L11n237

(Knotscape image)

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

Planar diagram presentation	X_10,1,11,2 X_3,14,4,15 X_22,5,9,6 X_6,9,7,10 X_20,12,21,11 X_18,14,19,13 X_12,20,13,19 X_7,16,8,17 X_15,4,16,5 X_17,8,18,1 X_2,21,3,22
Gauss code	{1, -11, -2, 9, 3, -4, -8, 10}, {4, -1, 5, -7, 6, 2, -9, 8, -10, -6, 7, -5, 11, -3}

A Braid Representative

A Morse Link Presentation

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

-4 is the signature of L11n237. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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