L9a15

L9a14

L9a16

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a15 at Knotilus!
	L9a15 is [math]\displaystyle{ 9^2_{15} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a15's Link Presentations]

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

L9a14 L9a16