L9n12

L9n11

L9n13

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n12 at Knotilus!
	L9n12 is [math]\displaystyle{ 9^2_{59} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n12's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,17,12,16 X7,15,8,14 X15,9,16,8 X13,5,14,18 X17,13,18,12 X2536 X4,9,1,10
Gauss code	{1, -8, 2, -9}, {8, -1, -4, 5, 9, -2, -3, 7, -6, 4, -5, 3, -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{-t(2)^5-t(1) t(2)^3+t(2)^3+t(1) t(2)^2-t(2)^2-t(1)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{11/2}-q^{9/2}+q^{7/2}-q^{5/2}+q^{3/2}-\sqrt{q}-\frac{1}{\sqrt{q}}-\frac{1}{q^{5/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z a^{-5} +2 z a^{-3} +2 a^{-3} z^{-1} -z^5 a^{-1} +a z^3-6 z^3 a^{-1} +4 a z-9 z a^{-1} +3 a z^{-1} -5 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-6} -3 z^2 a^{-6} + a^{-6} +z^5 a^{-5} -3 z^3 a^{-5} +z a^{-5} +z^4 a^{-4} -2 z^2 a^{-4} +z^3 a^{-3} -3 z a^{-3} +2 a^{-3} z^{-1} +z^6 a^{-2} -7 z^4 a^{-2} +13 z^2 a^{-2} -5 a^{-2} +a z^7+z^7 a^{-1} -7 a z^5-8 z^5 a^{-1} +15 a z^3+19 z^3 a^{-1} -12 a z-16 z a^{-1} +3 a z^{-1} +5 a^{-1} z^{-1} +z^6-7 z^4+12 z^2-5 }[/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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 12                   1 -1 10                     0 8               1 1   0 6           1 1       0 4           1 1       0 2       1 2 1         0 0         2           2 -2     1               1 -4 1                   1 -6 1                   1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ i=4 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ r=-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}^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/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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