L11n132

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_18,11,19,12 X_3,10,4,11 X_17,3,18,2 X_12,5,13,6 X₆₇₁₈ X_9,16,10,17 X_13,20,14,21 X_15,22,16,7 X_4,20,5,19 X_21,14,22,15
Gauss code	{1, 4, -3, -10, 5, -6}, {6, -1, -7, 3, 2, -5, -8, 11, -9, 7, -4, -2, 10, 8, -11, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 4 - v 2 u 3 + v 2 u 2 - v u 2 + u 2 - u + v$ (db)
Jones polynomial	$-\frac{1}{q^{5/2}}-\frac{1}{q^{7/2}}-\frac{1}{q^{11/2}}+\frac{1}{q^{13/2}}-\frac{1}{q^{15/2}}+\frac{1}{q^{17/2}}-\frac{1}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- z 3 a 9 -3 z a 9 -2 a 9 z -1 + z 5 a 7 + 6 z 3 a 7 + 11 z a 7 + 5 a 7 z -1 - z 5 a 5 -6 z 3 a 5 -9 z a 5 -3 a 5 z -1$ (db)
Kauffman polynomial	$- z 6 a 12 + 5 z 4 a 12 -6 z 2 a 12 + a 12 - z 7 a 11 + 5 z 5 a 11 -6 z 3 a 11 + z a 11 - z 6 a 10 + 4 z 4 a 10 -3 z 2 a 10 - z 5 a 9 + 4 z 3 a 9 -4 z a 9 + 2 a 9 z -1 + z 6 a 8 -7 z 4 a 8 + 13 z 2 a 8 -5 a 8 + z 7 a 7 -7 z 5 a 7 + 16 z 3 a 7 -15 z a 7 + 5 a 7 z -1 + z 6 a 6 -6 z 4 a 6 + 10 z 2 a 6 -5 a 6 - z 5 a 5 + 6 z 3 a 5 -10 z a 5 + 3 a 5 z -1$ (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 =

-3 is the signature of L11n132. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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