L11n437

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_12,3,13,4 X_13,20,14,21 X_9,18,10,19 X_21,10,22,11 X_19,14,20,7 X_5,17,6,16 X_17,22,18,15 X₂₇₃₈ X_4,11,5,12 X_15,1,16,6
Gauss code	{1, -9, 2, -10, -7, 11}, {9, -1, -4, 5, 10, -2, -3, 6}, {-11, 7, -8, 4, -6, 3, -5, 8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 3 - v w u 3 + w u 3 - v 2 w 2 u 2 + v w 2 u 2 + v 2 w u 2 - v u - w u + u - v w 2 - v 2 w + v w$ (db)
Jones polynomial	$1- q -1 + q -2 + q -4 + q -5 + q -7 - q -8 + q -9$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z 2 a 8 + a 8 z -2 + a 8 -2 a 6 z -2 -3 a 6 - z 6 a 4 -5 z 4 a 4 -5 z 2 a 4 + a 4 z -2 + z 4 a 2 + 4 z 2 a 2 + 2 a 2$ (db)
Kauffman polynomial	$z 6 a 10 -5 z 4 a 10 + 5 z 2 a 10 + z 7 a 9 -5 z 5 a 9 + 5 z 3 a 9 + z 8 a 8 -7 z 6 a 8 + 16 z 4 a 8 -17 z 2 a 8 - a 8 z -2 + 7 a 8 + z 7 a 7 -7 z 5 a 7 + 13 z 3 a 7 -9 z a 7 + 2 a 7 z -1 + z 8 a 6 -8 z 6 a 6 + 21 z 4 a 6 -25 z 2 a 6 -2 a 6 z -2 + 11 a 6 + z 7 a 5 -7 z 5 a 5 + 13 z 3 a 5 -9 z a 5 + 2 a 5 z -1 + z 6 a 4 -5 z 4 a 4 + 3 z 2 a 4 - a 4 z -2 + 3 a 4 + z 7 a 3 -5 z 5 a 3 + 5 z 3 a 3 + z 6 a 2 -5 z 4 a 2 + 6 z 2 a 2 -2 a 2$ (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 =

-3 is the signature of L11n437. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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