L11n436

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_18,8,19,7 X_5,16,6,17 X_15,10,16,11 X_3,15,4,14 X_11,5,12,4 X_2,20,3,19 X_20,9,21,10 X_13,7,14,12 X_17,22,18,13 X_6,21,1,22
Gauss code	{1, -7, -5, 6, -3, -11}, {2, -1, 8, 4, -6, 9}, {-9, 5, -4, 3, -10, -2, 7, -8, 11, 10}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 3 - v w u 3 + w u 3 - u 3 + v u 2 - v w u 2 + w u 2 - u 2 - v u + v w u - w u + u - v + v w - w + 1$ (db)
Jones polynomial	$- q 4 + 2 q 3 - q 2 + 2 q + 1 + q -2 - q -3 + 2 q -4 - q -5$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z 6 - z 4 a -2 + 5 z 4 - a 4 z 2 - a 2 z 2 -3 z 2 a -2 + 5 z 2 + a 2 z -2 + a -2 z -2 -2 z -2$ (db)
Kauffman polynomial	$a z 9 + z 9 a -1 + a 2 z 8 + 2 z 8 a -2 + 3 z 8 + a 3 z 7 -5 a z 7 -5 z 7 a -1 + z 7 a -3 + 2 a 4 z 6 -6 a 2 z 6 -12 z 6 a -2 -20 z 6 + a 5 z 5 -5 a 3 z 5 + 2 a z 5 + 3 z 5 a -1 -5 z 5 a -3 -8 a 4 z 4 + 8 a 2 z 4 + 20 z 4 a -2 + 36 z 4 -3 a 5 z 3 + 4 a 3 z 3 + 6 a z 3 + 4 z 3 a -1 + 5 z 3 a -3 + 4 a 4 z 2 -4 a 2 z 2 -12 z 2 a -2 -20 z 2 + 1-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 =

-1 is the signature of L11n436. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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