L10n16

L10n15

L10n17

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Link Presentations

[edit Notes on L10n16's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X9,14,10,15 X8493 X12,5,13,6 X20,13,5,14 X11,16,12,17 X15,10,16,11 X2,18,3,17
Gauss code	{1, -10, 5, -3}, {6, -1, 2, -5, -4, 9, -8, -6, 7, 4, -9, 8, 10, -2, 3, -7}

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