L9a50

L9a49

L9a51

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a50 at Knotilus!
	L9a50 is [math]\displaystyle{ 9^3_{1} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a50's Link Presentations]

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

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(3)^2 t(2)^2-t(3)^2 t(2)^2-t(1) t(3) t(2)^2+2 t(3) t(2)^2-t(2)^2-2 t(1) t(3)^2 t(2)+t(3)^2 t(2)-t(1) t(2)+2 t(1) t(3) t(2)-2 t(3) t(2)+2 t(2)+t(1) t(3)^2+t(1)-2 t(1) t(3)+t(3)-1}{\sqrt{t(1)} t(2) t(3)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^6+3 q^5-5 q^4+7 q^3+ q^{-3} -7 q^2-2 q^{-2} +8 q+5 q^{-1} -5 }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^4 a^{-4} -2 z^2 a^{-4} - a^{-4} +z^6 a^{-2} +4 z^4 a^{-2} +a^2 z^2+6 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} +2 a^2+4 a^{-2} -2 z^4-6 z^2-2 z^{-2} -5 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^8 a^{-2} +z^8+2 a z^7+6 z^7 a^{-1} +4 z^7 a^{-3} +a^2 z^6+8 z^6 a^{-2} +6 z^6 a^{-4} +3 z^6-6 a z^5-14 z^5 a^{-1} -3 z^5 a^{-3} +5 z^5 a^{-5} -4 a^2 z^4-27 z^4 a^{-2} -9 z^4 a^{-4} +3 z^4 a^{-6} -19 z^4+3 a z^3+3 z^3 a^{-1} -5 z^3 a^{-3} -4 z^3 a^{-5} +z^3 a^{-7} +6 a^2 z^2+23 z^2 a^{-2} +6 z^2 a^{-4} -z^2 a^{-6} +22 z^2+3 a z+5 z a^{-1} +3 z a^{-3} +z a^{-5} -4 a^2-8 a^{-2} -2 a^{-4} -9-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 13                   1 -1 11                 2   2 9               3 1   -2 7             4 2     2 5           4 4       0 3         4 3         1 1       3 6           3 -1     2 2             0 -3   1 4               3 -5   1                 -1 -7 1                   1

Integral Khovanov Homology (db, data source)

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