L11n154

L11n153

L11n155

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Link Presentations

[edit Notes on L11n154's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X7,16,8,17 X13,20,14,21 X15,22,16,7 X19,1,20,6 X18,11,19,12 X5,12,6,13 X21,14,22,15 X4,18,5,17
Gauss code	{1, -2, 3, -11, -9, 7}, {-4, -1, 2, -3, 8, 9, -5, 10, -6, 4, 11, -8, -7, 5, -10, 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^4-u^2 v^3+u^2 v^2-u^2 v-2 u v^4+3 u v^3-3 u v^2+3 u v-2 u-v^3+v^2-v+1}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{5}{q^{9/2}}+\frac{2}{q^{7/2}}-\frac{2}{q^{5/2}}+\frac{1}{q^{23/2}}-\frac{2}{q^{21/2}}+\frac{4}{q^{19/2}}-\frac{6}{q^{17/2}}+\frac{7}{q^{15/2}}-\frac{7}{q^{13/2}}+\frac{6}{q^{11/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 \left(-z^5\right)-4 a^9 z^3-5 a^9 z-2 a^9 z^{-1} +a^7 z^7+6 a^7 z^5+14 a^7 z^3+15 a^7 z+5 a^7 z^{-1} -2 a^5 z^5-9 a^5 z^3-11 a^5 z-3 a^5 z^{-1} }[/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^7-7 a^{11} z^5+9 a^{11} z^3-2 a^{11} z+2 a^{10} z^8-4 a^{10} z^6+5 a^{10} z^4+a^9 z^9-2 a^9 z^7+5 a^9 z^5-6 a^9 z^3+5 a^9 z-2 a^9 z^{-1} +3 a^8 z^8-11 a^8 z^6+21 a^8 z^4-20 a^8 z^2+5 a^8+a^7 z^9-5 a^7 z^7+17 a^7 z^5-30 a^7 z^3+20 a^7 z-5 a^7 z^{-1} +a^6 z^8-4 a^6 z^6+9 a^6 z^4-13 a^6 z^2+5 a^6+3 a^5 z^5-12 a^5 z^3+13 a^5 z-3 a^5 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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                   2 2 -6                 1 1 0 -8               4 1   3 -10             3 2     -1 -12           4 3       1 -14         3 3         0 -16       3 4           -1 -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}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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

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