L9a11

L9a10

L9a12

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a11 at Knotilus!
	L9a11 is [math]\displaystyle{ 9^2_{26} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a11's Link Presentations]

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

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 v^3-4 u v^2+6 u v-2 u-2 v^3+6 v^2-4 v+1}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{5}{q^{9/2}}+\frac{7}{q^{7/2}}+q^{5/2}-\frac{9}{q^{5/2}}-4 q^{3/2}+\frac{9}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{2}{q^{11/2}}+6 \sqrt{q}-\frac{8}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 z^{-1} -3 a^5 z-2 a^5 z^{-1} +3 a^3 z^3+4 a^3 z+2 a^3 z^{-1} -a z^5-2 a z^3+z^3 a^{-1} -3 a z-a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^4 z^8-a^2 z^8-2 a^5 z^7-6 a^3 z^7-4 a z^7-2 a^6 z^6-5 a^4 z^6-9 a^2 z^6-6 z^6-a^7 z^5+5 a^3 z^5-4 z^5 a^{-1} +4 a^6 z^4+12 a^4 z^4+17 a^2 z^4-z^4 a^{-2} +8 z^4+3 a^7 z^3+7 a^5 z^3+7 a^3 z^3+7 a z^3+4 z^3 a^{-1} -2 a^6 z^2-7 a^4 z^2-7 a^2 z^2-2 z^2-3 a^7 z-7 a^5 z-8 a^3 z-4 a z+a^4+a^7 z^{-1} +2 a^5 z^{-1} +2 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                 3   3 2               3 1   -2 0             5 3     2 -2           5 4       -1 -4         4 4         0 -6       3 5           2 -8     2 4             -2 -10   1 4               3 -12   1                 -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{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/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}^{5}\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^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/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}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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