L10n80

L10n79

L10n81

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Link Presentations

[edit Notes on L10n80's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,18,12,19 X7,14,8,15 X13,8,14,9 X15,13,16,20 X19,17,20,16 X17,12,18,5 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, -4, 5, 10, -2, -3, 8}, {-5, 4, -6, 7, -8, 3, -7, 6}

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