L11a231

L11a230

L11a232

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Link Presentations

[edit Notes on L11a231's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X22,13,7,14 X14,9,15,10 X10,21,11,22 X18,5,19,6 X20,16,21,15 X16,20,17,19 X2738 X4,11,5,12 X6,17,1,18
Gauss code	{1, -9, 2, -10, 6, -11}, {9, -1, 4, -5, 10, -2, 3, -4, 7, -8, 11, -6, 8, -7, 5, -3}

A Braid Representative	{{{braid_table}}}
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^3-7 u^2 v^2+6 u^2 v-2 u^2+u v^4-8 u v^3+13 u v^2-8 u v+u-2 v^4+6 v^3-7 v^2+2 v}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{21}{q^{9/2}}-\frac{19}{q^{7/2}}+\frac{14}{q^{5/2}}-\frac{9}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{3}{q^{19/2}}+\frac{7}{q^{17/2}}-\frac{13}{q^{15/2}}+\frac{17}{q^{13/2}}-\frac{21}{q^{11/2}}-\sqrt{q}+\frac{4}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{11} z^{-1} +4 a^9 z+3 a^9 z^{-1} -5 a^7 z^3-6 a^7 z-2 a^7 z^{-1} +2 a^5 z^5+2 a^5 z^3+a^5 z+a^3 z^5-a^3 z^3-2 a^3 z-a z^3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{12} z^6-3 a^{12} z^4+3 a^{12} z^2-a^{12}+3 a^{11} z^7-8 a^{11} z^5+7 a^{11} z^3-3 a^{11} z+a^{11} z^{-1} +4 a^{10} z^8-5 a^{10} z^6-6 a^{10} z^4+9 a^{10} z^2-3 a^{10}+4 a^9 z^9-2 a^9 z^7-11 a^9 z^5+15 a^9 z^3-12 a^9 z+3 a^9 z^{-1} +2 a^8 z^{10}+6 a^8 z^8-17 a^8 z^6+8 a^8 z^4+3 a^8 z^2-3 a^8+11 a^7 z^9-20 a^7 z^7+9 a^7 z^5+6 a^7 z^3-7 a^7 z+2 a^7 z^{-1} +2 a^6 z^{10}+12 a^6 z^8-34 a^6 z^6+31 a^6 z^4-8 a^6 z^2+7 a^5 z^9-7 a^5 z^7-2 a^5 z^5+5 a^5 z^3+10 a^4 z^8-19 a^4 z^6+15 a^4 z^4-5 a^4 z^2+8 a^3 z^7-13 a^3 z^5+6 a^3 z^3-2 a^3 z+4 a^2 z^6-5 a^2 z^4+a z^5-a z^3 }[/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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                       1 1 0                     3   -3 -2                   6 1   5 -4                 9 4     -5 -6               10 5       5 -8             11 9         -2 -10           10 10           0 -12         7 11             4 -14       6 10               -4 -16     2 8                 6 -18   1 5                   -4 -20   2                     2 -22 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=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/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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