L8a3

L8a2

L8a4

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a3 at Knotilus!
	L8a3 is $8^2_{9}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L8a3's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{2 u v^2-4 u v+u+v^3-4 v^2+2 v}{\sqrt{u} v^{3/2}}$ (db)
Jones polynomial	$-\frac{4}{q^{9/2}}+\frac{4}{q^{7/2}}-\frac{5}{q^{5/2}}-q^{3/2}+\frac{5}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{2}{q^{11/2}}+2 \sqrt{q}-\frac{4}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^7 z^{-1} -3 z a^5-2 a^5 z^{-1} +2 z^3 a^3+3 z a^3+2 a^3 z^{-1} +z^3 a-z a-a z^{-1} -z a^{-1}$ (db)
Kauffman polynomial	$-z^5 a^7+3 z^3 a^7-3 z a^7+a^7 z^{-1} -2 z^6 a^6+5 z^4 a^6-2 z^2 a^6-z^7 a^5-2 z^5 a^5+11 z^3 a^5-8 z a^5+2 a^5 z^{-1} -5 z^6 a^4+11 z^4 a^4-6 z^2 a^4+a^4-z^7 a^3-4 z^5 a^3+12 z^3 a^3-9 z a^3+2 a^3 z^{-1} -3 z^6 a^2+4 z^4 a^2-3 z^2 a^2-3 z^5 a+3 z^3 a-3 z a+a z^{-1} -2 z^4+z^2-z^3 a^{-1} +z a^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 χ 4                 1 1 2               1   -1 0             3 1   2 -2           3 2     -1 -4         2 2       0 -6       2 3         1 -8     2 2           0 -10   1 3             2 -12   1               -1 -14 1                 1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

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.

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