L9n16

L9n15

L9n17

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n16 at Knotilus!
	L9n16 is [math]\displaystyle{ 9^2_{51} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n16's Link Presentations]

Planar diagram presentation	X8192 X11,17,12,16 X3,10,4,11 X2,15,3,16 X12,5,13,6 X6718 X14,10,15,9 X18,14,7,13 X17,4,18,5
Gauss code	{1, -4, -3, 9, 5, -6}, {6, -1, 7, 3, -2, -5, 8, -7, 4, 2, -9, -8}

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