L11n332

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_11,16,12,17 X₃₈₄₉ X_17,2,18,3 X_5,14,6,15 X_18,7,19,8 X_15,12,16,5 X_13,20,14,21 X_9,13,10,22 X_21,11,22,10 X_4,19,1,20
Gauss code	{1, 4, -3, -11}, {-5, -1, 6, 3, -9, 10, -2, 7}, {-8, 5, -7, 2, -4, -6, 11, 8, -10, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 w u 3 + v w u 3 + v u 2 - v 3 w u 2 + 3 v 2 w u 2 - v w u 2 - u 2 + v 2 u -3 v u + v 3 w u - v 2 w u + u - v 2 + v$ (db)
Jones polynomial	$1-2 q -1 + 4 q -2 -5 q -3 + 6 q -4 -5 q -5 + 7 q -6 -3 q -7 + 2 q -8 - q -9$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$- a 10 + z 4 a 8 + 4 z 2 a 8 + a 8 z -2 + 2 a 8 - z 6 a 6 -4 z 4 a 6 -4 z 2 a 6 -2 a 6 z -2 -3 a 6 - z 6 a 4 -3 z 4 a 4 + a 4 z -2 + a 4 + z 4 a 2 + 3 z 2 a 2 + a 2$ (db)
Kauffman polynomial	$z 3 a 11 -2 z a 11 + 2 z 4 a 10 -4 z 2 a 10 + 3 a 10 + z 7 a 9 -4 z 5 a 9 + 10 z 3 a 9 -7 z a 9 + 2 z 8 a 8 -10 z 6 a 8 + 22 z 4 a 8 -19 z 2 a 8 - a 8 z -2 + 11 a 8 + z 9 a 7 -2 z 7 a 7 -3 z 5 a 7 + 14 z 3 a 7 -12 z a 7 + 2 a 7 z -1 + 4 z 8 a 6 -17 z 6 a 6 + 27 z 4 a 6 -22 z 2 a 6 -2 a 6 z -2 + 11 a 6 + z 9 a 5 - z 7 a 5 -6 z 5 a 5 + 10 z 3 a 5 -8 z a 5 + 2 a 5 z -1 + 2 z 8 a 4 -6 z 6 a 4 + 3 z 4 a 4 -3 z 2 a 4 - a 4 z -2 + 3 a 4 + 2 z 7 a 3 -7 z 5 a 3 + 5 z 3 a 3 - z a 3 + z 6 a 2 -4 z 4 a 2 + 4 z 2 a 2 - a 2$ (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 =

-4 is the signature of L11n332. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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