L9n7

L9n6

L9n8

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n7 at Knotilus!
	L9n7 is [math]\displaystyle{ 9^2_{48} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n7's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X5,12,6,13 X3849 X9,14,10,15 X13,10,14,11 X11,18,12,5 X15,2,16,3
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{2 u v^3-2 u v^2+u v+v^2-2 v+2}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{3}{q^{15/2}}-\frac{3}{q^{17/2}}+\frac{2}{q^{19/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{11} z^{-1} +z^3 a^9+3 z a^9+2 a^9 z^{-1} -z^5 a^7-3 z^3 a^7-z a^7-z^5 a^5-4 z^3 a^5-4 z a^5-a^5 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 3 a^{12} z^2-2 a^{12}+a^{11} z^5+a^{11} z^3-2 a^{11} z+a^{11} z^{-1} +2 a^{10} z^6-6 a^{10} z^4+11 a^{10} z^2-5 a^{10}+a^9 z^7-2 a^9 z^5+5 a^9 z^3-6 a^9 z+2 a^9 z^{-1} +3 a^8 z^6-8 a^8 z^4+7 a^8 z^2-3 a^8+a^7 z^7-2 a^7 z^5+a^6 z^6-2 a^6 z^4-a^6 z^2+a^6+a^5 z^5-4 a^5 z^3+4 a^5 z-a^5 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 -4               1 1 -6             1 1 0 -8           2     2 -10         1 1     0 -12       3 2       1 -14     1 2         1 -16   2 2           0 -18   1             1 -20 2               -2

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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} }[/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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