L11n318

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_14,5,15,6 X_20,11,21,12 X_22,17,11,18 X_16,21,17,22 X_10,13,5,14 X_19,8,20,9 X_7,18,8,19 X_2,9,3,10 X_4,16,1,15
Gauss code	{1, -10, 2, -11}, {3, -1, -9, 8, 10, -7}, {4, -2, 7, -3, 11, -6, 5, 9, -8, -4, 6, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v w u 4 + 2 v 2 u 3 -3 v u 3 - v 2 w u 3 + 3 v w u 3 - w u 3 + u 3 -2 v 2 u 2 + 5 v u 2 + 2 v 2 w u 2 -5 v w u 2 + 2 w u 2 -2 u 2 + v 2 u -3 v u - v 2 w u + 3 v w u -2 w u + u + v$ (db)
Jones polynomial	$3 q -2 -5 q -3 + 11 q -4 -12 q -5 + 15 q -6 -14 q -7 + 11 q -8 -8 q -9 + 4 q -10 - q -11$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$- z 2 a 10 - a 10 z -2 - a 10 + 3 z 4 a 8 + 8 z 2 a 8 + 4 a 8 z -2 + 8 a 8 -2 z 6 a 6 -9 z 4 a 6 -17 z 2 a 6 -5 a 6 z -2 -15 a 6 + 3 z 4 a 4 + 9 z 2 a 4 + 2 a 4 z -2 + 8 a 4$ (db)
Kauffman polynomial	$z 5 a 13 - z 3 a 13 + 4 z 6 a 12 -6 z 4 a 12 + z 2 a 12 + 7 z 7 a 11 -13 z 5 a 11 + 6 z 3 a 11 -2 z a 11 + a 11 z -1 + 6 z 8 a 10 -8 z 6 a 10 + 2 z 4 a 10 -3 z 2 a 10 - a 10 z -2 + 2 a 10 + 2 z 9 a 9 + 9 z 7 a 9 -28 z 5 a 9 + 27 z 3 a 9 -13 z a 9 + 5 a 9 z -1 + 11 z 8 a 8 -27 z 6 a 8 + 35 z 4 a 8 -27 z 2 a 8 -4 a 8 z -2 + 13 a 8 + 2 z 9 a 7 + 5 z 7 a 7 -20 z 5 a 7 + 33 z 3 a 7 -27 z a 7 + 9 a 7 z -1 + 5 z 8 a 6 -15 z 6 a 6 + 33 z 4 a 6 -37 z 2 a 6 -5 a 6 z -2 + 20 a 6 + 3 z 7 a 5 -6 z 5 a 5 + 13 z 3 a 5 -16 z a 5 + 5 a 5 z -1 + 6 z 4 a 4 -14 z 2 a 4 -2 a 4 z -2 + 10 a 4$ (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 =

-4 is the signature of L11n318. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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