L11n250

L11n249

L11n251

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n250 at Knotilus!

Link Presentations

[edit Notes on L11n250's Link Presentations]

Planar diagram presentation	X12,1,13,2 X9,19,10,18 X14,6,15,5 X6,12,7,11 X22,15,11,16 X20,8,21,7 X3948 X16,21,17,22 X4,18,5,17 X10,13,1,14 X19,3,20,2
Gauss code	{1, 11, -7, -9, 3, -4, 6, 7, -2, -10}, {4, -1, 10, -3, 5, -8, 9, 2, -11, -6, 8, -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^3-3 u^3 v^2+2 u^3 v-u^2 v^3+4 u^2 v^2-2 u^2 v-2 u v^2+4 u v-u+2 v^2-3 v+1}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{13/2}-4 q^{11/2}+6 q^{9/2}-8 q^{7/2}+8 q^{5/2}-9 q^{3/2}+7 \sqrt{q}-\frac{5}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{1}{q^{5/2}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^7 a^{-3} -2 z^5 a^{-1} +5 z^5 a^{-3} -z^5 a^{-5} +a z^3-7 z^3 a^{-1} +9 z^3 a^{-3} -4 z^3 a^{-5} +2 a z-5 z a^{-1} +8 z a^{-3} -4 z a^{-5} +z a^{-7} + a^{-3} z^{-1} - a^{-5} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^2 a^{-8} +4 z^3 a^{-7} -z a^{-7} +2 z^6 a^{-6} -z^4 a^{-6} +z^2 a^{-6} +5 z^7 a^{-5} -14 z^5 a^{-5} +14 z^3 a^{-5} -6 z a^{-5} + a^{-5} z^{-1} +5 z^8 a^{-4} -15 z^6 a^{-4} +12 z^4 a^{-4} -3 z^2 a^{-4} - a^{-4} +2 z^9 a^{-3} -17 z^5 a^{-3} +18 z^3 a^{-3} -7 z a^{-3} + a^{-3} z^{-1} +8 z^8 a^{-2} -30 z^6 a^{-2} +29 z^4 a^{-2} -8 z^2 a^{-2} +2 z^9 a^{-1} +a z^7-4 z^7 a^{-1} -4 a z^5-7 z^5 a^{-1} +5 a z^3+13 z^3 a^{-1} -2 a z-4 z a^{-1} +3 z^8-13 z^6+16 z^4-5 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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 14                   1 -1 12                 3   3 10               4 2   -2 8             4 2     2 6           4 4       0 4         5 4         1 2       3 5           2 0     2 4             -2 -2   1 3               2 -4   2                 -2 -6 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=4 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/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.

L11n249 L11n251