L11n323

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_5,14,6,15 X₃₈₄₉ X_15,2,16,3 X_16,7,17,8 X_9,18,10,19 X_11,21,12,20 X_19,22,20,13 X_13,12,14,5 X_4,17,1,18 X_21,11,22,10
Gauss code	{1, 4, -3, -10}, {-2, -1, 5, 3, -6, 11, -7, 9}, {-9, 2, -4, -5, 10, 6, -8, 7, -11, 8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 w u 3 + v 2 w u 3 + v 3 w u 2 - u - v + 1$ (db)
Jones polynomial	$q -1 - q -2 + 2 q -3 - q -4 + 2 q -5 - q -6 + 2 q -7 + q -9 - q -10$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$- z 2 a 10 -2 a 10 + z 6 a 8 + 6 z 4 a 8 + 10 z 2 a 8 + a 8 z -2 + 6 a 8 - z 8 a 6 -7 z 6 a 6 -16 z 4 a 6 -16 z 2 a 6 -2 a 6 z -2 -9 a 6 + z 6 a 4 + 6 z 4 a 4 + 10 z 2 a 4 + a 4 z -2 + 5 a 4$ (db)
Kauffman polynomial	$z a 13 + z 2 a 12 - a 12 + z a 11 + z 4 a 10 -3 z 2 a 10 + z 7 a 9 -5 z 5 a 9 + 6 z 3 a 9 -3 z a 9 + 2 z 8 a 8 -13 z 6 a 8 + 27 z 4 a 8 -24 z 2 a 8 - a 8 z -2 + 9 a 8 + z 9 a 7 -5 z 7 a 7 + 4 z 5 a 7 + 6 z 3 a 7 -8 z a 7 + 2 a 7 z -1 + 3 z 8 a 6 -20 z 6 a 6 + 42 z 4 a 6 -35 z 2 a 6 -2 a 6 z -2 + 13 a 6 + z 9 a 5 -6 z 7 a 5 + 9 z 5 a 5 -5 z a 5 + 2 a 5 z -1 + z 8 a 4 -7 z 6 a 4 + 16 z 4 a 4 -15 z 2 a 4 - a 4 z -2 + 6 a 4$ (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 =

-6 is the signature of L11n323. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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