L11n406

L11n405

L11n407

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n406 at Knotilus!

Link Presentations

[edit Notes on L11n406's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X11,22,12,19 X10,4,11,3 X5,21,6,20 X21,5,22,18 X19,12,20,13 X14,9,15,10 X2,14,3,13 X8,15,9,16
Gauss code	{1, -10, 5, -3}, {-8, 6, -7, 4}, {-6, -1, 2, -11, 9, -5, -4, 8, 10, -9, 11, -2, 3, 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{ 0 }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^3-q^2+2 q+ q^{-1} +2 q^{-4} - q^{-5} + q^{-6} - q^{-7} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^2 a^6-2 a^6+z^4 a^4+4 z^2 a^4+4 a^4+a^2 z^{-2} -z^4-4 z^2-2 z^{-2} -4+z^2 a^{-2} + a^{-2} z^{-2} +2 a^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^7-6 a^7 z^5+10 a^7 z^3-4 a^7 z+a^6 z^8-6 a^6 z^6+11 a^6 z^4-10 a^6 z^2+4 a^6+2 a^5 z^7-12 a^5 z^5+18 a^5 z^3-8 a^5 z+2 a^4 z^8-14 a^4 z^6+30 a^4 z^4-28 a^4 z^2+8 a^4+a^3 z^9-6 a^3 z^7+8 a^3 z^5-2 a^3 z^3+2 a^2 z^8-13 a^2 z^6+z^6 a^{-2} +22 a^2 z^4-5 z^4 a^{-2} -12 a^2 z^2+6 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -4 a^{-2} +a z^9-6 a z^7+z^7 a^{-1} +10 a z^5-4 z^5 a^{-1} -10 a z^3+8 a z+4 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +z^8-4 z^6-2 z^4+12 z^2+2 z^{-2} -7 }[/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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                         0 3                   2 1   1 1               3 1       2 -1             2 5 2       1 -3           1 2 2         1 -5         1 2 1           0 -7       2 2 2             2 -9       1                 1 -11   1 2 1                 0 -13                         0 -15 1                       -1

Integral Khovanov Homology (db, data source)

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

L11n405 L11n407