L11n383

L11n382

L11n384

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Link Presentations

[edit Notes on L11n383's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X10,6,11,5 X8493 X17,22,18,19 X11,20,12,21 X19,12,20,13 X21,18,22,5 X16,10,17,9 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(3)-1)^2 \left(t(3)^3+t(2)\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-7} +3 q^{-6} -3 q^{-5} +5 q^{-4} -4 q^{-3} +q^2+6 q^{-2} -2 q-4 q^{-1} +3 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^2 a^6+a^6 z^{-2} -a^6+2 z^4 a^4+6 z^2 a^4-2 a^4 z^{-2} +2 a^4-z^6 a^2-5 z^4 a^2-8 z^2 a^2+a^2 z^{-2} -3 a^2+z^4+3 z^2+2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 2 a^5 z^9+2 a^3 z^9+3 a^6 z^8+7 a^4 z^8+4 a^2 z^8+a^7 z^7-7 a^5 z^7-6 a^3 z^7+2 a z^7-15 a^6 z^6-36 a^4 z^6-21 a^2 z^6-4 a^7 z^5-a^5 z^5-5 a^3 z^5-8 a z^5+20 a^6 z^4+53 a^4 z^4+36 a^2 z^4+3 z^4+3 a^7 z^3+11 a^5 z^3+17 a^3 z^3+11 a z^3+2 z^3 a^{-1} -8 a^6 z^2-27 a^4 z^2-26 a^2 z^2+z^2 a^{-2} -6 z^2-a^7 z-a^5 z-5 a^3 z-7 a z-2 z a^{-1} -a^6+2 a^4+5 a^2+3-2 a^5 z^{-1} -2 a^3 z^{-1} +a^6 z^{-2} +2 a^4 z^{-2} +a^2 z^{-2} }[/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 1 2 χ 5                   1 1 3                 2 1 -1 1               2 1   1 -1             4 4 1   -1 -3           3 3 2     2 -5         3 5         2 -7       3 3 1         1 -9     2 5 1           2 -11   1 1               0 -13   2                 2 -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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/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}\oplus{\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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