L9a27

L9a26

L9a28

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a27 at Knotilus!
	L9a27 is [math]\displaystyle{ 9^2_{12} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a27's Link Presentations]

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

Integral Khovanov Homology (db, data source)

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

L9a26 L9a28