L11n378

L11n377

L11n379

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Link Presentations

[edit Notes on L11n378's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X15,1,16,4 X5,10,6,11 X3849 X22,18,19,17 X11,20,12,21 X19,12,20,13 X18,22,5,21 X9,16,10,17 X2,14,3,13
Gauss code	{1, -11, -5, 3}, {-8, 7, 9, -6}, {-4, -1, 2, 5, -10, 4, -7, 8, 11, -2, -3, 10, 6, -9}

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(1)-1) (t(2)-1) (t(3)-1)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-7} + q^{-6} -2 q^{-5} +3 q^{-4} -2 q^{-3} +3 q^{-2} - q^{-1} +3 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^2 a^6-a^6 z^{-2} -2 a^6+z^4 a^4+4 z^2 a^4+4 a^4 z^{-2} +7 a^4-3 z^2 a^2-5 a^2 z^{-2} -8 a^2+2 z^{-2} +3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^7-6 a^7 z^5+11 a^7 z^3-6 a^7 z+a^7 z^{-1} +a^6 z^8-5 a^6 z^6+7 a^6 z^4-4 a^6 z^2-a^6 z^{-2} +3 a^6+3 a^5 z^7-16 a^5 z^5+27 a^5 z^3-19 a^5 z+5 a^5 z^{-1} +a^4 z^8-4 a^4 z^6+6 a^4 z^4-11 a^4 z^2-4 a^4 z^{-2} +10 a^4+2 a^3 z^7-10 a^3 z^5+19 a^3 z^3-21 a^3 z+9 a^3 z^{-1} +a^2 z^6-a^2 z^4-7 a^2 z^2-5 a^2 z^{-2} +11 a^2+3 a z^3-8 a z+5 a z^{-1} -2 z^{-2} +5 }[/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   \ -7 -6 -5 -4 -3 -2 -1 0 χ 1               3 3 -1             2 4 2 -3           1   1 2 -5         1 2     1 -7       2 1       1 -9       1         1 -11   1 2           -1 -13                 0 -15 1               -1

Integral Khovanov Homology (db, data source)

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