L9n11

L9n10

L9n12

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n11 at Knotilus!
	L9n11 is [math]\displaystyle{ 9^2_{57} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n11's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,8,15,7 X15,18,16,5 X11,16,12,17 X17,12,18,13 X8,14,9,13 X2536 X4,9,1,10
Gauss code	{1, -8, 2, -9}, {8, -1, 3, -7, 9, -2, -5, 6, 7, -3, -4, 5, -6, 4}

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) t(2)^3-t(2)^3-2 t(1) t(2)^2+t(2)^2+t(1) t(2)-2 t(2)-t(1)+1}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\sqrt{q}+\frac{2}{\sqrt{q}}-\frac{3}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{3}{q^{11/2}}+\frac{1}{q^{13/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 \left(-z^3\right)-a^5 z+a^5 z^{-1} +a^3 z^5+3 a^3 z^3+a^3 z-a^3 z^{-1} -a z^3-2 a z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^2 a^8-3 z^3 a^7+z a^7-z^6 a^6+z^4 a^6-2 z^2 a^6-z^7 a^5+2 z^5 a^5-4 z^3 a^5+4 z a^5-a^5 z^{-1} -3 z^6 a^4+7 z^4 a^4-4 z^2 a^4+a^4-z^7 a^3+z^5 a^3+2 z^3 a^3+z a^3-a^3 z^{-1} -2 z^6 a^2+6 z^4 a^2-3 z^2 a^2-z^5 a+3 z^3 a-2 z a }[/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 χ 2               1 1 0             1   -1 -2           2 1   1 -4         2 2     0 -6       2 1       1 -8     1 2         1 -10   2 2           0 -12   2             2 -14 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=-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}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/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.

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