L9n3

(Knotscape image)

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

Visit L9n3's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_5,10,6,11 X₈₄₉₃ X_11,18,12,5 X_17,12,18,13 X_9,16,10,17 X_2,14,3,13
Gauss code	{1, -9, 5, -3}, {-4, -1, 2, -5, -8, 4, -6, 7, 9, -2, 3, 8, -7, 6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u + u + v -1$ (db)
Jones polynomial	$-\frac{1}{\sqrt{q}}-\frac{2}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{1}{q^{9/2}}+\frac{1}{q^{11/2}}-\frac{1}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 7 - a 7 z -1 + z 3 a 5 + 3 z a 5 + 2 a 5 z -1 - z a 3 - z a - a z -1$ (db)
Kauffman polynomial	$- z 6 a 8 + 5 z 4 a 8 -6 z 2 a 8 + 2 a 8 - z 7 a 7 + 5 z 5 a 7 -6 z 3 a 7 + 2 z a 7 - a 7 z -1 -2 z 6 a 6 + 10 z 4 a 6 -13 z 2 a 6 + 5 a 6 - z 7 a 5 + 5 z 5 a 5 -7 z 3 a 5 + 5 z a 5 -2 a 5 z -1 - z 6 a 4 + 5 z 4 a 4 -7 z 2 a 4 + 3 a 4 - z 3 a 3 + 2 z a 3 - a 2 - z a + a z -1$ (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 =

-1 is the signature of L9n3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -4$	$i = -2$	$i = 0$
$r = -7$		${\mathbb Z}$
$r = -6$		${\mathbb Z}_2$	${\mathbb Z}$
$r = -5$		${\mathbb Z}$
$r = -4$		${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}_2$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -1$		${\mathbb Z}_2$	${\mathbb Z}$
$r = 0$	${\mathbb Z}$	${\mathbb Z}^{2}$	${\mathbb Z}$