L9a38

L9a37

L9a39

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a38 at Knotilus!
	L9a38 is [math]\displaystyle{ 9^2_{5} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a38's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X4,9,5,10 X14,6,15,5 X18,14,9,13 X16,8,17,7 X6,16,7,15 X8,18,1,17
Gauss code	{1, -2, 3, -4, 5, -8, 7, -9}, {4, -1, 2, -3, 6, -5, 8, -7, 9, -6}

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)-1) (t(2)-1) (t(1) t(2)+1)^2}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -2 q^{9/2}+3 q^{7/2}-\frac{1}{q^{7/2}}-5 q^{5/2}+\frac{2}{q^{5/2}}+5 q^{3/2}-\frac{3}{q^{3/2}}+q^{11/2}-6 \sqrt{q}+\frac{4}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-1} +a z^5-6 z^5 a^{-1} +z^5 a^{-3} +4 a z^3-12 z^3 a^{-1} +4 z^3 a^{-3} +4 a z-8 z a^{-1} +4 z a^{-3} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-6} -2 z^2 a^{-6} +2 z^5 a^{-5} -4 z^3 a^{-5} +z a^{-5} +2 z^6 a^{-4} -3 z^4 a^{-4} +z^2 a^{-4} +2 z^7 a^{-3} +a^3 z^5-5 z^5 a^{-3} -3 a^3 z^3+8 z^3 a^{-3} +a^3 z-4 z a^{-3} +z^8 a^{-2} +2 a^2 z^6-z^6 a^{-2} -6 a^2 z^4+3 a^2 z^2+3 z^2 a^{-2} +2 a z^7+4 z^7 a^{-1} -6 a z^5-14 z^5 a^{-1} +6 a z^3+21 z^3 a^{-1} -4 a z-10 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +z^8-z^6-2 z^4+3 z^2-1 }[/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                 1   1 8               2 1   -1 6             3 1     2 4           2 2       0 2         4 3         1 0       2 4           2 -2     1 2             -1 -4   1 2               1 -6   1                 -1 -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}\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=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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.

L9a37 L9a39