L9a54

L9a53

L9a55

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a54 at Knotilus!
	L9a54 is [math]\displaystyle{ 9^3_{9} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a54's Link Presentations]

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

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)-1) (t(2)-1) (t(3)-1) \left(t(3)^2-t(3)+1\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^6+3 q^5-6 q^4+8 q^3+ q^{-3} -7 q^2-2 q^{-2} +9 q+5 q^{-1} -6 }[/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} -6 z^2+2 a^2+6 a^{-2} -2 a^{-4} -6+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^8 a^{-2} +z^8+2 a z^7+7 z^7 a^{-1} +5 z^7 a^{-3} +a^2 z^6+11 z^6 a^{-2} +8 z^6 a^{-4} +4 z^6-6 a z^5-16 z^5 a^{-1} -4 z^5 a^{-3} +6 z^5 a^{-5} -4 a^2 z^4-35 z^4 a^{-2} -14 z^4 a^{-4} +3 z^4 a^{-6} -22 z^4+4 a z^3+3 z^3 a^{-1} -7 z^3 a^{-3} -5 z^3 a^{-5} +z^3 a^{-7} +6 a^2 z^2+30 z^2 a^{-2} +10 z^2 a^{-4} +26 z^2+2 a z+6 z a^{-1} +6 z a^{-3} +2 z a^{-5} -4 a^2-12 a^{-2} -4 a^{-4} -11-2 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               4 1   -3 7             4 2     2 5           3 4       1 3         6 4         2 1       4 7           3 -1     1 2             -1 -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} }[/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}^{7}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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