L9n21

(Knotscape image)

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

Visit L9n21's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_3,11,4,10 X_7,15,8,14 X_13,5,14,8 X_16,11,17,12 X_18,15,9,16 X_12,17,13,18 X₂₅₃₆ X_9,1,10,4
Gauss code	{1, -8, -2, 9}, {8, -1, -3, 4}, {-9, 2, 5, -7, -4, 3, 6, -5, 7, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$u 3 - v u 2 - w u 2 + v u + w u - v w$ (db)
Jones polynomial	$q 3 + q + 2 + q -2 - q -3 + q -4 - q -5$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 2 a 4 - a 4 z -2 -2 a 4 + z 4 a 2 + 5 z 2 a 2 + 4 a 2 z -2 + 8 a 2 - z 4 -6 z 2 -5 z -2 -9 + z 2 a -2 + 2 a -2 z -2 + 3 a -2$ (db)
Kauffman polynomial	$a 4 z 6 + a 2 z 6 + z 6 a -2 + z 6 + a 5 z 5 + 2 a 3 z 5 + 2 a z 5 + z 5 a -1 -4 a 4 z 4 -6 a 2 z 4 -6 z 4 a -2 -8 z 4 -4 a 5 z 3 -10 a 3 z 3 -13 a z 3 -7 z 3 a -1 + 3 a 4 z 2 + 11 a 2 z 2 + 10 z 2 a -2 + 18 z 2 + 3 a 5 z + 13 a 3 z + 21 a z + 11 z a -1 -2 a 4 -10 a 2 -7 a -2 -14- a 5 z -1 -5 a 3 z -1 -9 a z -1 -5 a -1 z -1 + a 4 z -2 + 4 a 2 z -2 + 2 a -2 z -2 + 5 z -2$ (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 =

0 is the signature of L9n21. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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