L9a36

L9a35

L9a37

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a36 at Knotilus!
	L9a36 is [math]\displaystyle{ 9^2_{1} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a36's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,4,13,3 X18,12,9,11 X14,6,15,5 X16,8,17,7 X2,9,3,10 X4,14,5,13 X6,16,7,15 X8,18,1,17
Gauss code	{1, -6, 2, -7, 4, -8, 5, -9}, {6, -1, 3, -2, 7, -4, 8, -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^3 v^3-u^3 v^2-u^2 v^3+u^2 v^2-u^2 v-u v^2+u v-u-v+1}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -3 q^{9/2}+2 q^{7/2}-2 q^{5/2}+q^{3/2}+q^{19/2}-2 q^{17/2}+2 q^{15/2}-3 q^{13/2}+3 q^{11/2}-\sqrt{q} }[/math] (db)
Signature	5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^5 a^{-7} +4 z^3 a^{-7} +3 z a^{-7} -z^7 a^{-5} -6 z^5 a^{-5} -11 z^3 a^{-5} -7 z a^{-5} - a^{-5} z^{-1} +z^5 a^{-3} +5 z^3 a^{-3} +6 z a^{-3} + a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^8 a^{-4} -z^8 a^{-6} -z^7 a^{-3} -3 z^7 a^{-5} -2 z^7 a^{-7} +5 z^6 a^{-4} +3 z^6 a^{-6} -2 z^6 a^{-8} +6 z^5 a^{-3} +15 z^5 a^{-5} +7 z^5 a^{-7} -2 z^5 a^{-9} -6 z^4 a^{-4} +4 z^4 a^{-8} -2 z^4 a^{-10} -11 z^3 a^{-3} -21 z^3 a^{-5} -6 z^3 a^{-7} +2 z^3 a^{-9} -2 z^3 a^{-11} -2 z^2 a^{-6} +z^2 a^{-10} -z^2 a^{-12} +7 z a^{-3} +9 z a^{-5} +2 z a^{-7} +z a^{-9} +z a^{-11} + a^{-4} - a^{-3} z^{-1} - a^{-5} 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   \ -2 -1 0 1 2 3 4 5 6 7 χ 20                   1 -1 18                 1   1 16               1 1   0 14             2 1     1 12           2 2       0 10         1 1         0 8       1 2           1 6     1 1             0 4   1 2               1 2                     0 0 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=6 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/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=7 }[/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.

L9a35 L9a37