L9a39

L9a38

L9a40

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a39 at Knotilus!
	L9a39 is [math]\displaystyle{ 9^2_{2} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a39's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X14,5,15,6 X6,9,7,10 X16,8,17,7 X18,16,9,15 X8,18,1,17 X4,13,5,14
Gauss code	{1, -2, 3, -9, 4, -5, 6, -8}, {5, -1, 2, -3, 9, -4, 7, -6, 8, -7}

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(2)^3 t(1)^3-t(2)^2 t(1)^3-t(2)^3 t(1)^2+3 t(2)^2 t(1)^2-t(2) t(1)^2-t(2)^2 t(1)+3 t(2) t(1)-t(1)-t(2)+1}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{4}{q^{9/2}}-\frac{5}{q^{7/2}}+\frac{4}{q^{5/2}}+q^{3/2}-\frac{4}{q^{3/2}}-\frac{1}{q^{15/2}}+\frac{2}{q^{13/2}}-\frac{3}{q^{11/2}}-2 \sqrt{q}+\frac{2}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^3 z^7+a^5 z^5-6 a^3 z^5+a z^5+4 a^5 z^3-12 a^3 z^3+4 a z^3+4 a^5 z-9 a^3 z+3 a z+a^5 z^{-1} -a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^3 a^9+z a^9-2 z^4 a^8+2 z^2 a^8-2 z^5 a^7+z^3 a^7-2 z^6 a^6+2 z^4 a^6-z^2 a^6-2 z^7 a^5+5 z^5 a^5-7 z^3 a^5+5 z a^5-a^5 z^{-1} -z^8 a^4+z^6 a^4+3 z^4 a^4-4 z^2 a^4+a^4-4 z^7 a^3+16 z^5 a^3-20 z^3 a^3+10 z a^3-a^3 z^{-1} -z^8 a^2+2 z^6 a^2+3 z^4 a^2-4 z^2 a^2-2 z^7 a+9 z^5 a-11 z^3 a+4 z a-z^6+4 z^4-3 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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                   1 -1 2                 1   1 0               1 1   0 -2             3 1     2 -4           2 2       0 -6         3 2         1 -8       1 2           1 -10     2 3             -1 -12   1 2               1 -14   1                 -1 -16 1                   1

Integral Khovanov Homology (db, data source)

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

L9a38 L9a40