L9a44

L9a43

L9a45

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a44 at Knotilus!
	L9a44 is [math]\displaystyle{ 9^3_{3} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a44's Link Presentations]

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

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) t(3)^3+t(1) t(2) t(3)^3-2 t(2) t(3)^3+t(3)^3+2 t(1) t(3)^2-t(1) t(2) t(3)^2+2 t(2) t(3)^2-t(3)^2-2 t(1) t(3)+t(1) t(2) t(3)-2 t(2) t(3)+t(3)+2 t(1)-t(1) t(2)+t(2)-1}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^6+3 q^5-6 q^4+7 q^3-7 q^2+8 q-5+5 q^{-1} - q^{-2} + q^{-3} }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^6 a^{-2} +4 z^4 a^{-2} -z^4 a^{-4} -2 z^4+a^2 z^2+7 z^2 a^{-2} -2 z^2 a^{-4} -7 z^2+3 a^2+8 a^{-2} -2 a^{-4} -9+2 a^2 z^{-2} +4 a^{-2} z^{-2} - a^{-4} z^{-2} -5 z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^3 a^{-7} +3 z^4 a^{-6} +6 z^5 a^{-5} -6 z^3 a^{-5} +3 z a^{-5} - a^{-5} z^{-1} +7 z^6 a^{-4} -11 z^4 a^{-4} +6 z^2 a^{-4} + a^{-4} z^{-2} -2 a^{-4} +4 z^7 a^{-3} -z^5 a^{-3} -12 z^3 a^{-3} +13 z a^{-3} -5 a^{-3} z^{-1} +z^8 a^{-2} +a^2 z^6+7 z^6 a^{-2} -5 a^2 z^4-21 z^4 a^{-2} +9 a^2 z^2+16 z^2 a^{-2} +2 a^2 z^{-2} +4 a^{-2} z^{-2} -7 a^2-10 a^{-2} +a z^7+5 z^7 a^{-1} -a z^5-8 z^5 a^{-1} -6 a z^3-11 z^3 a^{-1} +11 a z+21 z a^{-1} -5 a z^{-1} -9 a^{-1} z^{-1} +z^8+z^6-12 z^4+19 z^2+5 z^{-2} -14 }[/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               4 1   -3 7             3 2     1 5           4 4       0 3         4 3         1 1       4 7           3 -1     1 1             0 -3     4               4 -5 1 1                 0 -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{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/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} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_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}^{3} }[/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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