L11n423

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_16,8,17,7 X_10,4,11,3 X_2,18,3,17 X_18,9,19,10 X_20,12,21,11 X_5,14,6,15 X_15,13,16,22 X_13,6,14,1 X_4,19,5,20 X_12,22,7,21
Gauss code	{1, -4, 3, -10, -7, 9}, {2, -1, 5, -3, 6, -11}, {-9, 7, -8, -2, 4, -5, 10, -6, 11, 8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{(w-1) \left(u^2 v-u^2+u v^2 w^2-u v w^2-u v w-u v+u-v^2 w^2+v w^2\right)}{u v w^{3/2}}$ (db)
Jones polynomial	$- q 4 + 3 q 3 -3 q 2 + 5 q -4 + 5 q -1 -3 q -2 + 3 q -3 - q -4$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 - a 2 z 4 - z 4 a -2 + 4 z 4 -2 a 2 z 2 -2 z 2 a -2 + 3 z 2 + a 2 + a -2 -2 + a 2 z -2 + a -2 z -2 -2 z -2$ (db)
Kauffman polynomial	$2 a z 9 + 2 z 9 a -1 + 3 a 2 z 8 + 3 z 8 a -2 + 6 z 8 + a 3 z 7 -8 a z 7 -8 z 7 a -1 + z 7 a -3 -15 a 2 z 6 -15 z 6 a -2 -30 z 6 -4 a 3 z 5 + 5 a z 5 + 5 z 5 a -1 -4 z 5 a -3 + 20 a 2 z 4 + 20 z 4 a -2 + 40 z 4 + 3 a 3 z 3 + a z 3 + z 3 a -1 + 3 z 3 a -3 -7 a 2 z 2 -7 z 2 a -2 -14 z 2 + 2 a z + 2 z a -1 -2 a 2 -2 a -2 -3-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 =

2 is the signature of L11n423. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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