L11a146

L11a145

L11a147

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a146 at Knotilus!

Link Presentations

[edit Notes on L11a146's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X16,8,17,7 X20,14,21,13 X22,16,7,15 X6,19,1,20 X18,11,19,12 X12,6,13,5 X14,22,15,21 X4,18,5,17
Gauss code	{1, -2, 3, -11, 9, -7}, {4, -1, 2, -3, 8, -9, 5, -10, 6, -4, 11, -8, 7, -5, 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{t(1)^2 t(2)^6-t(1) t(2)^6-2 t(1)^2 t(2)^5+3 t(1) t(2)^5-t(2)^5+2 t(1)^2 t(2)^4-5 t(1) t(2)^4+2 t(2)^4-2 t(1)^2 t(2)^3+5 t(1) t(2)^3-2 t(2)^3+2 t(1)^2 t(2)^2-5 t(1) t(2)^2+2 t(2)^2-t(1)^2 t(2)+3 t(1) t(2)-2 t(2)-t(1)+1}{t(1) t(2)^3} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{13/2}+3 q^{11/2}-5 q^{9/2}+9 q^{7/2}-12 q^{5/2}+13 q^{3/2}-14 \sqrt{q}+\frac{11}{\sqrt{q}}-\frac{9}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{9/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-3} -5 z^5 a^{-3} -7 z^3 a^{-3} -z a^{-3} + a^{-3} z^{-1} +z^9 a^{-1} -a z^7+7 z^7 a^{-1} -5 a z^5+17 z^5 a^{-1} -7 a z^3+15 z^3 a^{-1} -a z+z a^{-1} +2 a z^{-1} -3 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-7} -2 z^3 a^{-7} +3 z^6 a^{-6} -7 z^4 a^{-6} +3 z^2 a^{-6} +4 z^7 a^{-5} -8 z^5 a^{-5} +3 z^3 a^{-5} +4 z^8 a^{-4} +a^4 z^6-7 z^6 a^{-4} -3 a^4 z^4+2 z^4 a^{-4} +a^4 z^2+3 z^2 a^{-4} - a^{-4} +4 z^9 a^{-3} +3 a^3 z^7-12 z^7 a^{-3} -10 a^3 z^5+20 z^5 a^{-3} +7 a^3 z^3-13 z^3 a^{-3} -a^3 z+2 z a^{-3} + a^{-3} z^{-1} +2 z^{10} a^{-2} +4 a^2 z^8-3 z^8 a^{-2} -12 a^2 z^6+7 a^2 z^4+7 z^4 a^{-2} -3 a^{-2} +4 a z^9+8 z^9 a^{-1} -13 a z^7-32 z^7 a^{-1} +14 a z^5+53 z^5 a^{-1} -6 a z^3-31 z^3 a^{-1} -2 a z+z a^{-1} +2 a z^{-1} +3 a^{-1} z^{-1} +2 z^{10}-3 z^8-3 z^6+8 z^4-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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     2   -2 10                   3 1   2 8                 6 2     -4 6               6 3       3 4             7 6         -1 2           7 6           1 0         5 8             3 -2       4 6               -2 -4     2 6                 4 -6   1 3                   -2 -8   2                     2 -10 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=6 }[/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.

L11a145 L11a147