L11n204

L11n203

L11n205

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n204 at Knotilus!

Link Presentations

[edit Notes on L11n204's Link Presentations]

Planar diagram presentation	X10,1,11,2 X7,16,8,17 X11,18,12,19 X2,19,3,20 X3,12,4,13 X20,13,21,14 X14,5,15,6 X6,9,7,10 X15,22,16,9 X17,8,18,1 X21,4,22,5
Gauss code	{1, -4, -5, 11, 7, -8, -2, 10}, {8, -1, -3, 5, 6, -7, -9, 2, -10, 3, 4, -6, -11, 9}

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{\left(t(1) t(2)^2+1\right) \left(t(1)^2 t(2)^3+1\right)}{t(1)^{3/2} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{9/2}}-\frac{1}{q^{13/2}}+\frac{1}{q^{21/2}}-\frac{1}{q^{23/2}} }[/math] (db)
Signature	-8 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{13} \left(-z^3\right)-3 a^{13} z-a^{13} z^{-1} +a^{11} z^7+8 a^{11} z^5+20 a^{11} z^3+17 a^{11} z+3 a^{11} z^{-1} -a^9 z^9-9 a^9 z^7-28 a^9 z^5-36 a^9 z^3-18 a^9 z-2 a^9 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z a^{15}-a^{14}+z^3 a^{13}-3 z a^{13}+a^{13} z^{-1} -z^8 a^{12}+8 z^6 a^{12}-20 z^4 a^{12}+17 z^2 a^{12}-3 a^{12}-z^9 a^{11}+9 z^7 a^{11}-28 z^5 a^{11}+37 z^3 a^{11}-20 z a^{11}+3 a^{11} z^{-1} -z^8 a^{10}+8 z^6 a^{10}-20 z^4 a^{10}+17 z^2 a^{10}-3 a^{10}-z^9 a^9+9 z^7 a^9-28 z^5 a^9+36 z^3 a^9-18 z a^9+2 a^9 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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -8                 1 1 -10                 1 1 -12             1     1 -14         1         1 -16         1 1       0 -18     1 1           0 -20       1           -1 -22 1 1               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=-10 }[/math] [math]\displaystyle{ i=-8 }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/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=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/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=-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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