L9n22

L9n21

L9n23

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n22 at Knotilus!
	L9n22 is [math]\displaystyle{ 9^3_{15} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n22's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X11,17,12,16 X15,9,16,18 X17,13,18,12 X2536 X4,9,1,10
Gauss code	{1, -8, 2, -9}, {8, -1, 3, -4}, {9, -2, -5, 7, 4, -3, -6, 5, -7, 6}

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+u w^3-u w^2+u w-u+v w^3-v w^2+v w-v-w^3}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -2 q+3-3 q^{-1} +4 q^{-2} -2 q^{-3} +4 q^{-4} - q^{-5} + q^{-6} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ 2 a^6 z^{-2} +a^6-3 z^2 a^4-5 a^4 z^{-2} -8 a^4+2 z^4 a^2+8 z^2 a^2+4 a^2 z^{-2} +10 a^2-2 z^2- z^{-2} -3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^6 z^6-5 a^6 z^4+9 a^6 z^2+2 a^6 z^{-2} -7 a^6+a^5 z^7-2 a^5 z^5-5 a^5 z^3+11 a^5 z-5 a^5 z^{-1} +5 a^4 z^6-20 a^4 z^4+23 a^4 z^2+5 a^4 z^{-2} -14 a^4+a^3 z^7+2 a^3 z^5-17 a^3 z^3+21 a^3 z-9 a^3 z^{-1} +4 a^2 z^6-14 a^2 z^4+16 a^2 z^2+4 a^2 z^{-2} -10 a^2+4 a z^5-12 a z^3+13 a z+3 z a^{-1} -5 a z^{-1} - a^{-1} z^{-1} +z^4+2 z^2+ z^{-2} -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   \ -6 -5 -4 -3 -2 -1 0 1 χ 3               2 -2 1             1   1 -1           3 3   0 -3         1 1 1   1 -5       1 3       2 -7     3 1         2 -9   1 4           3 -11                 0 -13 1               1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{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.

L9n21 L9n23