L9a55

L9a54

L9n1

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a55 at Knotilus!
	L9a55 is [math]\displaystyle{ 9^4_{1} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a55's Link Presentations]

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

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)-t(2) t(3) t(1)+t(3) t(1)-t(2) t(4) t(1)+t(2) t(3) t(4) t(1)-2 t(3) t(4) t(1)+2 t(4) t(1)-2 t(1)-2 t(2)+2 t(2) t(3)-t(3)+t(2) t(4)-2 t(2) t(3) t(4)+2 t(3) t(4)-t(4)+1}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} \sqrt{t(4)}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{5/2}-4 q^{3/2}+6 \sqrt{q}-\frac{8}{\sqrt{q}}+\frac{7}{q^{3/2}}-\frac{10}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{1}{q^{11/2}}-\frac{1}{q^{13/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 z^{-1} +a^7 z^{-3} -3 z a^5-5 a^5 z^{-1} -3 a^5 z^{-3} +3 z^3 a^3+6 z a^3+7 a^3 z^{-1} +3 a^3 z^{-3} -z^5 a-2 z^3 a-3 z a-3 a z^{-1} -a z^{-3} +z^3 a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^4 z^8-a^2 z^8-a^5 z^7-5 a^3 z^7-4 a z^7-a^6 z^6-a^4 z^6-6 a^2 z^6-6 z^6-a^7 z^5-3 a^5 z^5+4 a^3 z^5+2 a z^5-4 z^5 a^{-1} -2 a^4 z^4+7 a^2 z^4-z^4 a^{-2} +8 z^4+4 a^7 z^3+12 a^5 z^3+8 a^3 z^3+4 a z^3+4 z^3 a^{-1} +6 a^6 z^2+12 a^4 z^2+6 a^2 z^2-6 a^7 z-14 a^5 z-14 a^3 z-6 a z-8 a^6-15 a^4-8 a^2+4 a^7 z^{-1} +9 a^5 z^{-1} +9 a^3 z^{-1} +4 a z^{-1} +3 a^6 z^{-2} +6 a^4 z^{-2} +3 a^2 z^{-2} -a^7 z^{-3} -3 a^5 z^{-3} -3 a^3 z^{-3} -a z^{-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   \ -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           6 7       1 -4         4 1         3 -6       1 6           5 -8     4 4             0 -10   1 5               4 -12                     0 -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}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2 }[/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.

L9a54 L9n1