L11n211

L11n210

L11n212

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n211 at Knotilus!

Link Presentations

[edit Notes on L11n211's Link Presentations]

Planar diagram presentation	X10,1,11,2 X15,21,16,20 X5,14,6,15 X3,12,4,13 X13,4,14,5 X2,19,3,20 X16,7,17,8 X8,9,1,10 X18,12,19,11 X22,18,9,17 X21,6,22,7
Gauss code	{1, -6, -4, 5, -3, 11, 7, -8}, {8, -1, 9, 4, -5, 3, -2, -7, 10, -9, 6, 2, -11, -10}

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^3 v^2+u^2 v^5-2 u^2 v^4+3 u^2 v^3-3 u^2 v^2+2 u^2 v+2 u v^4-3 u v^3+3 u v^2-2 u v+u+v^3}{u^{3/2} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{8}{q^{9/2}}-\frac{8}{q^{7/2}}+\frac{5}{q^{5/2}}-\frac{3}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{2}{q^{17/2}}-\frac{5}{q^{15/2}}+\frac{7}{q^{13/2}}-\frac{8}{q^{11/2}}+\frac{1}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^5 a^7+4 z^3 a^7+6 z a^7+2 a^7 z^{-1} -z^7 a^5-6 z^5 a^5-14 z^3 a^5-13 z a^5-3 a^5 z^{-1} +z^5 a^3+3 z^3 a^3+3 z a^3+a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^5 a^{11}+3 z^3 a^{11}-2 z a^{11}-2 z^6 a^{10}+4 z^4 a^{10}-z^2 a^{10}-3 z^7 a^9+7 z^5 a^9-6 z^3 a^9+3 z a^9-2 z^8 a^8+2 z^6 a^8+z^2 a^8-z^9 a^7+z^7 a^7-6 z^5 a^7+11 z^3 a^7-8 z a^7+2 a^7 z^{-1} -3 z^8 a^6+7 z^6 a^6-14 z^4 a^6+12 z^2 a^6-3 a^6-z^9 a^5+4 z^7 a^5-17 z^5 a^5+25 z^3 a^5-16 z a^5+3 a^5 z^{-1} -z^8 a^4+3 z^6 a^4-11 z^4 a^4+12 z^2 a^4-3 a^4-3 z^5 a^3+5 z^3 a^3-3 z a^3+a^3 z^{-1} -z^4 a^2+2 z^2 a^2-a^2 }[/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 1 χ 0                   1 -1 -2                 2   2 -4               4 2   -2 -6             4 1     3 -8           4 4       0 -10         4 4         0 -12       3 4           1 -14     2 4             -2 -16     3               3 -18 1 2                 -1 -20 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{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/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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