L9n20

L9n19

L9n21

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n20 at Knotilus!
	L9n20 is [math]\displaystyle{ 9^3_{16} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n20's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X11,17,12,16 X15,9,16,18 X17,13,18,12 X2536 X9,1,10,4
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{-2 u v w^2+2 u v w-u v+u w^2-u w+v w^2-v w+w^3-2 w^2+2 w}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^8+2 q^7-4 q^6+5 q^5-4 q^4+6 q^3-3 q^2+3 q }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -2 z^4 a^{-4} +3 z^2 a^{-2} -7 z^2 a^{-4} +3 z^2 a^{-6} +5 a^{-2} -10 a^{-4} +6 a^{-6} - a^{-8} +2 a^{-2} z^{-2} -5 a^{-4} z^{-2} +4 a^{-6} z^{-2} - a^{-8} z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^7 a^{-5} +z^7 a^{-7} +4 z^6 a^{-4} +6 z^6 a^{-6} +2 z^6 a^{-8} +3 z^5 a^{-3} +5 z^5 a^{-5} +3 z^5 a^{-7} +z^5 a^{-9} -11 z^4 a^{-4} -16 z^4 a^{-6} -5 z^4 a^{-8} -6 z^3 a^{-3} -18 z^3 a^{-5} -15 z^3 a^{-7} -3 z^3 a^{-9} +6 z^2 a^{-2} +18 z^2 a^{-4} +15 z^2 a^{-6} +3 z^2 a^{-8} +11 z a^{-3} +21 z a^{-5} +13 z a^{-7} +3 z a^{-9} -7 a^{-2} -14 a^{-4} -10 a^{-6} -2 a^{-8} -5 a^{-3} z^{-1} -9 a^{-5} z^{-1} -5 a^{-7} z^{-1} - a^{-9} z^{-1} +2 a^{-2} z^{-2} +5 a^{-4} z^{-2} +4 a^{-6} z^{-2} + a^{-8} z^{-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   \ 0 1 2 3 4 5 6 7 χ 17               1 -1 15             1   1 13           3 1   -2 11         2 1     1 9       3 4       1 7     3 1         2 5     3           3 3 3 3             0 1 3               3

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ i=3 }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=7 }[/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.

L9n19 L9n21