L10n67

L10n66

L10n68

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Link Presentations

[edit Notes on L10n67's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,16,12,17 X13,19,14,18 X17,20,18,9 X19,13,20,12 X8,16,5,15 X14,8,15,7 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, 8, -7}, {10, -2, -3, 6, -4, -8, 7, 3, -5, 4, -6, 5}

A Braid Representative	{{{braid_table}}}
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) t(3)^3-2 t(1) t(3)^2+t(1) t(2) t(3)^2-4 t(2) t(3)^2+t(3)^2+4 t(1) t(3)-t(1) t(2) t(3)+2 t(2) t(3)-t(3)-t(1)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 2 q^2-3 q+6-6 q^{-1} +6 q^{-2} -5 q^{-3} +5 q^{-4} -2 q^{-5} + q^{-6} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^6 z^{-2} +a^6-3 z^2 a^4-3 a^4 z^{-2} -5 a^4+2 z^4 a^2+6 z^2 a^2+4 a^2 z^{-2} +8 a^2-4 z^2-3 z^{-2} -6+ a^{-2} z^{-2} +2 a^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^6 z^6-4 a^6 z^4+6 a^6 z^2+a^6 z^{-2} -4 a^6+2 a^5 z^7-6 a^5 z^5+3 a^5 z^3+a^5 z-a^5 z^{-1} +a^4 z^8+3 a^4 z^6-21 a^4 z^4+25 a^4 z^2+3 a^4 z^{-2} -14 a^4+6 a^3 z^7-18 a^3 z^5+11 a^3 z^3+a^3 z-a^3 z^{-1} +a^2 z^8+6 a^2 z^6-30 a^2 z^4+39 a^2 z^2+3 z^2 a^{-2} +4 a^2 z^{-2} + a^{-2} z^{-2} -21 a^2-4 a^{-2} +4 a z^7-11 a z^5+z^5 a^{-1} +9 a z^3+z^3 a^{-1} +a z+z a^{-1} -a z^{-1} - a^{-1} z^{-1} +4 z^6-13 z^4+23 z^2+3 z^{-2} -14 }[/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   \ -6 -5 -4 -3 -2 -1 0 1 2 χ 5                 2 2 3               1   -1 1             5 2   3 -1           4 4     0 -3         2 3 1     0 -5       3 4         1 -7     2 2           0 -9   1 4             3 -11   1               -1 -13 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=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/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} }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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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