L11n379

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_5,10,6,11 X₃₈₄₉ X_22,18,19,17 X_20,12,21,11 X_12,20,13,19 X_18,22,5,21 X_9,16,10,17 X_13,2,14,3
Gauss code	{1, 11, -5, -3}, {8, -7, 9, -6}, {-4, -1, 2, 5, -10, 4, 7, -8, -11, -2, 3, 10, 6, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + v u 3 - w u 2 + w$ (db)
Jones polynomial	$q 2 + 1 + q -1 + q -2 + q -3 - q -4$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- a 6 z -2 - a 6 + z 4 a 4 + 6 z 2 a 4 + 4 a 4 z -2 + 8 a 4 - z 6 a 2 -7 z 4 a 2 -15 z 2 a 2 -5 a 2 z -2 -13 a 2 + z 4 + 5 z 2 + 2 z -2 + 6$ (db)
Kauffman polynomial	$a 2 z 8 + z 8 + a 3 z 7 + a z 7 - a 4 z 6 -9 a 2 z 6 -8 z 6 - a 5 z 5 -9 a 3 z 5 -8 a z 5 + 7 a 4 z 4 + 28 a 2 z 4 + 21 z 4 + 6 a 5 z 3 + 26 a 3 z 3 + 20 a z 3 -15 a 4 z 2 -37 a 2 z 2 -22 z 2 -9 a 5 z -27 a 3 z -18 a z + a 6 + 12 a 4 + 21 a 2 + 11 + a 7 z -1 + 5 a 5 z -1 + 9 a 3 z -1 + 5 a z -1 - a 6 z -2 -4 a 4 z -2 -5 a 2 z -2 -2 z -2$ (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 =

-3 is the signature of L11n379. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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