L11a188

L11a187

L11a189

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Link Presentations

[edit Notes on L11a188's Link Presentations]

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

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^5-u^2 v^4+u^2 v^3-u^2 v^2+u^2 v-u^2+u v^6-3 u v^5+3 u v^4-3 u v^3+3 u v^2-3 u v+u-v^6+v^5-v^4+v^3-v^2+v}{u v^3} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{8}{q^{9/2}}+\frac{5}{q^{7/2}}-\frac{4}{q^{5/2}}+\frac{2}{q^{3/2}}+\frac{1}{q^{23/2}}-\frac{2}{q^{21/2}}+\frac{4}{q^{19/2}}-\frac{6}{q^{17/2}}+\frac{8}{q^{15/2}}-\frac{9}{q^{13/2}}+\frac{8}{q^{11/2}}-\frac{1}{\sqrt{q}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 \left(-z^5\right)-4 a^9 z^3-4 a^9 z-2 a^9 z^{-1} +a^7 z^7+5 a^7 z^5+9 a^7 z^3+10 a^7 z+5 a^7 z^{-1} +a^5 z^7+4 a^5 z^5+2 a^5 z^3-4 a^5 z-3 a^5 z^{-1} -a^3 z^5-4 a^3 z^3-3 a^3 z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{14} z^4-2 a^{14} z^2+2 a^{13} z^5-3 a^{13} z^3+3 a^{12} z^6-6 a^{12} z^4+5 a^{12} z^2-a^{12}+3 a^{11} z^7-6 a^{11} z^5+6 a^{11} z^3+3 a^{10} z^8-8 a^{10} z^6+10 a^{10} z^4-2 a^{10} z^2+3 a^9 z^9-13 a^9 z^7+28 a^9 z^5-28 a^9 z^3+12 a^9 z-2 a^9 z^{-1} +a^8 z^{10}-11 a^8 z^6+25 a^8 z^4-22 a^8 z^2+5 a^8+5 a^7 z^9-25 a^7 z^7+50 a^7 z^5-53 a^7 z^3+25 a^7 z-5 a^7 z^{-1} +a^6 z^{10}-a^6 z^8-9 a^6 z^6+18 a^6 z^4-15 a^6 z^2+5 a^6+2 a^5 z^9-8 a^5 z^7+9 a^5 z^5-9 a^5 z^3+10 a^5 z-3 a^5 z^{-1} +2 a^4 z^8-9 a^4 z^6+10 a^4 z^4-2 a^4 z^2+a^3 z^7-5 a^3 z^5+7 a^3 z^3-3 a^3 z }[/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 0                       1 1 -2                     1   -1 -4                   3 1   2 -6                 3 2     -1 -8               5 2       3 -10             4 4         0 -12           5 4           1 -14         3 4             1 -16       3 5               -2 -18     1 3                 2 -20   1 3                   -2 -22   1                     1 -24 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/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}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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