L9n8

(Knotscape image)

See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L9n8's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_5,12,6,13 X₈₄₉₃ X_9,14,10,15 X_13,10,14,11 X_11,18,12,5 X_2,16,3,15
Gauss code	{1, -9, 5, -3}, {-4, -1, 2, -5, -6, 7, -8, 4, -7, 6, 9, -2, 3, 8}

A Braid Representative

A Morse Link Presentation

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

The coefficients of the monomials

t r q j

are shown, along with their alternating sums

χ

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j -2 r = s + 1

j -2 r = s -1

, where

s =

-3 is the signature of L9n8. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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