L8a10

L8a9

L8a11

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a10 at Knotilus!
	L8a10 is [math]\displaystyle{ 8^2_{5} }[/math] in the Rolfsen table of links.

Symmetric version

Mongolian ornament

Link Presentations

[edit Notes on L8a10's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X12,15,13,16 X14,5,15,6 X4,13,5,14 X16,11,7,12 X2738 X6,9,1,10
Gauss code	{1, -7, 2, -5, 4, -8}, {7, -1, 8, -2, 6, -3, 5, -4, 3, -6}

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 t(2) t(1)^2-t(1)^2+2 t(2)^2 t(1)-3 t(2) t(1)+2 t(1)-t(2)^2+2 t(2)}{t(1) t(2)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{4}{q^{9/2}}-\frac{5}{q^{11/2}}+\frac{4}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{2}{q^{17/2}}-\frac{1}{q^{19/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 z-a^7 z^3+a^7 z^{-1} -2 a^5 z^3-3 a^5 z-a^5 z^{-1} -a^3 z^3-a^3 z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^5 a^{11}+3 z^3 a^{11}-2 z a^{11}-2 z^6 a^{10}+6 z^4 a^{10}-4 z^2 a^{10}-z^7 a^9+4 z^3 a^9-z a^9-4 z^6 a^8+8 z^4 a^8-3 z^2 a^8-z^7 a^7-2 z^5 a^7+6 z^3 a^7-4 z a^7+a^7 z^{-1} -2 z^6 a^6+2 z^2 a^6-a^6-3 z^5 a^5+4 z^3 a^5-4 z a^5+a^5 z^{-1} -2 z^4 a^4+z^2 a^4-z^3 a^3+z a^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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -2                 1 1 -4               2 1 -1 -6             2     2 -8           2 2     0 -10         3 2       1 -12       1 2         1 -14     2 3           -1 -16   1 2             1 -18   1               -1 -20 1                 1

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=-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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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.

L8a9 L8a11