L11n22

L11n21

L11n23

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n22 at Knotilus!

Link Presentations

[edit Notes on L11n22's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X9,21,10,20 X3849 X21,18,22,19 X11,14,12,15 X5,13,6,12 X13,5,14,22 X19,11,20,10 X15,2,16,3
Gauss code	{1, 11, -5, -3}, {-8, -1, 2, 5, -4, 10, -7, 8, -9, 7, -11, -2, 3, 6, -10, 4, -6, 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{t(1) t(2)^5-4 t(1) t(2)^4+4 t(1) t(2)^3+4 t(2)^2-4 t(2)+1}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{3}{q^{9/2}}-\frac{6}{q^{7/2}}-q^{5/2}+\frac{6}{q^{5/2}}+2 q^{3/2}-\frac{5}{q^{3/2}}+\frac{1}{q^{13/2}}-\frac{3}{q^{11/2}}-4 \sqrt{q}+\frac{5}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^7+z^5 a^5+4 z^3 a^5+3 z a^5+2 a^5 z^{-1} -z^7 a^3-5 z^5 a^3-8 z^3 a^3-8 z a^3-4 a^3 z^{-1} +2 z^5 a+8 z^3 a+7 z a+3 a z^{-1} -z^3 a^{-1} -3 z a^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^8 z^2+3 a^7 z^3-3 a^7 z+a^6 z^6-a^6 z^4+a^6 z^2+a^6+3 a^5 z^7-12 a^5 z^5+20 a^5 z^3-15 a^5 z+2 a^5 z^{-1} +3 a^4 z^8-11 a^4 z^6+11 a^4 z^4-4 a^4 z^2+2 a^4+a^3 z^9+2 a^3 z^7-23 a^3 z^5+37 a^3 z^3-23 a^3 z+4 a^3 z^{-1} +5 a^2 z^8-21 a^2 z^6+23 a^2 z^4-9 a^2 z^2+3 a^2+a z^9+z^7 a^{-1} -16 a z^5-5 z^5 a^{-1} +28 a z^3+8 z^3 a^{-1} -16 a z-5 z a^{-1} +3 a z^{-1} + a^{-1} z^{-1} +2 z^8-9 z^6+11 z^4-5 z^2+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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                   1 1 4                 1   -1 2               3 1   2 0             2 1     -1 -2         1 4 3       0 -4         4 3         -1 -6       3 3           0 -8   1 2 4             3 -10   2 2               0 -12   2                 2 -14 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=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/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.

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