L10n100

L10n99

L10n101

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n100 at Knotilus!

Link Presentations

[edit Notes on L10n100's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X11,18,12,19 X20,16,17,15 X16,20,9,19 X17,12,18,13 X2536 X9,1,10,4
Gauss code	{1, -9, -2, 10}, {9, -1, -3, 4}, {-8, 5, 7, -6}, {-10, 2, -5, 8, -4, 3, 6, -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{(u-1) (v-1) (w-1) (x-1)}{\sqrt{u} \sqrt{v} \sqrt{w} \sqrt{x}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -2 q^{9/2}+3 q^{7/2}-7 q^{5/2}+4 q^{3/2}-7 \sqrt{q}+\frac{4}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{1}{q^{5/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ - a^{-5} z^{-3} - a^{-5} z^{-1} -z^3 a^{-3} +3 a^{-3} z^{-3} +z a^{-3} +4 a^{-3} z^{-1} +z^5 a^{-1} -a z^3+2 z^3 a^{-1} +a z^{-3} -3 a^{-1} z^{-3} -z a^{-1} +2 a z^{-1} -5 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^7 a^{-1} -z^7 a^{-3} -5 z^6 a^{-2} -z^6 a^{-4} -4 z^6-4 a z^5-6 z^5 a^{-1} -2 z^5 a^{-3} -a^2 z^4+6 z^4 a^{-2} +5 z^4+6 a z^3+10 z^3 a^{-1} +z^3 a^{-3} -3 z^3 a^{-5} -6 z^2 a^{-2} -5 z^2 a^{-4} -z^2+z a^{-1} +5 z a^{-3} +4 z a^{-5} +11 a^{-2} +6 a^{-4} +6-3 a z^{-1} -6 a^{-1} z^{-1} -6 a^{-3} z^{-1} -3 a^{-5} z^{-1} -6 a^{-2} z^{-2} -3 a^{-4} z^{-2} -3 z^{-2} +a z^{-3} +3 a^{-1} z^{-3} +3 a^{-3} z^{-3} + a^{-5} z^{-3} }[/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   \ -3 -2 -1 0 1 2 3 4 χ 10               2 2 8             3 2 -1 6           4     4 4           3     3 2       7 4       3 0     3 6         3 -2   1 1           0 -4   3             3 -6 1               -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{6}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\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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