L8a20

L8a19

L8a21

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a20 at Knotilus!
	L8a20 is [math]\displaystyle{ 8^3_{4} }[/math] in the Rolfsen table of links.

Depiction obtained with knotilus

Link Presentations

[edit Notes on L8a20's Link Presentations]

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

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{(w-1) \left(u v w-u v-2 u w-2 v w-w^2+w\right)}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^4-2 q^3+5 q^2-5 q+6-5 q^{-1} +5 q^{-2} -2 q^{-3} + q^{-4} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^4+ a^{-4} -2 a^2 z^2+a^2 z^{-2} -2 z^2 a^{-2} + a^{-2} z^{-2} +z^4-2 z^{-2} -2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^4 z^4+z^4 a^{-4} -2 a^4 z^2-2 z^2 a^{-4} +a^4+ a^{-4} +2 a^3 z^5+2 z^5 a^{-3} -2 a^3 z^3-2 z^3 a^{-3} +3 a^2 z^6+3 z^6 a^{-2} -5 a^2 z^4-5 z^4 a^{-2} +5 a^2 z^2+5 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -4 a^2-4 a^{-2} +a z^7+z^7 a^{-1} +5 a z^5+5 z^5 a^{-1} -12 a z^3-12 z^3 a^{-1} +8 a z+8 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +6 z^6-12 z^4+14 z^2+2 z^{-2} -9 }[/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 χ 9                 1 1 7               2 1 -1 5             3     3 3           2 2     0 1         4 3       1 -1       3 4         1 -3     2 2           0 -5     3             3 -7 1 2               -1 -9 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=1 }[/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}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/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.

L8a19 L8a21