L11n212

L11n211

L11n213

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n212 at Knotilus!

Link Presentations

[edit Notes on L11n212's Link Presentations]

Planar diagram presentation	X10,1,11,2 X13,21,14,20 X3,12,4,13 X2,19,3,20 X14,5,15,6 X16,7,17,8 X8,9,1,10 X18,12,19,11 X6,15,7,16 X22,18,9,17 X21,4,22,5
Gauss code	{1, -4, -3, 11, 5, -9, 6, -7}, {7, -1, 8, 3, -2, -5, 9, -6, 10, -8, 4, 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 v+1) \left(u^2 v+u v^4-2 u v^3+2 u v^2-2 u v+u+v^3\right)}{u^{3/2} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{1}{\sqrt{q}}-\frac{3}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{7}{q^{9/2}}-\frac{7}{q^{11/2}}+\frac{5}{q^{13/2}}-\frac{4}{q^{15/2}}+\frac{2}{q^{17/2}}-\frac{1}{q^{19/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 z^5+4 a^7 z^3+5 a^7 z+2 a^7 z^{-1} -a^5 z^7-6 a^5 z^5-13 a^5 z^3-11 a^5 z-3 a^5 z^{-1} +a^3 z^5+3 a^3 z^3+2 a^3 z+a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{11} z^5-3 a^{11} z^3+a^{11} z+2 a^{10} z^6-5 a^{10} z^4+a^{10} z^2+3 a^9 z^7-10 a^9 z^5+11 a^9 z^3-5 a^9 z+2 a^8 z^8-5 a^8 z^6+4 a^8 z^4+a^7 z^9-2 a^7 z^7+3 a^7 z^5-3 a^7 z^3+6 a^7 z-2 a^7 z^{-1} +3 a^6 z^8-11 a^6 z^6+20 a^6 z^4-12 a^6 z^2+3 a^6+a^5 z^9-5 a^5 z^7+17 a^5 z^5-23 a^5 z^3+14 a^5 z-3 a^5 z^{-1} +a^4 z^8-4 a^4 z^6+12 a^4 z^4-13 a^4 z^2+3 a^4+3 a^3 z^5-6 a^3 z^3+2 a^3 z-a^3 z^{-1} +a^2 z^4-2 a^2 z^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               3 2   -1 -6             3 1     2 -8           4 3       -1 -10         3 3         0 -12       2 4           2 -14     2 3             -1 -16     2               2 -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}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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