L11n230

L11n229

L11n231

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n230 at Knotilus!

Link Presentations

[edit Notes on L11n230's Link Presentations]

Planar diagram presentation	X10,1,11,2 X11,17,12,16 X5,21,6,20 X3,12,4,13 X7,14,8,15 X13,6,14,7 X17,9,18,22 X21,19,22,18 X8,9,1,10 X19,5,20,4 X15,2,16,3
Gauss code	{1, 11, -4, 10, -3, 6, -5, -9}, {9, -1, -2, 4, -6, 5, -11, 2, -7, 8, -10, 3, -8, 7}

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