L11n378

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_15,1,16,4 X_5,10,6,11 X₃₈₄₉ X_22,18,19,17 X_11,20,12,21 X_19,12,20,13 X_18,22,5,21 X_9,16,10,17 X_2,14,3,13
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 $u$ , $v$ , $w$ , ...)	$v u - v w u + w u - u - v + v w - w + 1$ (db)
Jones polynomial	$3- q -1 + 3 q -2 -2 q -3 + 3 q -4 -2 q -5 + q -6 - q -7$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 2 a 6 - a 6 z -2 -2 a 6 + z 4 a 4 + 4 z 2 a 4 + 4 a 4 z -2 + 7 a 4 -3 z 2 a 2 -5 a 2 z -2 -8 a 2 + 2 z -2 + 3$ (db)
Kauffman polynomial	$a 6 z 8 + a 4 z 8 + a 7 z 7 + 3 a 5 z 7 + 2 a 3 z 7 -5 a 6 z 6 -4 a 4 z 6 + a 2 z 6 -6 a 7 z 5 -16 a 5 z 5 -10 a 3 z 5 + 7 a 6 z 4 + 6 a 4 z 4 - a 2 z 4 + 11 a 7 z 3 + 27 a 5 z 3 + 19 a 3 z 3 + 3 a z 3 -4 a 6 z 2 -11 a 4 z 2 -7 a 2 z 2 -6 a 7 z -19 a 5 z -21 a 3 z -8 a z + 3 a 6 + 10 a 4 + 11 a 2 + 5 + a 7 z -1 + 5 a 5 z -1 + 9 a 3 z -1 + 5 a z -1 - a 6 z -2 -4 a 4 z -2 -5 a 2 z -2 -2 z -2$ (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 =

0 is the signature of L11n378. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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