L11n214

L11n213

L11n215

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Link Presentations

[edit Notes on L11n214's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X9,18,10,19 X17,22,18,9 X21,1,22,8 X20,13,21,14 X5,14,6,15 X7,16,8,17 X15,6,16,7 X4,20,5,19
Gauss code	{1, -2, 3, -11, -8, 10, -9, 6}, {-4, -1, 2, -3, 7, 8, -10, 9, -5, 4, 11, -7, -6, 5}

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{t(2)^3 t(1)^3-t(2)^2 t(1)^3-3 t(2)^3 t(1)^2+5 t(2)^2 t(1)^2-4 t(2) t(1)^2+t(1)^2+t(2)^3 t(1)-4 t(2)^2 t(1)+5 t(2) t(1)-3 t(1)-t(2)+1}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{7}{q^{9/2}}+\frac{3}{q^{7/2}}-\frac{2}{q^{5/2}}+\frac{1}{q^{23/2}}-\frac{3}{q^{21/2}}+\frac{6}{q^{19/2}}-\frac{9}{q^{17/2}}+\frac{10}{q^{15/2}}-\frac{10}{q^{13/2}}+\frac{9}{q^{11/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 \left(-z^5\right)-3 a^9 z^3-3 a^9 z-a^9 z^{-1} +a^7 z^7+5 a^7 z^5+10 a^7 z^3+10 a^7 z+3 a^7 z^{-1} -2 a^5 z^5-8 a^5 z^3-9 a^5 z-2 a^5 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{14} z^4-a^{14} z^2+3 a^{13} z^5-3 a^{13} z^3+a^{13} z+5 a^{12} z^6-6 a^{12} z^4+3 a^{12} z^2+5 a^{11} z^7-5 a^{11} z^5+a^{11} z^3+a^{11} z+3 a^{10} z^8-3 a^{10} z^4+a^{10}+a^9 z^9+3 a^9 z^7-2 a^9 z^5-6 a^9 z^3+4 a^9 z-a^9 z^{-1} +4 a^8 z^8-7 a^8 z^6+8 a^8 z^4-11 a^8 z^2+3 a^8+a^7 z^9-2 a^7 z^7+9 a^7 z^5-20 a^7 z^3+14 a^7 z-3 a^7 z^{-1} +a^6 z^8-2 a^6 z^6+4 a^6 z^4-7 a^6 z^2+3 a^6+3 a^5 z^5-10 a^5 z^3+10 a^5 z-2 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                 2 1 -1 -8               5 1   4 -10             4 2     -2 -12           6 5       1 -14         5 5         0 -16       4 5           -1 -18     2 5             3 -20   1 4               -3 -22   2                 2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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

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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