4 z^3 a^{-7} -2 z a^{-7} + a^{-7} z^{-1} +3 z^{10} a^{-6} +14 z^8 a^{-6} -44 z^6 a^{-6} +41 z^4 a^{-6} -9 z^2 a^{-6} -3 a^{-6} +16 z^9 a^{-5} -32 z^7 a^{-5} +17 z^5 a^{-5} +3 z^3 a^{-5} -5 z a^{-5} +3 a^{-5} z^{-1} +3 z^{10} a^{-4} +10 z^8 a^{-4} -36 z^6 a^{-4} +30 z^4 a^{-4} -6 z^2 a^{-4} -3 a^{-4} +7 z^9 a^{-3} -9 z^7 a^{-3} -6 z^5 a^{-3} +10 z^3 a^{-3} -6 z a^{-3} +2 a^{-3} z^{-1} +7 z^8 a^{-2} -14 z^6 a^{-2} +6 z^4 a^{-2} -z^2 a^{-2} +4 z^7 a^{-1} -9 z^5 a^{-1} +6 z^3 a^{-1} -2 z a^{-1} +z^6-2 z^4+z^2 }[/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   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 20                       1 1 18                     3   -3 16                   6 1   5 14                 10 3     -7 12               11 6       5 10             13 10         -3 8           12 11           1 6         9 14             5 4       7 11               -4 2     3 10                 7 0   1 6                   -5 -2   3                     3 -4 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=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=8 }[/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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