L8a18

L8a17

L8a19

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a18 at Knotilus!
	L8a18 is [math]\displaystyle{ 8^3_{1} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L8a18's Link Presentations]

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

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 w^2+u v w-u w+u-v^2 w^2+v^2 w-v w+v-1}{\sqrt{u} v w} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^8-2 q^7+3 q^6-3 q^5+4 q^4-2 q^3+3 q^2-q+1 }[/math] (db)
Signature	4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^4 a^{-6} +3 z^2 a^{-6} + a^{-6} z^{-2} +2 a^{-6} -z^6 a^{-4} -5 z^4 a^{-4} -8 z^2 a^{-4} -2 a^{-4} z^{-2} -6 a^{-4} +z^4 a^{-2} +4 z^2 a^{-2} + a^{-2} z^{-2} +4 a^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^2 a^{-10} +2 z^3 a^{-9} +3 z^4 a^{-8} -3 z^2 a^{-8} + a^{-8} +3 z^5 a^{-7} -4 z^3 a^{-7} +3 z^6 a^{-6} -8 z^4 a^{-6} +6 z^2 a^{-6} + a^{-6} z^{-2} -3 a^{-6} +z^7 a^{-5} -7 z^3 a^{-5} +6 z a^{-5} -2 a^{-5} z^{-1} +4 z^6 a^{-4} -16 z^4 a^{-4} +18 z^2 a^{-4} +2 a^{-4} z^{-2} -8 a^{-4} +z^7 a^{-3} -3 z^5 a^{-3} -z^3 a^{-3} +6 z a^{-3} -2 a^{-3} z^{-1} +z^6 a^{-2} -5 z^4 a^{-2} +8 z^2 a^{-2} + a^{-2} z^{-2} -5 a^{-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   \ -2 -1 0 1 2 3 4 5 6 χ 17                 1 1 15               2 1 -1 13             1     1 11           2 2     0 9         2 1       1 7       1 3         2 5     2 1           1 3   1 3             2 1                   0 -1 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=5 }[/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}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/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}^{2} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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.

L8a17 L8a19