L9n23

L9n22

L9n24

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n23 at Knotilus!
	L9n23 is [math]\displaystyle{ 9^3_{13} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n23's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X13,2,14,3 X14,7,15,8 X11,16,12,17 X9,11,10,18 X17,5,18,10 X4,15,1,16
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{-t(1) t(3)^2 t(2)^2+t(1) t(3) t(2)^2+t(1) t(3)^2 t(2)-t(2)-t(3)+1}{\sqrt{t(1)} t(2) t(3)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 1- q^{-1} +2 q^{-2} - q^{-3} +3 q^{-4} - q^{-5} +2 q^{-6} - q^{-7} }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^8+z^4 a^6+4 z^2 a^6+a^6 z^{-2} +3 a^6-z^6 a^4-5 z^4 a^4-7 z^2 a^4-2 a^4 z^{-2} -5 a^4+z^4 a^2+4 z^2 a^2+a^2 z^{-2} +3 a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z a^9+2 z^2 a^8-2 a^8+z^5 a^7-3 z^3 a^7+3 z a^7+2 z^6 a^6-9 z^4 a^6+13 z^2 a^6+a^6 z^{-2} -8 a^6+z^7 a^5-3 z^5 a^5-z^3 a^5+5 z a^5-2 a^5 z^{-1} +3 z^6 a^4-14 z^4 a^4+18 z^2 a^4+2 a^4 z^{-2} -9 a^4+z^7 a^3-4 z^5 a^3+2 z^3 a^3+3 z a^3-2 a^3 z^{-1} +z^6 a^2-5 z^4 a^2+7 z^2 a^2+a^2 z^{-2} -4 a^2 }[/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 1               1 1 -1                 0 -3           2 1   1 -5       1 1 1     1 -7       3 1       2 -9   1 1 2         2 -11   2 1           1 -13   1             1 -15 1               -1

Integral Khovanov Homology (db, data source)

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

L9n22 L9n24