L9a45

L9a44

L9a46

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a45 at Knotilus!
	L9a45 is [math]\displaystyle{ 9^3_{7} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a45's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X18,12,9,11 X16,14,17,13 X8,16,5,15 X14,8,15,7 X12,18,13,17 X2536 X4,9,1,10
Gauss code	{1, -8, 2, -9}, {8, -1, 6, -5}, {9, -2, 3, -7, 4, -6, 5, -4, 7, -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{-2 t(2) t(1)+2 t(2) t(3) t(1)-2 t(3) t(1)+3 t(1)+2 t(2)-3 t(2) t(3)+2 t(3)-2}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^5+3 q^4-4 q^3+5 q^2-6 q+7-4 q^{-1} +4 q^{-2} - q^{-3} + q^{-4} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^4 z^{-2} +a^4-2 z^2 a^2-2 a^2 z^{-2} -3 a^2+z^4+z^2+ z^{-2} +2+z^4 a^{-2} +z^2 a^{-2} -z^2 a^{-4} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-5} -2 z^3 a^{-5} +3 z^6 a^{-4} +a^4 z^4-8 z^4 a^{-4} -3 a^4 z^2+3 z^2 a^{-4} -a^4 z^{-2} +3 a^4+3 z^7 a^{-3} +a^3 z^5-8 z^5 a^{-3} +5 z^3 a^{-3} -3 a^3 z+2 a^3 z^{-1} +z^8 a^{-2} +a^2 z^6+z^6 a^{-2} +2 a^2 z^4-5 z^4 a^{-2} -6 a^2 z^2+2 z^2 a^{-2} -2 a^2 z^{-2} +5 a^2+a z^7+4 z^7 a^{-1} +a z^5-9 z^5 a^{-1} +7 z^3 a^{-1} -3 a z+2 a z^{-1} +z^8-z^6+4 z^4-4 z^2- z^{-2} +3 }[/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 11                   1 -1 9                 2   2 7               2 1   -1 5             3 2     1 3           3 2       -1 1         4 3         1 -1       3 6           3 -3     1 1             0 -5     3               3 -7 1 1                 0 -9 1                   1

Integral Khovanov Homology (db, data source)

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

L9a44 L9a46