L9a23

L9a22

L9a24

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a23 at Knotilus!
	L9a23 is [math]\displaystyle{ 9^2_{22} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a23's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X18,16,7,15 X14,5,15,6 X4,13,5,14 X12,18,13,17 X16,12,17,11 X2738 X6,9,1,10
Gauss code	{1, -8, 2, -5, 4, -9}, {8, -1, 9, -2, 7, -6, 5, -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(2)^4+t(1)^2 t(2)^3-3 t(1) t(2)^3+2 t(2)^3-2 t(1)^2 t(2)^2+5 t(1) t(2)^2-2 t(2)^2+2 t(1)^2 t(2)-3 t(1) t(2)+t(2)+t(1)}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{5/2}-3 q^{3/2}+5 \sqrt{q}-\frac{7}{\sqrt{q}}+\frac{8}{q^{3/2}}-\frac{8}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z^3+2 a^5 z+2 a^5 z^{-1} -a^3 z^5-3 a^3 z^3-5 a^3 z-3 a^3 z^{-1} -a z^5-2 a z^3+z^3 a^{-1} -a z+a z^{-1} +z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^4 z^8-a^2 z^8-3 a^5 z^7-6 a^3 z^7-3 a z^7-2 a^6 z^6-4 a^4 z^6-6 a^2 z^6-4 z^6-a^7 z^5+8 a^5 z^5+13 a^3 z^5+a z^5-3 z^5 a^{-1} +4 a^6 z^4+13 a^4 z^4+15 a^2 z^4-z^4 a^{-2} +5 z^4+3 a^7 z^3-10 a^5 z^3-15 a^3 z^3+2 a z^3+4 z^3 a^{-1} -a^6 z^2-11 a^4 z^2-13 a^2 z^2+z^2 a^{-2} -2 z^2-2 a^7 z+8 a^5 z+11 a^3 z-z a^{-1} +3 a^4+3 a^2+1-2 a^5 z^{-1} -3 a^3 z^{-1} -a 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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 6                   1 -1 4                 2   2 2               3 1   -2 0             4 2     2 -2           5 4       -1 -4         3 3         0 -6       3 5           2 -8     2 3             -1 -10     3               3 -12 1 2                 -1 -14 1                   1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/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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