L9n2

L9n1

L9n3

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n2 at Knotilus!
	L9n2 is [math]\displaystyle{ 9^2_{46} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n2's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X15,1,16,4 X5,10,6,11 X3849 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	{{{braid_table}}}
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)-1) (t(2)-1)}{\sqrt{t(1)} \sqrt{t(2)}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{2}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{3}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{2}{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+2 a^5 z+2 a^5 z^{-1} +a^3 z^3+a^3 z-2 a z-a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^6 a^8+5 z^4 a^8-7 z^2 a^8+2 a^8-z^7 a^7+4 z^5 a^7-4 z^3 a^7+2 z a^7-a^7 z^{-1} -3 z^6 a^6+12 z^4 a^6-13 z^2 a^6+5 a^6-z^7 a^5+2 z^5 a^5+3 z a^5-2 a^5 z^{-1} -2 z^6 a^4+6 z^4 a^4-6 z^2 a^4+3 a^4-2 z^5 a^3+4 z^3 a^3-2 z a^3-z^4 a^2-a^2-3 z a+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               2 2 -2             2 2 0 -4           1     1 -6         1 2     1 -8       2 1       1 -10       1         1 -12   1 2           -1 -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=-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}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}^{2} }[/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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