L10n45

L10n44

L10n46

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n45 at Knotilus!

Link Presentations

[edit Notes on L10n45's Link Presentations]

Planar diagram presentation	X8192 X18,9,19,10 X6718 X20,14,7,13 X5,13,6,12 X3,10,4,11 X15,5,16,4 X11,16,12,17 X14,20,15,19 X17,2,18,3
Gauss code	{1, 10, -6, 7, -5, -3}, {3, -1, 2, 6, -8, 5, 4, -9, -7, 8, -10, -2, 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{t(2)^2 t(1)^2-t(2) t(1)^2-t(2) t(1)-t(2)+1}{t(1) t(2)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{5/2}+q^{3/2}-\sqrt{q}+\frac{1}{\sqrt{q}}-\frac{1}{q^{3/2}}-\frac{1}{q^{7/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z^{-1} -a^3 z^3-4 a^3 z-2 a^3 z^{-1} +a z^5+5 a z^3-z^3 a^{-1} +6 a z+2 a z^{-1} -3 z a^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^5 z+a^5 z^{-1} -a^4 z^2+a^3 z^7-6 a^3 z^5+10 a^3 z^3-7 a^3 z+2 a^3 z^{-1} +a^2 z^8-6 a^2 z^6+10 a^2 z^4-6 a^2 z^2+a^2+2 a z^7+z^7 a^{-1} -12 a z^5-6 z^5 a^{-1} +20 a z^3+10 z^3 a^{-1} -11 a z-5 z a^{-1} +2 a z^{-1} + a^{-1} z^{-1} +z^8-6 z^6+10 z^4-5 z^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   \ -2 -1 0 1 2 3 4 χ 6             1 1 4               0 2         1 1   0 0     1 1       0 -2     1 1       0 -4 1 1 1         1 -6 1             1 -8 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{ i=0 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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.

L10n44 L10n46