L9n9

(Knotscape image)

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

Visit L9n9's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_7,14,8,15 X_15,18,16,5 X_11,16,12,17 X_17,12,18,13 X_13,8,14,9 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -8, 2, -9}, {8, -1, -3, 7, 9, -2, -5, 6, -7, 3, -4, 5, -6, 4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 - u 3 + u 2 + v u - v -1$ (db)
Jones polynomial	$-\frac{1}{q^{5/2}}-\frac{1}{q^{11/2}}-\frac{1}{q^{15/2}}+\frac{1}{q^{17/2}}-\frac{1}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- a 11 z -1 + 2 z a 9 + 2 a 9 z -1 + z a 7 - z 5 a 5 -5 z 3 a 5 -5 z a 5 - a 5 z -1$ (db)
Kauffman polynomial	$- z 6 a 12 + 5 z 4 a 12 -6 z 2 a 12 + 2 a 12 - z 7 a 11 + 5 z 5 a 11 -6 z 3 a 11 + 3 z a 11 - a 11 z -1 -2 z 6 a 10 + 11 z 4 a 10 -15 z 2 a 10 + 5 a 10 - z 7 a 9 + 6 z 5 a 9 -10 z 3 a 9 + 7 z a 9 -2 a 9 z -1 - z 6 a 8 + 6 z 4 a 8 -8 z 2 a 8 + 3 a 8 + z 3 a 7 - z a 7 + z 2 a 6 - a 6 - z 5 a 5 + 5 z 3 a 5 -5 z a 5 + a 5 z -1$ (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 L9n9. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -6$	$i = -4$	$i = -2$
$r = -9$		${\mathbb Z}$
$r = -8$		${\mathbb Z}_2$	${\mathbb Z}$
$r = -7$		${\mathbb Z}$
$r = -6$		${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}$	${\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}_2$	${\mathbb Z}^{3}$	${\mathbb Z}$
$r = -3$	${\mathbb Z}$	${\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}_2$	${\mathbb Z}$
$r = -1$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$