L11n386

L11n385

L11n387

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Link Presentations

[edit Notes on L11n386's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X15,1,16,4 X5,12,6,13 X3849 X9,16,10,17 X17,19,18,22 X11,20,12,21 X19,10,20,11 X21,5,22,18 X2,14,3,13
Gauss code	{1, -11, -5, 3}, {-9, 8, -10, 7}, {-4, -1, 2, 5, -6, 9, -8, 4, 11, -2, -3, 6, -7, 10}

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{(u-1) (v-1) (w-1) \left(w^2-w+1\right)}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-8} +2 q^{-7} -4 q^{-6} +6 q^{-5} -8 q^{-4} +9 q^{-3} -6 q^{-2} +2 q+7 q^{-1} -3 }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^4 a^6-3 z^2 a^6-a^6 z^{-2} -3 a^6+z^6 a^4+5 z^4 a^4+11 z^2 a^4+4 a^4 z^{-2} +10 a^4-3 z^4 a^2-10 z^2 a^2-5 a^2 z^{-2} -11 a^2+2 z^2+2 z^{-2} +4 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^9 z^5-3 a^9 z^3+a^9 z+2 a^8 z^6-5 a^8 z^4+a^8 z^2+3 a^7 z^7-9 a^7 z^5+9 a^7 z^3-5 a^7 z+a^7 z^{-1} +3 a^6 z^8-10 a^6 z^6+14 a^6 z^4-9 a^6 z^2-a^6 z^{-2} +5 a^6+a^5 z^9+2 a^5 z^7-17 a^5 z^5+35 a^5 z^3-22 a^5 z+5 a^5 z^{-1} +5 a^4 z^8-20 a^4 z^6+39 a^4 z^4-34 a^4 z^2-4 a^4 z^{-2} +16 a^4+a^3 z^9-8 a^3 z^5+26 a^3 z^3-26 a^3 z+9 a^3 z^{-1} +2 a^2 z^8-8 a^2 z^6+23 a^2 z^4-32 a^2 z^2-5 a^2 z^{-2} +17 a^2+a z^7-a z^5+3 a z^3-10 a z+5 a z^{-1} +3 z^4-8 z^2-2 z^{-2} +7 }[/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 3                   2 2 1                 1   -1 -1               6 2   4 -3             4 5     1 -5           5 2       3 -7         3 4         1 -9       3 5           -2 -11     1 3             2 -13   1 3               -2 -15   1                 1 -17 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{ 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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{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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