L11n418

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 w u 3 + v w u 3 - v 2 u 2 - w 2 u 2 + v u 2 + 2 v 2 w u 2 -2 v w u 2 + w u 2 + v 2 u - v w 2 u + w 2 u - v 2 w u + 2 v w u -2 w u - v w + w$ (db)
Jones polynomial	$-1 + 3 q -1 -4 q -2 + 7 q -3 -6 q -4 + 7 q -5 -5 q -6 + 4 q -7 -2 q -8 + q -9$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z 2 a 8 + a 8 z -2 + 2 a 8 -2 z 4 a 6 -7 z 2 a 6 -2 a 6 z -2 -8 a 6 + z 6 a 4 + 5 z 4 a 4 + 10 z 2 a 4 + a 4 z -2 + 7 a 4 - z 4 a 2 -2 z 2 a 2 - a 2$ (db)
Kauffman polynomial	$z 6 a 10 -4 z 4 a 10 + 4 z 2 a 10 - a 10 + 2 z 7 a 9 -7 z 5 a 9 + 5 z 3 a 9 + 2 z 8 a 8 -6 z 6 a 8 + 4 z 4 a 8 -4 z 2 a 8 - a 8 z -2 + 4 a 8 + z 9 a 7 - z 7 a 7 -4 z 5 a 7 + 6 z 3 a 7 -6 z a 7 + 2 a 7 z -1 + 4 z 8 a 6 -16 z 6 a 6 + 28 z 4 a 6 -27 z 2 a 6 -2 a 6 z -2 + 12 a 6 + z 9 a 5 -2 z 7 a 5 + z 5 a 5 + 6 z 3 a 5 -8 z a 5 + 2 a 5 z -1 + 2 z 8 a 4 -9 z 6 a 4 + 23 z 4 a 4 -23 z 2 a 4 - a 4 z -2 + 10 a 4 + z 7 a 3 -2 z 5 a 3 + 6 z 3 a 3 -3 z a 3 + 3 z 4 a 2 -4 z 2 a 2 + 2 a 2 + z 3 a - z a$ (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 L11n418. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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