L9a1

L8n8

L9a2

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a1 at Knotilus!
	L9a1 is [math]\displaystyle{ 9^2_{32} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a1's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X10,6,11,5 X8493 X18,12,5,11 X12,18,13,17 X16,10,17,9 X2,14,3,13
Gauss code	{1, -9, 5, -3}, {4, -1, 2, -5, 8, -4, 6, -7, 9, -2, 3, -8, 7, -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{(t(1)-1) (t(2)-1) \left(2 t(2)^2-3 t(2)+2\right)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{13/2}+3 q^{11/2}-5 q^{9/2}+8 q^{7/2}-10 q^{5/2}+9 q^{3/2}-9 \sqrt{q}+\frac{6}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{1}{q^{5/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^5 a^{-1} +z^5 a^{-3} -a z^3+z^3 a^{-1} +2 z^3 a^{-3} -z^3 a^{-5} -z a^{-1} +2 z a^{-3} -z a^{-5} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 z^8 a^{-2} -2 z^8 a^{-4} -5 z^7 a^{-1} -9 z^7 a^{-3} -4 z^7 a^{-5} -5 z^6 a^{-2} -2 z^6 a^{-4} -3 z^6 a^{-6} -6 z^6-4 a z^5+4 z^5 a^{-1} +18 z^5 a^{-3} +9 z^5 a^{-5} -z^5 a^{-7} -a^2 z^4+11 z^4 a^{-2} +10 z^4 a^{-4} +7 z^4 a^{-6} +7 z^4+4 a z^3-13 z^3 a^{-3} -7 z^3 a^{-5} +2 z^3 a^{-7} -4 z^2 a^{-2} -7 z^2 a^{-4} -4 z^2 a^{-6} -z^2+2 z a^{-1} +4 z a^{-3} +2 z a^{-5} +1-a z^{-1} - a^{-1} 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   \ -3 -2 -1 0 1 2 3 4 5 6 χ 14                   1 1 12                 2   -2 10               3 1   2 8             5 2     -3 6           5 3       2 4         4 5         1 2       5 5           0 0     3 6             3 -2   1 3               -2 -4   3                 3 -6 1                   -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=6 }[/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.

L8n8 L9a2