L9a5

L9a4

L9a6

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a5 at Knotilus!
	L9a5 is [math]\displaystyle{ 9^2_{30} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a5's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,8,15,7 X18,16,5,15 X16,12,17,11 X12,18,13,17 X8,14,9,13 X2536 X4,9,1,10
Gauss code	{1, -8, 2, -9}, {8, -1, 3, -7, 9, -2, 5, -6, 7, -3, 4, -5, 6, -4}

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)^3-3 t(2)^3-4 t(1) t(2)^2+5 t(2)^2+5 t(1) t(2)-4 t(2)-3 t(1)+1}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -3 q^{9/2}+6 q^{7/2}-\frac{1}{q^{7/2}}-8 q^{5/2}+\frac{2}{q^{5/2}}+9 q^{3/2}-\frac{6}{q^{3/2}}+q^{11/2}-9 \sqrt{q}+\frac{7}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^5 a^{-1} -2 a z^3+z^3 a^{-1} -2 z^3 a^{-3} +a^3 z-2 a z-z a^{-1} -z a^{-3} +z a^{-5} +a^3 z^{-1} -2 a^{-1} z^{-1} + a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^8 a^{-2} -z^8-3 a z^7-7 z^7 a^{-1} -4 z^7 a^{-3} -2 a^2 z^6-10 z^6 a^{-2} -5 z^6 a^{-4} -7 z^6-a^3 z^5+5 a z^5+11 z^5 a^{-1} +2 z^5 a^{-3} -3 z^5 a^{-5} +3 a^2 z^4+24 z^4 a^{-2} +7 z^4 a^{-4} -z^4 a^{-6} +19 z^4+3 a^3 z^3-3 a z^3-7 z^3 a^{-1} +2 z^3 a^{-3} +3 z^3 a^{-5} -19 z^2 a^{-2} -5 z^2 a^{-4} +z^2 a^{-6} -13 z^2-3 a^3 z+2 a z+6 z a^{-1} -z a^{-5} -a^2+5 a^{-2} +2 a^{-4} +3+a^3 z^{-1} -2 a^{-1} z^{-1} - a^{-3} 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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 12                   1 -1 10                 2   2 8               4 1   -3 6             4 2     2 4           5 4       -1 2         4 4         0 0       4 6           2 -2     2 3             -1 -4     4               4 -6 1 2                 -1 -8 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=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=5 }[/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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