L11n48

L11n47

L11n49

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Link Presentations

[edit Notes on L11n48's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X19,1,20,4 X5,14,6,15 X3849 X9,16,10,17 X15,10,16,11 X11,20,12,21 X13,22,14,5 X21,12,22,13 X2,18,3,17
Gauss code	{1, -11, -5, 3}, {-4, -1, 2, 5, -6, 7, -8, 10, -9, 4, -7, 6, 11, -2, -3, 8, -10, 9}

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{(t(1)-1) (t(2)-1) \left(t(2)^2+1\right)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{3/2}}-\frac{2}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{3}{q^{11/2}}+\frac{2}{q^{13/2}}-\frac{2}{q^{15/2}}+\frac{2}{q^{17/2}}-\frac{1}{q^{19/2}}+\frac{1}{q^{21/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 \left(-z^3\right)-3 a^9 z-2 a^9 z^{-1} +a^7 z^5+5 a^7 z^3+8 a^7 z+4 a^7 z^{-1} -a^5 z^3-2 a^5 z-a^5 z^{-1} -a^3 z^3-3 a^3 z-a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{12} z^6-5 a^{12} z^4+6 a^{12} z^2-2 a^{12}+a^{11} z^7-4 a^{11} z^5+2 a^{11} z^3+a^{11} z+a^{10} z^8-4 a^{10} z^6+2 a^{10} z^4+2 a^{10} z^2-a^{10}+a^9 z^9-6 a^9 z^7+13 a^9 z^5-15 a^9 z^3+8 a^9 z-2 a^9 z^{-1} +2 a^8 z^8-12 a^8 z^6+24 a^8 z^4-20 a^8 z^2+6 a^8+a^7 z^9-7 a^7 z^7+18 a^7 z^5-22 a^7 z^3+15 a^7 z-4 a^7 z^{-1} +a^6 z^8-7 a^6 z^6+17 a^6 z^4-15 a^6 z^2+5 a^6+a^5 z^5-4 a^5 z^3+5 a^5 z-a^5 z^{-1} +a^4 z^2-a^4+a^3 z^3-3 a^3 z+a^3 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 -2                   1 1 -4                 1 2 1 -6               1   1 2 -8             2 2     0 -10           2 1       1 -12         1 3 1       1 -14       2 2           0 -16       1 1           0 -18   1 2               -1 -20                     0 -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=-6 }[/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 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\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}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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