L10n29

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L10n30

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Link Presentations

[edit Notes on L10n29's Link Presentations]

Planar diagram presentation	X6172 X3,10,4,11 X7,14,8,15 X15,20,16,5 X9,17,10,16 X19,9,20,8 X13,19,14,18 X17,13,18,12 X2536 X11,4,12,1
Gauss code	{1, -9, -2, 10}, {9, -1, -3, 6, -5, 2, -10, 8, -7, 3, -4, 5, -8, 7, -6, 4}

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(1) t(2)^5-3 t(1) t(2)^4+t(2)^4+4 t(1) t(2)^3-4 t(2)^3-4 t(1) t(2)^2+4 t(2)^2+t(1) t(2)-3 t(2)+1}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{7/2}+3 q^{5/2}-6 q^{3/2}+8 \sqrt{q}-\frac{9}{\sqrt{q}}+\frac{9}{q^{3/2}}-\frac{8}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{3}{q^{9/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a z^7-a^3 z^5+5 a z^5-z^5 a^{-1} -4 a^3 z^3+9 a z^3-3 z^3 a^{-1} +a^5 z-7 a^3 z+7 a z-3 z a^{-1} +2 a^5 z^{-1} -4 a^3 z^{-1} +3 a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 a^2 z^8-2 z^8-5 a^3 z^7-9 a z^7-4 z^7 a^{-1} -3 a^4 z^6-3 a^2 z^6-3 z^6 a^{-2} -3 z^6+13 a^3 z^5+21 a z^5+7 z^5 a^{-1} -z^5 a^{-3} +3 a^4 z^4+11 a^2 z^4+6 z^4 a^{-2} +14 z^4-6 a^5 z^3-23 a^3 z^3-21 a z^3-2 z^3 a^{-1} +2 z^3 a^{-3} -4 a^4 z^2-11 a^2 z^2-3 z^2 a^{-2} -10 z^2+8 a^5 z+17 a^3 z+12 a z+2 z a^{-1} -z a^{-3} +2 a^4+3 a^2+ a^{-2} +3-2 a^5 z^{-1} -4 a^3 z^{-1} -3 a z^{-1} - a^{-1} 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   \ -4 -3 -2 -1 0 1 2 3 4 χ 8                 1 1 6               2   -2 4             4 1   3 2           4 2     -2 0         5 4       1 -2       5 5         0 -4     3 4           -1 -6   2 5             3 -8 1 3               -2 -10 3                 3

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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