L9n3

L9n2

L9n4

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n3 at Knotilus!
	L9n3 is [math]\displaystyle{ 9^2_{47} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n3's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X8493 X11,18,12,5 X17,12,18,13 X9,16,10,17 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{(u-1) (v-1)}{\sqrt{u} \sqrt{v}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{\sqrt{q}}-\frac{2}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{1}{q^{9/2}}+\frac{1}{q^{11/2}}-\frac{1}{q^{13/2}}+\frac{1}{q^{15/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 (-z)-a^7 z^{-1} +a^5 z^3+3 a^5 z+2 a^5 z^{-1} -a^3 z-a z-a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^8 z^6-5 a^8 z^4+6 a^8 z^2-2 a^8+a^7 z^7-5 a^7 z^5+6 a^7 z^3-2 a^7 z+a^7 z^{-1} +2 a^6 z^6-10 a^6 z^4+13 a^6 z^2-5 a^6+a^5 z^7-5 a^5 z^5+7 a^5 z^3-5 a^5 z+2 a^5 z^{-1} +a^4 z^6-5 a^4 z^4+7 a^4 z^2-3 a^4+a^3 z^3-2 a^3 z+a^2+a z-a 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 0               1 1 -2             1 2 1 -4           1   1 2 -6         1 2     1 -8       1 1       0 -10       1 1       0 -12   1 1           0 -14                 0 -16 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{ i=0 }[/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} }[/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} }[/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}^{2}\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}^{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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