L11n424

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_7,16,8,17 X_5,14,6,15 X_3,10,4,11 X_13,4,14,5 X_2,18,3,17 X_18,9,19,10 X_21,7,22,12 X_11,13,12,22 X_15,20,16,21 X_19,1,20,6
Gauss code	{1, -6, -4, 5, -3, 11}, {-2, -1, 7, 4, -9, 8}, {-5, 3, -10, 2, 6, -7, -11, 10, -8, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 3 + v u 3 + v 2 w u 3 - v w u 3 + v 2 u 2 - v w 2 u 2 -2 v u 2 -2 v 2 w u 2 + 5 v w u 2 - w u 2 + 2 v w 2 u - w 2 u + v u + v 2 w u -5 v w u + 2 w u - v w 2 + w 2 + v w - w$ (db)
Jones polynomial	$2 q -4 + 8 q -1 -9 q -2 + 12 q -3 -10 q -4 + 9 q -5 -6 q -6 + 3 q -7 - q -8$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$- z 4 a 6 -2 z 2 a 6 -2 a 6 + z 6 a 4 + 4 z 4 a 4 + 8 z 2 a 4 + a 4 z -2 + 6 a 4 -3 z 4 a 2 -8 z 2 a 2 -2 a 2 z -2 -7 a 2 + 2 z 2 + z -2 + 3$ (db)
Kauffman polynomial	$z 5 a 9 -2 z 3 a 9 + 3 z 6 a 8 -6 z 4 a 8 + z 2 a 8 + 5 z 7 a 7 -12 z 5 a 7 + 8 z 3 a 7 -2 z a 7 + 5 z 8 a 6 -14 z 6 a 6 + 18 z 4 a 6 -12 z 2 a 6 + 4 a 6 + 2 z 9 a 5 -8 z 5 a 5 + 14 z 3 a 5 -6 z a 5 + 8 z 8 a 4 -28 z 6 a 4 + 49 z 4 a 4 -38 z 2 a 4 - a 4 z -2 + 13 a 4 + 2 z 9 a 3 -4 z 7 a 3 + 6 z 5 a 3 + 2 z 3 a 3 -7 z a 3 + 2 a 3 z -1 + 3 z 8 a 2 -11 z 6 a 2 + 28 z 4 a 2 -32 z 2 a 2 -2 a 2 z -2 + 13 a 2 + z 7 a + z 5 a -2 z 3 a -3 z a + 2 a z -1 + 3 z 4 -7 z 2 - z -2 + 5$ (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 =

-2 is the signature of L11n424. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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