L11n179

L11n178

L11n180

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n179 at Knotilus!

Link Presentations

[edit Notes on L11n179's Link Presentations]

Planar diagram presentation	X8192 X3,10,4,11 X16,7,17,8 X20,10,21,9 X22,15,7,16 X14,6,15,5 X18,14,19,13 X12,20,13,19 X11,4,12,5 X6,18,1,17 X2,21,3,22
Gauss code	{1, -11, -2, 9, 6, -10}, {3, -1, 4, 2, -9, -8, 7, -6, 5, -3, 10, -7, 8, -4, 11, -5}

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{t(1) t(2)^4-t(2)^4+t(1)^2 t(2)^3-4 t(1) t(2)^3+2 t(2)^3-2 t(1)^2 t(2)^2+7 t(1) t(2)^2-2 t(2)^2+2 t(1)^2 t(2)-4 t(1) t(2)+t(2)-t(1)^2+t(1)}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{6}{q^{9/2}}+\frac{8}{q^{7/2}}+q^{5/2}-\frac{10}{q^{5/2}}-4 q^{3/2}+\frac{10}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{3}{q^{11/2}}+6 \sqrt{q}-\frac{9}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^5+z a^5+a^5 z^{-1} -z^5 a^3-2 z^3 a^3-3 z a^3-2 a^3 z^{-1} -z^5 a-z^3 a+z a+2 a z^{-1} +z^3 a^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^3 z^9-a z^9-4 a^4 z^8-5 a^2 z^8-z^8-5 a^5 z^7-5 a^3 z^7-3 a^6 z^6+9 a^4 z^6+12 a^2 z^6-a^7 z^5+13 a^5 z^5+18 a^3 z^5-4 z^5 a^{-1} +6 a^6 z^4-8 a^4 z^4-15 a^2 z^4-z^4 a^{-2} -2 z^4+2 a^7 z^3-11 a^5 z^3-23 a^3 z^3-5 a z^3+5 z^3 a^{-1} -a^6 z^2+3 a^4 z^2+6 a^2 z^2+z^2 a^{-2} +3 z^2+5 a^5 z+11 a^3 z+7 a z+z a^{-1} -a^2-a^5 z^{-1} -2 a^3 z^{-1} -2 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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 6                   1 -1 4                 3   3 2               3 1   -2 0             6 3     3 -2           5 4       -1 -4         5 5         0 -6       4 6           2 -8     2 4             -2 -10   1 4               3 -12   2                 -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{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/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}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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.

L11n178 L11n180