L8a9

L8a8

L8a10

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a9 at Knotilus!
	L8a9 is [math]\displaystyle{ 8^2_{8} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L8a9's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X16,10,7,9 X2738 X4,16,5,15 X12,5,13,6 X14,11,15,12 X6,13,1,14
Gauss code	{1, -4, 2, -5, 6, -8}, {4, -1, 3, -2, 7, -6, 8, -7, 5, -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)^2 t(1)^2-2 t(2) t(1)^2+t(1)^2-2 t(2)^2 t(1)+5 t(2) t(1)-2 t(1)+t(2)^2-2 t(2)+1}{t(1) t(2)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{3}{q^{9/2}}+\frac{4}{q^{7/2}}+q^{5/2}-\frac{6}{q^{5/2}}-3 q^{3/2}+\frac{6}{q^{3/2}}+\frac{1}{q^{11/2}}+4 \sqrt{q}-\frac{6}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 (-z)+2 a^3 z^3+3 a^3 z+a^3 z^{-1} -a z^5-3 a z^3+z^3 a^{-1} -4 a z-a z^{-1} +z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^6 z^4-a^6 z^2+3 a^5 z^5-5 a^5 z^3+2 a^5 z+3 a^4 z^6-3 a^4 z^4+a^3 z^7+6 a^3 z^5-13 a^3 z^3+7 a^3 z-a^3 z^{-1} +6 a^2 z^6-8 a^2 z^4+z^4 a^{-2} +2 a^2 z^2-z^2 a^{-2} +a^2+a z^7+6 a z^5+3 z^5 a^{-1} -13 a z^3-5 z^3 a^{-1} +7 a z+2 z a^{-1} -a z^{-1} +3 z^6-3 z^4 }[/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   \ -5 -4 -3 -2 -1 0 1 2 3 χ 6                 1 -1 4               2   2 2             2 1   -1 0           4 2     2 -2         3 3       0 -4       3 3         0 -6     2 4           2 -8   1 2             -1 -10   2               2 -12 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=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/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} }[/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.

L8a8 L8a10