L11a166

L11a165

L11a167

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Link Presentations

[edit Notes on L11a166's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X6718 X16,13,17,14 X14,6,15,5 X4,16,5,15 X20,11,21,12 X22,18,7,17 X18,22,19,21 X12,19,13,20
Gauss code	{1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 8, -11, 5, -6, 7, -5, 9, -10, 11, -8, 10, -9}

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