L9a34

L9a33

L9a35

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a34 at Knotilus!
	L9a34 is [math]\displaystyle{ 9^2_{6} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a34's Link Presentations]

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

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(1)+t(2)-1) (t(1) t(2)+1) (t(2) t(1)-t(1)-t(2))}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{6}{q^{9/2}}-\frac{6}{q^{7/2}}+\frac{5}{q^{5/2}}-\frac{4}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{2}{q^{15/2}}+\frac{3}{q^{13/2}}-\frac{6}{q^{11/2}}-\sqrt{q}+\frac{2}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^7-2 z a^7+z^5 a^5+3 z^3 a^5+3 z a^5+a^5 z^{-1} +z^5 a^3+2 z^3 a^3-z a^3-a^3 z^{-1} -z^3 a-2 z a }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^4 a^{10}+2 z^2 a^{10}-2 z^5 a^9+4 z^3 a^9-2 z a^9-2 z^6 a^8+2 z^4 a^8-2 z^7 a^7+3 z^5 a^7-3 z^3 a^7-z a^7-z^8 a^6-z^2 a^6-4 z^7 a^5+10 z^5 a^5-12 z^3 a^5+6 z a^5-a^5 z^{-1} -z^8 a^4+2 z^4 a^4-z^2 a^4+a^4-2 z^7 a^3+4 z^5 a^3-2 z^3 a^3+3 z a^3-a^3 z^{-1} -2 z^6 a^2+5 z^4 a^2-2 z^2 a^2-z^5 a+3 z^3 a-2 z a }[/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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                   1 1 0                 1   -1 -2               3 1   2 -4             3 2     -1 -6           3 2       1 -8         3 3         0 -10       3 3           0 -12     1 4             3 -14   1 2               -1 -16   1                 1 -18 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=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/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}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}_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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