L11n210

L11n209

L11n211

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n210 at Knotilus!

Link Presentations

[edit Notes on L11n210's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X5,14,6,15 X22,18,9,17 X4,19,5,20 X21,6,22,7 X16,7,17,8 X8,9,1,10 X18,14,19,13 X15,21,16,20
Gauss code	{1, -2, 3, -6, -4, 7, 8, -9}, {9, -1, 2, -3, 10, 4, -11, -8, 5, -10, 6, 11, -7, -5}

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-u^2 v^4+u^2 v^3-u^2 v^2+u^2 v+u v^4-u v^3+u v^2-u v+u+v^3}{u^{3/2} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{1}{\sqrt{q}}-\frac{2}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{4}{q^{9/2}}-\frac{4}{q^{11/2}}+\frac{3}{q^{13/2}}-\frac{2}{q^{15/2}}+\frac{1}{q^{17/2}}-\frac{1}{q^{19/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^5 a^7+5 z^3 a^7+7 z a^7+2 a^7 z^{-1} -z^7 a^5-7 z^5 a^5-17 z^3 a^5-15 z a^5-3 a^5 z^{-1} +z^5 a^3+4 z^3 a^3+4 z a^3+a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{11} z^5-4 a^{11} z^3+3 a^{11} z+a^{10} z^6-3 a^{10} z^4+a^{10} z^2+a^9 z^7-3 a^9 z^5+2 a^9 z^3-a^9 z+a^8 z^8-4 a^8 z^6+6 a^8 z^4-3 a^8 z^2+a^7 z^9-6 a^7 z^7+15 a^7 z^5-16 a^7 z^3+10 a^7 z-2 a^7 z^{-1} +2 a^6 z^8-11 a^6 z^6+23 a^6 z^4-15 a^6 z^2+3 a^6+a^5 z^9-7 a^5 z^7+21 a^5 z^5-28 a^5 z^3+18 a^5 z-3 a^5 z^{-1} +a^4 z^8-6 a^4 z^6+15 a^4 z^4-14 a^4 z^2+3 a^4+2 a^3 z^5-6 a^3 z^3+4 a^3 z-a^3 z^{-1} +a^2 z^4-3 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                 1   1 -4               2 2   0 -6             2       2 -8           2 2       0 -10         2 2         0 -12       1 2           1 -14     1 2             -1 -16     1               1 -18 1 1                 0 -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} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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