L8n1

L8a21

L8n2

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8n1 at Knotilus!
	L8n1 is [math]\displaystyle{ 8^2_{16} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L8n1's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X9,12,10,13 X3849 X5,11,6,10 X11,5,12,16 X13,2,14,3
Gauss code	{1, 8, -5, -3}, {-6, -1, 2, 5, -4, 6, -7, 4, -8, -2, 3, 7}

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{u v^3-2 u v^2-2 v+1}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{2}{q^{9/2}}-\frac{2}{q^{7/2}}+\frac{2}{q^{5/2}}-\frac{2}{q^{3/2}}-\frac{2}{q^{11/2}}-\sqrt{q}+\frac{1}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 z^{-1} -a^5 z^3-3 a^5 z-2 a^5 z^{-1} +a^3 z^5+4 a^3 z^3+4 a^3 z+2 a^3 z^{-1} -a z^3-3 a z-a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 3 a^7 z-a^7 z^{-1} +a^6 z^4+2 a^5 z^5-6 a^5 z^3+7 a^5 z-2 a^5 z^{-1} +a^4 z^6-2 a^4 z^4+a^4 z^2-a^4+3 a^3 z^5-10 a^3 z^3+8 a^3 z-2 a^3 z^{-1} +a^2 z^6-3 a^2 z^4+a^2 z^2+a z^5-4 a z^3+4 a z-a 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 χ 2             1 1 0               0 -2         2 1   1 -4       1 1     0 -6     1 1       0 -8   1 1         0 -10 1 1           0 -12 2             2

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/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 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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.

L8a21 L8n2