L9n26

L9n25

L9n27

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n26 at Knotilus!
	L9n26 is [math]\displaystyle{ 9^3_{19} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n26's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,17,12,16 X7,14,8,15 X13,8,14,9 X15,13,16,18 X17,5,18,12 X2536 X4,9,1,10
Gauss code	{1, -8, 2, -9}, {-5, 4, -6, 3, -7, 6}, {8, -1, -4, 5, 9, -2, -3, 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(2)-1) (t(3)-1) \left(t(3)^2-t(1)\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-6} - q^{-5} +3 q^{-4} -2 q^{-3} +3 q^{-2} -q-2 q^{-1} +3 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^6 z^{-2} +a^6-2 a^4 z^2-2 a^4 z^{-2} -4 a^4+a^2 z^4+3 a^2 z^2+a^2 z^{-2} +3 a^2-z^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^6 z^6-5 a^6 z^4+8 a^6 z^2+a^6 z^{-2} -5 a^6+a^5 z^7-3 a^5 z^5-a^5 z^3+5 a^5 z-2 a^5 z^{-1} +4 a^4 z^6-17 a^4 z^4+20 a^4 z^2+2 a^4 z^{-2} -10 a^4+a^3 z^7-a^3 z^5-6 a^3 z^3+7 a^3 z-2 a^3 z^{-1} +3 a^2 z^6-12 a^2 z^4+15 a^2 z^2+a^2 z^{-2} -7 a^2+2 a z^5-5 a z^3+3 a z+z a^{-1} +3 z^2-1 }[/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   \ -6 -5 -4 -3 -2 -1 0 1 χ 3               1 -1 1             2   2 -1           2 3   1 -3         1 1 1   1 -5       1 2       1 -7     2 1         1 -9   1 3           2 -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}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}^{2} }[/math] [math]\displaystyle{ r=1 }[/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.

L9n25 L9n27