L9a48

L9a47

L9a49

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a48 at Knotilus!
	L9a48 is [math]\displaystyle{ 9^3_{5} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a48's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X18,14,11,13 X8,16,9,15 X14,8,15,7 X16,10,17,9 X10,18,5,17 X2536 X4,11,1,12
Gauss code	{1, -8, 2, -9}, {8, -1, 5, -4, 6, -7}, {9, -2, 3, -5, 4, -6, 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{u v^2 w^2-u v^2 w-u v w^2+2 u v w-u v-u w+2 u-2 v^2 w^2+v^2 w+v w^2-2 v w+v+w-1}{\sqrt{u} v w} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^6+3 q^5-4 q^4+6 q^3-6 q^2+6 q-4+4 q^{-1} - q^{-2} + q^{-3} }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^4 a^{-4} -2 z^2 a^{-4} +z^6 a^{-2} +4 z^4 a^{-2} +a^2 z^2+5 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} +3 a^2+3 a^{-2} -2 z^4-7 z^2-2 z^{-2} -6 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^8 a^{-2} +z^8+a z^7+4 z^7 a^{-1} +3 z^7 a^{-3} +a^2 z^6+2 z^6 a^{-2} +4 z^6 a^{-4} -z^6-2 a z^5-10 z^5 a^{-1} -4 z^5 a^{-3} +4 z^5 a^{-5} -5 a^2 z^4-8 z^4 a^{-2} -4 z^4 a^{-4} +3 z^4 a^{-6} -6 z^4-3 a z^3+2 z^3 a^{-1} +z^3 a^{-3} -3 z^3 a^{-5} +z^3 a^{-7} +8 a^2 z^2+5 z^2 a^{-2} -2 z^2 a^{-6} +11 z^2+6 a z+6 z a^{-1} -5 a^2-3 a^{-2} + a^{-4} -8-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2} }[/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 13                   1 -1 11                 2   2 9               2 1   -1 7             4 2     2 5           4 4       0 3         2 2         0 1       3 5           2 -1     1 1             0 -3     3               3 -5 1 1                 0 -7 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=3 }[/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}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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.

L9a47 L9a49