L9a24

L9a23

L9a25

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a24 at Knotilus!
	L9a24 is [math]\displaystyle{ 9^2_{21} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a24's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X14,6,15,5 X16,11,17,12 X18,13,7,14 X12,17,13,18 X4,16,5,15 X2738 X6,9,1,10
Gauss code	{1, -8, 2, -7, 3, -9}, {8, -1, 9, -2, 4, -6, 5, -3, 7, -4, 6, -5}

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{u^2 v^3-u^2 v^2+u^2 v-u^2+u v^4-3 u v^3+3 u v^2-3 u v+u-v^4+v^3-v^2+v}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\sqrt{q}+\frac{2}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{6}{q^{9/2}}-\frac{6}{q^{11/2}}+\frac{4}{q^{13/2}}-\frac{2}{q^{15/2}}+\frac{1}{q^{17/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^7-2 z a^7-a^7 z^{-1} +z^5 a^5+3 z^3 a^5+4 z a^5+3 a^5 z^{-1} +z^5 a^3+2 z^3 a^3-z a^3-2 a^3 z^{-1} -z^3 a-2 z a }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{10} z^4-2 a^{10} z^2+2 a^9 z^5-3 a^9 z^3+3 a^8 z^6-6 a^8 z^4+5 a^8 z^2-a^8+3 a^7 z^7-8 a^7 z^5+13 a^7 z^3-7 a^7 z+a^7 z^{-1} +a^6 z^8+2 a^6 z^6-8 a^6 z^4+11 a^6 z^2-3 a^6+5 a^5 z^7-15 a^5 z^5+22 a^5 z^3-14 a^5 z+3 a^5 z^{-1} +a^4 z^8+a^4 z^6-6 a^4 z^4+6 a^4 z^2-3 a^4+2 a^3 z^7-4 a^3 z^5+3 a^3 z^3-5 a^3 z+2 a^3 z^{-1} +2 a^2 z^6-5 a^2 z^4+2 a^2 z^2+a z^5-3 a z^3+2 a z }[/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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                   1 1 0                 1   -1 -2               3 1   2 -4             3 2     -1 -6           4 2       2 -8         3 4         1 -10       3 3           0 -12     1 3             2 -14   1 3               -2 -16   1                 1 -18 1                   -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/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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