L11n182

L11n181

L11n183

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n182 at Knotilus!

Link Presentations

[edit Notes on L11n182's Link Presentations]

Planar diagram presentation	X8192 X9,21,10,20 X14,5,15,6 X16,8,17,7 X3,10,4,11 X22,14,7,13 X18,12,19,11 X12,18,13,17 X6,15,1,16 X4,21,5,22 X19,2,20,3
Gauss code	{1, 11, -5, -10, 3, -9}, {4, -1, -2, 5, 7, -8, 6, -3, 9, -4, 8, -7, -11, 2, 10, -6}

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^2 v^3-3 u^2 v^2+3 u^2 v+2 u v^4-5 u v^3+7 u v^2-5 u v+2 u+3 v^3-3 v^2+v}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 2 q^{5/2}-5 q^{3/2}+8 \sqrt{q}-\frac{11}{\sqrt{q}}+\frac{12}{q^{3/2}}-\frac{12}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z^3+a^5 z+a^5 z^{-1} -a^3 z^5-a^3 z^3-2 a z^5-6 a z^3+2 z^3 a^{-1} -7 a z-2 a z^{-1} +3 z a^{-1} + a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^5-2 a^7 z^3+3 a^6 z^6-5 a^6 z^4+6 a^5 z^7-16 a^5 z^5+16 a^5 z^3-8 a^5 z+a^5 z^{-1} +5 a^4 z^8-10 a^4 z^6+6 a^4 z^4+a^4 z^2-a^4+2 a^3 z^9+2 a^3 z^7-10 a^3 z^5+10 a^3 z^3-2 a^3 z+8 a^2 z^8-21 a^2 z^6+27 a^2 z^4+3 z^4 a^{-2} -14 a^2 z^2-5 z^2 a^{-2} +3 a^2+2 a^{-2} +2 a z^9-3 a z^7+z^7 a^{-1} +10 a z^5+3 z^5 a^{-1} -15 a z^3-7 z^3 a^{-1} +9 a z+3 z a^{-1} -2 a z^{-1} - a^{-1} z^{-1} +3 z^8-8 z^6+19 z^4-20 z^2+5 }[/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               5 2   -3 0             6 3     3 -2           7 6       -1 -4         5 5         0 -6       4 7           3 -8     3 5             -2 -10     4               4 -12 1 3                 -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=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\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^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\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^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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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