L9n22

(Knotscape image)

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

Visit L9n22's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_14,7,15,8 X_8,13,5,14 X_11,17,12,16 X_15,9,16,18 X_17,13,18,12 X₂₅₃₆ X_4,9,1,10
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 $u$ , $v$ , $w$ , ...)	$- v u 3 - w u 3 + u 3 + v u 2 + w u 2 - v u - w u + v - v w + w$ (db)
Jones polynomial	$-2 q + 3-3 q -1 + 4 q -2 -2 q -3 + 4 q -4 - q -5 + q -6$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$2 a 6 z -2 + a 6 -3 z 2 a 4 -5 a 4 z -2 -8 a 4 + 2 z 4 a 2 + 8 z 2 a 2 + 4 a 2 z -2 + 10 a 2 -2 z 2 - z -2 -3$ (db)
Kauffman polynomial	$a 5 z 7 + a 3 z 7 + a 6 z 6 + 5 a 4 z 6 + 4 a 2 z 6 -2 a 5 z 5 + 2 a 3 z 5 + 4 a z 5 -5 a 6 z 4 -20 a 4 z 4 -14 a 2 z 4 + z 4 -5 a 5 z 3 -17 a 3 z 3 -12 a z 3 + 9 a 6 z 2 + 23 a 4 z 2 + 16 a 2 z 2 + 2 z 2 + 11 a 5 z + 21 a 3 z + 13 a z + 3 z a -1 -7 a 6 -14 a 4 -10 a 2 -2-5 a 5 z -1 -9 a 3 z -1 -5 a z -1 - a -1 z -1 + 2 a 6 z -2 + 5 a 4 z -2 + 4 a 2 z -2 + z -2$ (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 L9n22. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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