L9a17

L9a16

L9a18

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a17 at Knotilus!
	L9a17 is [math]\displaystyle{ 9^2_{27} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a17's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
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) \left(2 t(2)^2-t(2)+2\right)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 7 q^{9/2}-7 q^{7/2}+5 q^{5/2}-4 q^{3/2}+q^{17/2}-3 q^{15/2}+5 q^{13/2}-6 q^{11/2}+\sqrt{q}-\frac{1}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^5 a^{-3} -z^5 a^{-5} +z^3 a^{-1} -3 z^3 a^{-3} -2 z^3 a^{-5} +z^3 a^{-7} +3 z a^{-1} -4 z a^{-3} +z a^{-7} +2 a^{-1} z^{-1} -3 a^{-3} z^{-1} + a^{-5} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^8 a^{-4} -z^8 a^{-6} -z^7 a^{-3} -4 z^7 a^{-5} -3 z^7 a^{-7} -z^6 a^{-2} +z^6 a^{-4} -2 z^6 a^{-6} -4 z^6 a^{-8} -z^5 a^{-1} -z^5 a^{-3} +7 z^5 a^{-5} +4 z^5 a^{-7} -3 z^5 a^{-9} +z^4 a^{-2} -4 z^4 a^{-4} +2 z^4 a^{-6} +6 z^4 a^{-8} -z^4 a^{-10} +4 z^3 a^{-1} +7 z^3 a^{-3} -5 z^3 a^{-5} -4 z^3 a^{-7} +4 z^3 a^{-9} +3 z^2 a^{-2} +7 z^2 a^{-4} +z^2 a^{-6} -2 z^2 a^{-8} +z^2 a^{-10} -5 z a^{-1} -7 z a^{-3} +2 z a^{-7} -3 a^{-2} -3 a^{-4} - a^{-6} +2 a^{-1} z^{-1} +3 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 18                   1 -1 16                 2   2 14               3 1   -2 12             3 2     1 10           4 3       -1 8         3 3         0 6       2 4           2 4     2 3             -1 2   1 4               3 0                     0 -2 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=4 }[/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}^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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.

L9a16 L9a18