L11n198

L11n197

L11n199

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n198 at Knotilus!

Link Presentations

[edit Notes on L11n198's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,3,13,4 X5,14,6,15 X7,17,8,16 X15,21,16,20 X18,14,19,13 X21,6,22,7 X22,18,9,17 X19,5,20,4 X2,9,3,10 X8,11,1,12
Gauss code	{1, -10, 2, 9, -3, 7, -4, -11}, {10, -1, 11, -2, 6, 3, -5, 4, 8, -6, -9, 5, -7, -8}

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(2)^2 t(1)^3-t(2) t(1)^3+t(2)^3 t(1)^2-2 t(2)^2 t(1)^2+2 t(2) t(1)^2+2 t(2)^2 t(1)-2 t(2) t(1)+t(1)-t(2)^2+t(2)}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-2 q^{7/2}+3 q^{5/2}-4 q^{3/2}+4 \sqrt{q}-\frac{5}{\sqrt{q}}+\frac{4}{q^{3/2}}-\frac{3}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{1}{q^{9/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^3 z^3+z^3 a^{-3} +3 a^3 z+2 z a^{-3} +2 a^3 z^{-1} -a z^5-z^5 a^{-1} -4 a z^3-3 z^3 a^{-1} -6 a z-z a^{-1} -3 a z^{-1} + a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a z^9-z^9 a^{-1} -a^2 z^8-2 z^8 a^{-2} -3 z^8+5 a z^7+3 z^7 a^{-1} -2 z^7 a^{-3} +5 a^2 z^6+8 z^6 a^{-2} -z^6 a^{-4} +14 z^6-2 a^3 z^5-11 a z^5-z^5 a^{-1} +8 z^5 a^{-3} -a^4 z^4-11 a^2 z^4-8 z^4 a^{-2} +4 z^4 a^{-4} -22 z^4-a^5 z^3+5 a^3 z^3+13 a z^3-7 z^3 a^{-3} +a^4 z^2+8 a^2 z^2+4 z^2 a^{-2} -3 z^2 a^{-4} +14 z^2+2 a^5 z-6 a^3 z-11 a z-z a^{-1} +2 z a^{-3} -3 a^2- a^{-2} -3+2 a^3 z^{-1} +3 a z^{-1} + a^{-1} z^{-1} }[/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 10                   1 -1 8                 1   1 6               2 1   -1 4             2 1     1 2           2 2       0 0         3 2         1 -2       2 3           1 -4     1 2             -1 -6     2               2 -8 1 1                 0 -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=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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.

L11n197 L11n199