L11n339

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_18,12,19,11 X_7,14,8,15 X_13,8,14,9 X_19,22,20,13 X_15,20,16,21 X_21,16,22,17 X_12,18,5,17 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, 3, -9}, {-5, 4, -7, 8, 9, -3, -6, 7, -8, 6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 3 - v u 3 - v 2 w u 3 + v w u 3 - v 3 u 2 + v u 2 + v 2 w u 2 - w u 2 + v 3 u - v u - v 2 w u + w u - v 2 + v + v 2 w - v w$ (db)
Jones polynomial	$1-2 q -1 + 4 q -2 -3 q -3 + 4 q -4 -2 q -5 + 3 q -6 - q -8 + q -9 - q -10$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$- a 10 z -2 - a 10 + 2 z 2 a 8 + 4 a 8 z -2 + 5 a 8 -2 z 2 a 6 -5 a 6 z -2 -7 a 6 - z 6 a 4 -4 z 4 a 4 -3 z 2 a 4 + 2 a 4 z -2 + a 4 + z 4 a 2 + 3 z 2 a 2 + 2 a 2$ (db)
Kauffman polynomial	$z 7 a 11 -6 z 5 a 11 + 10 z 3 a 11 -6 z a 11 + a 11 z -1 + z 8 a 10 -6 z 6 a 10 + 9 z 4 a 10 -6 z 2 a 10 - a 10 z -2 + 4 a 10 + 2 z 7 a 9 -16 z 5 a 9 + 33 z 3 a 9 -22 z a 9 + 5 a 9 z -1 + z 8 a 8 -9 z 6 a 8 + 22 z 4 a 8 -22 z 2 a 8 -4 a 8 z -2 + 15 a 8 + 2 z 7 a 7 -15 z 5 a 7 + 34 z 3 a 7 -30 z a 7 + 9 a 7 z -1 + z 8 a 6 -5 z 6 a 6 + 11 z 4 a 6 -19 z 2 a 6 -5 a 6 z -2 + 16 a 6 + 3 z 7 a 5 -12 z 5 a 5 + 15 z 3 a 5 -13 z a 5 + 5 a 5 z -1 + z 8 a 4 - z 6 a 4 -6 z 4 a 4 + 2 z 2 a 4 -2 a 4 z -2 + 4 a 4 + 2 z 7 a 3 -7 z 5 a 3 + 4 z 3 a 3 + z a 3 + z 6 a 2 -4 z 4 a 2 + 5 z 2 a 2 -2 a 2$ (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 L11n339. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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