L9n8

L9n7

L9n9

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n8 at Knotilus!
	L9n8 is [math]\displaystyle{ 9^2_{56} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n8's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X5,12,6,13 X8493 X9,14,10,15 X13,10,14,11 X11,18,12,5 X2,16,3,15
Gauss code	{1, -9, 5, -3}, {-4, -1, 2, -5, -6, 7, -8, 4, -7, 6, 9, -2, 3, 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-1) (v-1) \left(v^2-v+1\right)}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{4}{q^{9/2}}-\frac{5}{q^{7/2}}+\frac{3}{q^{5/2}}-\frac{4}{q^{3/2}}+\frac{2}{q^{13/2}}-\frac{3}{q^{11/2}}-\sqrt{q}+\frac{2}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^7 z^{-1} -z^3 a^5+2 a^5 z^{-1} +z^5 a^3+3 z^3 a^3+2 z a^3-z^3 a-2 z a-a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -3 z^2 a^8+2 a^8-z^5 a^7-z^3 a^7-a^7 z^{-1} -2 z^6 a^6+4 z^4 a^6-8 z^2 a^6+5 a^6-z^7 a^5-z^5 a^5+4 z^3 a^5-z a^5-2 a^5 z^{-1} -4 z^6 a^4+9 z^4 a^4-6 z^2 a^4+3 a^4-z^7 a^3-z^5 a^3+8 z^3 a^3-4 z a^3-2 z^6 a^2+5 z^4 a^2-z^2 a^2-a^2-z^5 a+3 z^3 a-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   \ -5 -4 -3 -2 -1 0 1 2 χ 2               1 1 0             1   -1 -2           3 1   2 -4         2 3     1 -6       3 1       2 -8     1 2         1 -10   2 3           -1 -12   1             1 -14 2               -2

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}^{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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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