L11n326

(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_13,18,14,19 X_9,21,10,20 X_19,5,20,12 X_11,13,12,22 X_21,11,22,10 X_4,17,1,18
Gauss code	{1, 4, -3, -11}, {-2, -1, 5, 3, -7, 10, -9, 8}, {-6, 2, -4, -5, 11, 6, -8, 7, -10, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 w u 3 + v 2 w u 3 - v u 2 + v 3 w u 2 -2 v 2 w u 2 + v w u 2 - v 2 u + 2 v u + v 2 w u - u - v + 1$ (db)
Jones polynomial	$q 3 -2 q 2 + 3 q -3 + 5 q -1 -4 q -2 + 5 q -3 -2 q -4 + 2 q -5 - q -6$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$- a 2 z 8 + a 4 z 6 -7 a 2 z 6 + z 6 + 6 a 4 z 4 -17 a 2 z 4 + 5 z 4 - a 6 z 2 + 11 a 4 z 2 -18 a 2 z 2 + 7 z 2 -2 a 6 + 7 a 4 -9 a 2 + 4 + a 4 z -2 -2 a 2 z -2 + z -2$ (db)
Kauffman polynomial	$a 3 z 9 + a z 9 + 2 a 4 z 8 + 4 a 2 z 8 + 2 z 8 + a 5 z 7 -3 a 3 z 7 -2 a z 7 + 2 z 7 a -1 -12 a 4 z 6 -21 a 2 z 6 + z 6 a -2 -8 z 6 -5 a 5 z 5 -3 a 3 z 5 -6 a z 5 -8 z 5 a -1 + 2 a 6 z 4 + 28 a 4 z 4 + 37 a 2 z 4 -4 z 4 a -2 + 7 z 4 + a 7 z 3 + 11 a 5 z 3 + 17 a 3 z 3 + 13 a z 3 + 6 z 3 a -1 -5 a 6 z 2 -27 a 4 z 2 -28 a 2 z 2 + 3 z 2 a -2 -3 z 2 -2 a 7 z -7 a 5 z -12 a 3 z -8 a z - z a -1 + 3 a 6 + 11 a 4 + 11 a 2 - a -2 + 3 + 2 a 3 z -1 + 2 a z -1 - a 4 z -2 -2 a 2 z -2 - z -2$ (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 L11n326. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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