L11n236

(Knotscape image)

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

Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_14,3,15,4 X_20,13,21,14 X_12,21,13,22 X_22,5,9,6 X_7,16,8,17 X_15,18,16,19 X_17,8,18,1 X_6,9,7,10 X_4,19,5,20
Gauss code	{1, -2, 3, -11, 6, -10, -7, 9}, {10, -1, 2, -5, 4, -3, -8, 7, -9, 8, 11, -4, 5, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 3 u 3 + v 2 u 3 -2 v u 2 + u 2 + v 3 u -2 v 2 u + v -2$ (db)
Jones polynomial	$-\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{2}{q^{11/2}}+\frac{1}{q^{19/2}}-\frac{1}{q^{21/2}}+\frac{1}{q^{23/2}}-\frac{1}{q^{25/2}}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$- z a 13 + z 5 a 11 + 6 z 3 a 11 + 6 z a 11 - z 7 a 9 -6 z 5 a 9 -9 z 3 a 9 -3 z a 9 + a 9 z -1 - z 7 a 7 -6 z 5 a 7 -10 z 3 a 7 -6 z a 7 - a 7 z -1$ (db)
Kauffman polynomial	$- z 5 a 15 + 4 z 3 a 15 -3 z a 15 - z 6 a 14 + 4 z 4 a 14 -2 z 2 a 14 - z 5 a 13 + 5 z 3 a 13 -3 z a 13 - z 8 a 12 + 7 z 6 a 12 -14 z 4 a 12 + 10 z 2 a 12 - z 9 a 11 + 7 z 7 a 11 -15 z 5 a 11 + 15 z 3 a 11 -7 z a 11 -2 z 8 a 10 + 13 z 6 a 10 -22 z 4 a 10 + 10 z 2 a 10 - z 9 a 9 + 6 z 7 a 9 -9 z 5 a 9 + 4 z 3 a 9 - z a 9 - a 9 z -1 - z 8 a 8 + 5 z 6 a 8 -4 z 4 a 8 -2 z 2 a 8 + a 8 - z 7 a 7 + 6 z 5 a 7 -10 z 3 a 7 + 6 z a 7 - a 7 z -1$ (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 =

-6 is the signature of L11n236. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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