L9a20

L9a19

L9a21

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a20 at Knotilus!
	L9a20 is [math]\displaystyle{ 9^2_{42} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a20's Link Presentations]

Planar diagram presentation	X8192 X16,11,17,12 X10,4,11,3 X2,15,3,16 X12,5,13,6 X6718 X14,10,15,9 X18,14,7,13 X4,18,5,17
Gauss code	{1, -4, 3, -9, 5, -6}, {6, -1, 7, -3, 2, -5, 8, -7, 4, -2, 9, -8}

A Braid Representative	{{{braid_table}}}
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)^2 t(2)^4-t(1) t(2)^4-3 t(1)^2 t(2)^3+4 t(1) t(2)^3-t(2)^3+3 t(1)^2 t(2)^2-7 t(1) t(2)^2+3 t(2)^2-t(1)^2 t(2)+4 t(1) t(2)-3 t(2)-t(1)+1}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{7/2}+4 q^{5/2}-7 q^{3/2}+9 \sqrt{q}-\frac{12}{\sqrt{q}}+\frac{11}{q^{3/2}}-\frac{10}{q^{5/2}}+\frac{7}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{1}{q^{11/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^3 z^5-2 a^3 z^3+a^3 z^{-1} +a z^7+4 a z^5-z^5 a^{-1} +4 a z^3-2 z^3 a^{-1} -a z-a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -3 a^2 z^8-3 z^8-7 a^3 z^7-13 a z^7-6 z^7 a^{-1} -7 a^4 z^6-6 a^2 z^6-4 z^6 a^{-2} -3 z^6-4 a^5 z^5+8 a^3 z^5+25 a z^5+12 z^5 a^{-1} -z^5 a^{-3} -a^6 z^4+8 a^4 z^4+16 a^2 z^4+7 z^4 a^{-2} +14 z^4+3 a^5 z^3-2 a^3 z^3-12 a z^3-6 z^3 a^{-1} +z^3 a^{-3} -2 a^4 z^2-6 a^2 z^2-2 z^2 a^{-2} -6 z^2-a^3 z-a z-a^2+a^3 z^{-1} +a 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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                   1 1 6                 3   -3 4               4 1   3 2             5 3     -2 0           7 4       3 -2         5 6         1 -4       5 6           -1 -6     3 6             3 -8   1 4               -3 -10   3                 3 -12 1                   -1

Integral Khovanov Homology (db, data source)

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

L9a19 L9a21