L9a53

L9a52

L9a54

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a53 at Knotilus!
	L9a53 is [math]\displaystyle{ 9^3_{12} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a53's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X18,10,15,9 X8493 X16,5,17,6 X14,17,5,18 X10,16,11,15 X2,12,3,11
Gauss code	{1, -9, 5, -3}, {8, -6, 7, -4}, {6, -1, 2, -5, 4, -8, 9, -2, 3, -7}

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)-1) (t(2)-1) (t(3)-1)^3}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^4-4 q^3+8 q^2-9 q+12-10 q^{-1} +10 q^{-2} -6 q^{-3} +3 q^{-4} - q^{-5} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^4 z^2-a^4+2 a^2 z^4+z^4 a^{-2} +4 a^2 z^2+z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} +3 a^2+ a^{-2} -z^6-3 z^4-4 z^2-2 z^{-2} -3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 2 a^2 z^8+2 z^8+4 a^3 z^7+11 a z^7+7 z^7 a^{-1} +3 a^4 z^6+7 a^2 z^6+8 z^6 a^{-2} +12 z^6+a^5 z^5-5 a^3 z^5-17 a z^5-7 z^5 a^{-1} +4 z^5 a^{-3} -6 a^4 z^4-22 a^2 z^4-11 z^4 a^{-2} +z^4 a^{-4} -28 z^4-2 a^5 z^3-a^3 z^3+3 a z^3-2 z^3 a^{-3} +5 a^4 z^2+17 a^2 z^2+5 z^2 a^{-2} +17 z^2+a^5 z+3 a^3 z+3 a z+z a^{-1} -2 a^4-6 a^2-2 a^{-2} -5-2 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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                   1 1 7                 3   -3 5               5 1   4 3             4 3     -1 1           8 5       3 -1         6 8         2 -3       4 4           0 -5     2 6             4 -7   1 4               -3 -9   2                 2 -11 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=1 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/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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