L9n1

L9a55

L9n2

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n1 at Knotilus!
	L9n1 is [math]\displaystyle{ 9^2_{45} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n1's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X3849 X11,18,12,5 X17,12,18,13 X9,16,10,17 X13,2,14,3
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{2 t(1) t(2)^3-t(1) t(2)^2-t(2)+2}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{2}{q^{9/2}}+\frac{1}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{2}{q^{15/2}}-\frac{1}{q^{17/2}}+\frac{1}{q^{19/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{11} z^{-1} +z^3 a^9+4 z a^9+2 a^9 z^{-1} -z^5 a^7-4 z^3 a^7-3 z a^7-z^5 a^5-4 z^3 a^5-3 z a^5-a^5 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{12} z^2-2 a^{12}+a^{11} z^3-a^{11} z+a^{11} z^{-1} +a^{10} z^6-5 a^{10} z^4+10 a^{10} z^2-5 a^{10}+a^9 z^7-5 a^9 z^5+9 a^9 z^3-5 a^9 z+2 a^9 z^{-1} +2 a^8 z^6-8 a^8 z^4+9 a^8 z^2-3 a^8+a^7 z^7-4 a^7 z^5+4 a^7 z^3-a^7 z+a^6 z^6-3 a^6 z^4+a^6+a^5 z^5-4 a^5 z^3+3 a^5 z-a^5 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   \ -7 -6 -5 -4 -3 -2 -1 0 χ -4               1 1 -6             1 1 0 -8           1     1 -10       1 1 1     1 -12       3 1       2 -14       1         1 -16   1 2           -1 -18                 0 -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=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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.

L9a55 L9n2