L11n243

L11n242

L11n244

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n243 at Knotilus!

Link Presentations

[edit Notes on L11n243's Link Presentations]

Planar diagram presentation	X12,1,13,2 X16,7,17,8 X5,1,6,10 X3746 X9,5,10,4 X20,14,21,13 X22,17,11,18 X18,21,19,22 X14,20,15,19 X2,11,3,12 X8,15,9,16
Gauss code	{1, -10, -4, 5, -3, 4, 2, -11, -5, 3}, {10, -1, 6, -9, 11, -2, 7, -8, 9, -6, 8, -7}

A Braid Representative	{{{braid_table}}}
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{2 u^3 v^2-4 u^3 v+u^3-2 u^2 v^2+2 u^2 v+2 u v^2-2 u v+v^3-4 v^2+2 v}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{7/2}-3 q^{5/2}+5 q^{3/2}-7 \sqrt{q}+\frac{7}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{6}{q^{5/2}}-\frac{5}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z+2 a^5 z^{-1} -3 a^3 z^3-8 a^3 z-5 a^3 z^{-1} +z a^{-3} + a^{-3} z^{-1} +2 a z^5+8 a z^3-3 z^3 a^{-1} +11 a z+6 a z^{-1} -7 z a^{-1} -4 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^5 z^7-5 a^5 z^5+9 a^5 z^3-7 a^5 z+2 a^5 z^{-1} +2 a^4 z^8-8 a^4 z^6+8 a^4 z^4-a^4 z^2+z^2 a^{-4} -a^4- a^{-4} +a^3 z^9+2 a^3 z^7-23 a^3 z^5+36 a^3 z^3+3 z^3 a^{-3} -22 a^3 z-3 z a^{-3} +5 a^3 z^{-1} + a^{-3} z^{-1} +6 a^2 z^8-22 a^2 z^6+2 z^6 a^{-2} +18 a^2 z^4-5 z^4 a^{-2} -2 a^2 z^2+8 z^2 a^{-2} -a^2-3 a^{-2} +a z^9+6 a z^7+5 z^7 a^{-1} -38 a z^5-20 z^5 a^{-1} +54 a z^3+30 z^3 a^{-1} -30 a z-18 z a^{-1} +6 a z^{-1} +4 a^{-1} z^{-1} +4 z^8-12 z^6+5 z^4+6 z^2-3 }[/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 8                   1 -1 6                 2   2 4               3 1   -2 2             4 2     2 0           4 4       0 -2         3 4 1       0 -4       3 4           1 -6     2 3             -1 -8   1 4               3 -10   1                 -1 -12 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{ i=2 }[/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}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/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.

L11n242 L11n244