L11n237

L11n236

L11n238

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Link Presentations

[edit Notes on L11n237's Link Presentations]

Planar diagram presentation	X10,1,11,2 X3,14,4,15 X22,5,9,6 X6,9,7,10 X20,12,21,11 X18,14,19,13 X12,20,13,19 X7,16,8,17 X15,4,16,5 X17,8,18,1 X2,21,3,22
Gauss code	{1, -11, -2, 9, 3, -4, -8, 10}, {4, -1, 5, -7, 6, 2, -9, 8, -10, -6, 7, -5, 11, -3}

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