L8n8

L8n7

L9a1

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8n8 at Knotilus!
	L8n8 is [math]\displaystyle{ 8^4_{3} }[/math] in the Rolfsen table of links.

Detail from an 18th century royal decree, Vietnam.

Link Presentations

[edit Notes on L8n8's Link Presentations]

Planar diagram presentation	X6172 X2536 X16,11,13,12 X3,11,4,10 X9,1,10,4 X7,15,8,14 X13,5,14,8 X12,15,9,16
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, 3, -8}, {-7, 6, 8, -3}

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

L8n7 L9a1