L8a15

L8a14

L8a16

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a15 at Knotilus!
	L8a15 is [math]\displaystyle{ 8^3_{3} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L8a15's Link Presentations]

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

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{u v w-2 u v-2 u w+2 u-2 v w+2 v+2 w-1}{\sqrt{u} \sqrt{v} \sqrt{w}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-7} - q^{-6} +4 q^{-5} -4 q^{-4} +6 q^{-3} -4 q^{-2} +q+4 q^{-1} -3 }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^8 z^{-2} -2 a^6 z^{-2} -3 a^6+3 a^4 z^2+a^4 z^{-2} +3 a^4-a^2 z^4-a^2 z^2+z^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^8-3 z^2 a^8-a^8 z^{-2} +3 a^8+z^5 a^7-3 z a^7+2 a^7 z^{-1} +z^6 a^6+2 z^4 a^6-6 z^2 a^6-2 a^6 z^{-2} +5 a^6+z^7 a^5+2 z^3 a^5-3 z a^5+2 a^5 z^{-1} +4 z^6 a^4-5 z^4 a^4-a^4 z^{-2} +3 a^4+z^7 a^3+2 z^5 a^3-3 z^3 a^3+3 z^6 a^2-5 z^4 a^2+2 z^2 a^2+3 z^5 a-5 z^3 a+z^4-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   \ -6 -5 -4 -3 -2 -1 0 1 2 χ 3                 1 1 1               2   -2 -1             2 1   1 -3           3 3     0 -5         3 1       2 -7       1 3         2 -9     3 3           0 -11   1 4             3 -13                   0 -15 1                 1

Integral Khovanov Homology (db, data source)

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

L8a14 L8a16