L10n102

L10n101

L10n103

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n102 at Knotilus!

Link Presentations

[edit Notes on L10n102's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X11,18,12,19 X15,20,16,17 X19,16,20,9 X17,12,18,13 X2536 X4,9,1,10
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{(x-1) \left(u v w x+u v w-u w+u x^2-v w+v x^2-x^2-x\right)}{\sqrt{u} \sqrt{v} \sqrt{w} x^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{1}{q^{9/2}}-\frac{1}{q^{5/2}}-\frac{1}{q^{23/2}}+\frac{1}{q^{21/2}}-\frac{4}{q^{19/2}}+\frac{2}{q^{17/2}}-\frac{4}{q^{15/2}}+\frac{1}{q^{13/2}}-\frac{3}{q^{11/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{13} z^{-3} -4 a^{11} z^{-1} -3 a^{11} z^{-3} +5 z a^9+7 a^9 z^{-1} +3 a^9 z^{-3} -z^3 a^7-z a^7-2 a^7 z^{-1} -a^7 z^{-3} -z^5 a^5-5 z^3 a^5-4 z a^5-a^5 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^7 a^{13}+6 z^5 a^{13}-13 z^3 a^{13}+13 z a^{13}-6 a^{13} z^{-1} +a^{13} z^{-3} -z^8 a^{12}+3 z^6 a^{12}+4 z^4 a^{12}-16 z^2 a^{12}-3 a^{12} z^{-2} +13 a^{12}-5 z^7 a^{11}+26 z^5 a^{11}-41 z^3 a^{11}+31 z a^{11}-14 a^{11} z^{-1} +3 a^{11} z^{-3} -z^8 a^{10}+19 z^4 a^{10}-36 z^2 a^{10}-6 a^{10} z^{-2} +24 a^{10}-4 z^7 a^9+22 z^5 a^9-36 z^3 a^9+26 z a^9-12 a^9 z^{-1} +3 a^9 z^{-3} -3 z^6 a^8+16 z^4 a^8-21 z^2 a^8-3 a^8 z^{-2} +11 a^8+z^5 a^7-3 z^3 a^7+4 z a^7-3 a^7 z^{-1} +a^7 z^{-3} +z^4 a^6-z^2 a^6-a^6-z^5 a^5+5 z^3 a^5-4 z a^5+a^5 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   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                     1 1 -6                     1 1 -8               2 1     -1 -10             2         2 -12           3 6 1       2 -14         1 1 3         3 -16       1 3             2 -18     3 1               2 -20   1 4                 3 -22                       0 -24 1                     1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-10 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ r=0 }[/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.

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