L11n381

L11n380

L11n382

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n381 at Knotilus!

Link Presentations

[edit Notes on L11n381's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X8493 X22,18,19,17 X11,20,12,21 X19,12,20,13 X18,22,5,21 X9,16,10,17 X2,14,3,13
Gauss code	{1, -11, 5, -3}, {-8, 7, 9, -6}, {-4, -1, 2, -5, -10, 4, -7, 8, 11, -2, 3, 10, 6, -9}

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], ...)	0 (db)
Jones polynomial	[math]\displaystyle{ - q^{-7} + q^{-6} - q^{-5} + q^{-3} + q^{-2} +q+2 q^{-1} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^6 \left(-z^2\right)-a^6 z^{-2} -2 a^6+a^4 z^4+5 a^4 z^2+4 a^4 z^{-2} +7 a^4-a^2 z^4-5 a^2 z^2-5 a^2 z^{-2} -8 a^2+z^2+2 z^{-2} +3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^6 z^8+a^4 z^8+a^7 z^7+2 a^5 z^7+a^3 z^7-6 a^6 z^6-7 a^4 z^6-a^2 z^6-6 a^7 z^5-14 a^5 z^5-8 a^3 z^5+9 a^6 z^4+15 a^4 z^4+7 a^2 z^4+z^4+10 a^7 z^3+28 a^5 z^3+20 a^3 z^3+2 a z^3-4 a^6 z^2-15 a^4 z^2-15 a^2 z^2-4 z^2-6 a^7 z-19 a^5 z-21 a^3 z-8 a z+3 a^6+10 a^4+11 a^2+5+a^7 z^{-1} +5 a^5 z^{-1} +9 a^3 z^{-1} +5 a z^{-1} -a^6 z^{-2} -4 a^4 z^{-2} -5 a^2 z^{-2} -2 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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 3                   1 1 1               1     1 -1             1 4 1   2 -3           1 1 3     3 -5         1 3         2 -7       1 1 1         1 -9       1 2           -1 -11   1 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} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/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.

L11n380 L11n382