L9n24

L9n23

L9n25

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n24 at Knotilus!
	L9n24 is [math]\displaystyle{ 9^3_{14} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n24's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X2,14,3,13 X14,7,15,8 X11,16,12,17 X9,11,10,18 X17,5,18,10 X15,1,16,4
Gauss code	{1, -4, -3, 9}, {-2, -1, 5, 3, -7, 8}, {-6, 2, 4, -5, -9, 6, -8, 7}

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 v w^2-2 u v w-u w^2+u w+v^2 (-w)+v^2+2 v w-v}{\sqrt{u} v w} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 2 q^2-2 q+4-3 q^{-1} +4 q^{-2} -2 q^{-3} +2 q^{-4} - q^{-5} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^2 a^4-a^4+z^4 a^2+3 z^2 a^2+a^2 z^{-2} +4 a^2-3 z^2-2 z^{-2} -5+ a^{-2} z^{-2} +2 a^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^5 z^5-3 a^5 z^3+a^5 z+2 a^4 z^6-7 a^4 z^4+6 a^4 z^2-2 a^4+a^3 z^7-a^3 z^5-4 a^3 z^3+3 a^3 z+4 a^2 z^6-14 a^2 z^4+17 a^2 z^2+3 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -8 a^2-4 a^{-2} +a z^7-a z^5+z^5 a^{-1} -2 a z^3-z^3 a^{-1} +5 a z+3 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +2 z^6-7 z^4+14 z^2+2 z^{-2} -9 }[/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 5               2 2 3             1 1 0 1           3 1   2 -1         2 3     1 -3       2 1       1 -5     1 3         2 -7   1 1           0 -9   1             1 -11 1               -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/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}^{3}\oplus{\mathbb Z}_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}^{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}\oplus{\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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