L10n101

L10n100

L10n102

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n101 at Knotilus!

Link Presentations

[edit Notes on L10n101's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X18,12,19,11 X20,16,17,15 X16,20,9,19 X12,18,13,17 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	{{{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(4)-1) \left(-t(4)^2+t(1) t(4)+t(2) t(4)-t(1) t(3) t(4)+t(1) t(2) t(3) t(4)-t(2) t(3) t(4)-t(4)+t(1) t(2) t(3)\right)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} t(4)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 2 q^{9/2}-4 q^{7/2}+2 q^{5/2}-4 q^{3/2}-q^{17/2}-q^{13/2}-3 q^{11/2}+2 \sqrt{q}-\frac{1}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ - a^{-9} z^{-3} - a^{-9} z^{-1} +3 a^{-7} z^{-3} +2 z a^{-7} +5 a^{-7} z^{-1} -3 a^{-5} z^{-3} -3 z a^{-5} -6 a^{-5} z^{-1} -z^5 a^{-3} -3 z^3 a^{-3} + a^{-3} z^{-3} -z a^{-3} + a^{-3} z^{-1} +z^3 a^{-1} +2 z a^{-1} + a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^7 a^{-9} -7 z^5 a^{-9} +15 z^3 a^{-9} - a^{-9} z^{-3} -14 z a^{-9} +6 a^{-9} z^{-1} +z^6 a^{-8} -9 z^4 a^{-8} +19 z^2 a^{-8} +3 a^{-8} z^{-2} -13 a^{-8} +z^7 a^{-7} -10 z^5 a^{-7} +28 z^3 a^{-7} -3 a^{-7} z^{-3} -27 z a^{-7} +14 a^{-7} z^{-1} +2 z^6 a^{-6} -14 z^4 a^{-6} +33 z^2 a^{-6} +6 a^{-6} z^{-2} -24 a^{-6} +z^7 a^{-5} -4 z^5 a^{-5} +10 z^3 a^{-5} -3 a^{-5} z^{-3} -16 z a^{-5} +12 a^{-5} z^{-1} +3 z^6 a^{-4} -10 z^4 a^{-4} +14 z^2 a^{-4} +3 a^{-4} z^{-2} -11 a^{-4} +z^7 a^{-3} -6 z^3 a^{-3} - a^{-3} z^{-3} +3 a^{-3} z^{-1} +2 z^6 a^{-2} -5 z^4 a^{-2} + a^{-2} +z^5 a^{-1} -3 z^3 a^{-1} +3 z a^{-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   \ -2 -1 0 1 2 3 4 5 6 7 8 χ 18                     1 1 16                     1 1 14                 1     1 12             4         4 10           3 5 1       1 8         2             2 6         3 1           2 4     4 2               2 2   1 3                 2 0   1                   -1 -2 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{ i=4 }[/math] [math]\displaystyle{ r=-2 }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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.

L10n100 L10n102