L9n17

L9n16

L9n18

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n17 at Knotilus!
	L9n17 is [math]\displaystyle{ 9^2_{52} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n17's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X7,14,8,15 X13,18,14,7 X17,1,18,6 X16,11,17,12 X5,12,6,13 X4,16,5,15
Gauss code	{1, -2, 3, -9, -8, 6}, {-4, -1, 2, -3, 7, 8, -5, 4, 9, -7, -6, 5}

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{t(2)^2 t(1)^2-t(2) t(1)^2-2 t(2)^2 t(1)+3 t(2) t(1)-2 t(1)-t(2)+1}{t(1) t(2)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{4}{q^{9/2}}-\frac{4}{q^{7/2}}+\frac{2}{q^{5/2}}-\frac{2}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{2}{q^{15/2}}+\frac{3}{q^{13/2}}-\frac{4}{q^{11/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^7-2 z a^7-a^7 z^{-1} +z^5 a^5+4 z^3 a^5+6 z a^5+3 a^5 z^{-1} -2 z^3 a^3-5 z a^3-2 a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^4 a^{10}+2 z^2 a^{10}-2 z^5 a^9+4 z^3 a^9-z a^9-2 z^6 a^8+4 z^4 a^8-3 z^2 a^8+a^8-z^7 a^7+z^5 a^7-2 z^3 a^7+2 z a^7-a^7 z^{-1} -3 z^6 a^6+7 z^4 a^6-9 z^2 a^6+3 a^6-z^7 a^5+3 z^5 a^5-9 z^3 a^5+9 z a^5-3 a^5 z^{-1} -z^6 a^4+2 z^4 a^4-4 z^2 a^4+3 a^4-3 z^3 a^3+6 z a^3-2 a^3 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 -2               2 2 -4             1 1 0 -6           3 1   2 -8         2 2     0 -10       2 2       0 -12     1 2         1 -14   1 2           -1 -16   1             1 -18 1               -1

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=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{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=0 }[/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.

L9n16 L9n18