L10n55

L10n54

L10n56

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Link Presentations

[edit Notes on L10n55's Link Presentations]

Planar diagram presentation	X10,1,11,2 X11,17,12,16 X8,9,1,10 X17,9,18,20 X3,12,4,13 X7,14,8,15 X13,6,14,7 X5,18,6,19 X19,4,20,5 X15,2,16,3
Gauss code	{1, 10, -5, 9, -8, 7, -6, -3}, {3, -1, -2, 5, -7, 6, -10, 2, -4, 8, -9, 4}

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