L11n158

L11n157

L11n159

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n158 at Knotilus!

Link Presentations

[edit Notes on L11n158's Link Presentations]

Planar diagram presentation	X8192 X16,7,17,8 X3,10,4,11 X2,15,3,16 X14,10,15,9 X11,19,12,18 X12,5,13,6 X6,21,1,22 X20,14,21,13 X22,17,7,18 X19,4,20,5
Gauss code	{1, -4, -3, 11, 7, -8}, {2, -1, 5, 3, -6, -7, 9, -5, 4, -2, 10, 6, -11, -9, 8, -10}

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{2 u^2 v-u^2+u v^4-u v^3+u v^2-u v+u-v^4+2 v^3}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{4}{q^{9/2}}+\frac{3}{q^{7/2}}-\frac{3}{q^{5/2}}+\frac{2}{q^{3/2}}-\frac{1}{q^{17/2}}+\frac{2}{q^{15/2}}-\frac{3}{q^{13/2}}+\frac{3}{q^{11/2}}-\frac{1}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 z^3+2 a^7 z+a^7 z^{-1} -a^5 z^5-4 a^5 z^3-5 a^5 z-a^5 z^{-1} +a^3 z^3+a^3 z-a z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^7 a^9+5 z^5 a^9-7 z^3 a^9+3 z a^9-2 z^8 a^8+10 z^6 a^8-13 z^4 a^8+4 z^2 a^8-z^9 a^7+3 z^7 a^7+2 z^5 a^7-5 z^3 a^7-z a^7+a^7 z^{-1} -3 z^8 a^6+14 z^6 a^6-17 z^4 a^6+7 z^2 a^6-a^6-z^9 a^5+4 z^7 a^5-4 z^5 a^5+4 z^3 a^5-3 z a^5+a^5 z^{-1} -z^8 a^4+4 z^6 a^4-4 z^4 a^4+2 z^2 a^4-z^5 a^3+2 z^3 a^3-z^2 a^2-z a }[/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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ 0                 1 1 -2               2 1 -1 -4             1     1 -6           2 2     0 -8         2 1       1 -10       1 2         1 -12     2 2           0 -14   1 2             1 -16   1               -1 -18 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=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}^{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=-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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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.

L11n157 L11n159