L11n344

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_2,16,3,15 X_3,10,4,11 X_5,14,6,15 X_11,22,12,13 X_13,12,14,5 X_21,1,22,4 X_20,17,21,18 X_16,7,17,8 X_8,20,9,19 X_18,10,19,9
Gauss code	{1, -2, -3, 7}, {-4, -1, 9, -10, 11, 3, -5, 6}, {-6, 4, 2, -9, 8, -11, 10, -8, -7, 5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 w u 3 + v w u 3 + v 3 u 2 -3 v 2 u 2 + 2 v u 2 + 3 v 2 w u 2 -3 v w u 2 + w u 2 - v 3 u + 3 v 2 u -3 v u -2 v 2 w u + 3 v w u - w u - v 2 + v$ (db)
Jones polynomial	$2 q -3 + 7 q -1 -9 q -2 + 11 q -3 -9 q -4 + 9 q -5 -6 q -6 + 3 q -7 - q -8$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$- z 4 a 6 -2 z 2 a 6 -2 a 6 + z 6 a 4 + 4 z 4 a 4 + 8 z 2 a 4 + a 4 z -2 + 7 a 4 -3 z 4 a 2 -9 z 2 a 2 -2 a 2 z -2 -9 a 2 + 2 z 2 + z -2 + 4$ (db)
Kauffman polynomial	$z 5 a 9 -2 z 3 a 9 + z a 9 + 3 z 6 a 8 -6 z 4 a 8 + 3 z 2 a 8 - a 8 + 4 z 7 a 7 -6 z 5 a 7 - z 3 a 7 + z a 7 + 3 z 8 a 6 -2 z 6 a 6 -4 z 4 a 6 + 2 z 2 a 6 + z 9 a 5 + 4 z 7 a 5 -10 z 5 a 5 + 9 z 3 a 5 -3 z a 5 + 5 z 8 a 4 -12 z 6 a 4 + 21 z 4 a 4 -21 z 2 a 4 - a 4 z -2 + 9 a 4 + z 9 a 3 + z 7 a 3 -4 z 5 a 3 + 11 z 3 a 3 -8 z a 3 + 2 a 3 z -1 + 2 z 8 a 2 -7 z 6 a 2 + 22 z 4 a 2 -28 z 2 a 2 -2 a 2 z -2 + 13 a 2 + z 7 a - z 5 a + 3 z 3 a -5 z a + 2 a z -1 + 3 z 4 -8 z 2 - z -2 + 6$ (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 L11n344. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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