L11n224

(Knotscape image)

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

Planar diagram presentation	X_10,1,11,2 X_3,12,4,13 X_16,9,17,10 X_20,12,21,11 X_22,15,9,16 X_5,14,6,15 X_7,18,8,19 X_13,4,14,5 X_17,6,18,7 X_19,8,20,1 X_2,21,3,22
Gauss code	{1, -11, -2, 8, -6, 9, -7, 10}, {3, -1, 4, 2, -8, 6, 5, -3, -9, 7, -10, -4, 11, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 3 - v 2 u 3 + v u 3 + v 2 u 2 -2 v u 2 -2 v 2 u + v u + v 2 - v -1$ (db)
Jones polynomial	$-\frac{1}{q^{7/2}}-\frac{2}{q^{13/2}}+\frac{3}{q^{15/2}}-\frac{3}{q^{17/2}}+\frac{3}{q^{19/2}}-\frac{3}{q^{21/2}}+\frac{2}{q^{23/2}}-\frac{1}{q^{25/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$z 3 a 11 + z a 11 + 2 z 3 a 9 + 4 z a 9 + a 9 z -1 - z 7 a 7 -7 z 5 a 7 -14 z 3 a 7 -9 z a 7 - a 7 z -1$ (db)
Kauffman polynomial	$- z 5 a 15 + 3 z 3 a 15 -2 z a 15 -2 z 6 a 14 + 6 z 4 a 14 -3 z 2 a 14 - z 7 a 13 + z 5 a 13 + 3 z 3 a 13 - z a 13 -2 z 6 a 12 + 3 z 4 a 12 + z 2 a 12 -3 z 5 a 11 + 6 z 3 a 11 -3 z a 11 - z 4 a 10 - z 2 a 10 + 2 z 5 a 9 -8 z 3 a 9 + 5 z a 9 - a 9 z -1 + 2 z 4 a 8 -5 z 2 a 8 + a 8 - z 7 a 7 + 7 z 5 a 7 -14 z 3 a 7 + 9 z a 7 - a 7 z -1$ (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 =

-5 is the signature of L11n224. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -8$	$i = -6$	$i = -4$
$r = -10$		${\mathbb Z}$
$r = -9$		${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -8$		${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}^{2}$
$r = -7$		${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -6$		${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -5$	${\mathbb Z}$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -4$		${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}^{2}$
$r = -3$	${\mathbb Z}$	${\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}_2$	${\mathbb Z}$
$r = -1$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$