L9a35

L9a34

L9a36

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a35 at Knotilus!
	L9a35 is [math]\displaystyle{ 9^2_{9} }[/math] in the Rolfsen table of links.

More symmetric depiction

Link Presentations

[edit Notes on L9a35's Link Presentations]

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

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

Integral Khovanov Homology (db, data source)

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

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