L11n374

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_2,9,3,10 X_12,3,13,4 X_10,5,11,6 X_16,11,5,12 X_4,15,1,16 X_13,20,14,21 X_7,19,8,18 X_17,9,18,8 X_19,22,20,17 X_21,14,22,15
Gauss code	{1, -2, 3, -6}, {-9, 8, -10, 7, -11, 10}, {4, -1, -8, 9, 2, -4, 5, -3, -7, 11, 6, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$2 v u 3 -2 v w u 3 -3 v u 2 + 3 v w u 2 - w u 2 + u 2 + v u - v w u + 3 w u -3 u -2 w + 2$ (db)
Jones polynomial	$1-2 q -1 + 6 q -2 -6 q -3 + 8 q -4 -8 q -5 + 8 q -6 -5 q -7 + 3 q -8 - q -9$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$- a 10 + z 4 a 8 + 4 z 2 a 8 + 3 a 8 - z 6 a 6 -4 z 4 a 6 -5 z 2 a 6 + a 6 z -2 -2 a 6 - z 6 a 4 -3 z 4 a 4 -2 z 2 a 4 -2 a 4 z -2 -3 a 4 + z 4 a 2 + 3 z 2 a 2 + a 2 z -2 + 3 a 2$ (db)
Kauffman polynomial	$z 3 a 11 -2 z a 11 + 3 z 4 a 10 -4 z 2 a 10 + a 10 + 2 z 7 a 9 -7 z 5 a 9 + 14 z 3 a 9 -6 z a 9 + 3 z 8 a 8 -12 z 6 a 8 + 22 z 4 a 8 -12 z 2 a 8 + 3 a 8 + z 9 a 7 + 2 z 7 a 7 -14 z 5 a 7 + 20 z 3 a 7 -6 z a 7 + 5 z 8 a 6 -15 z 6 a 6 + 14 z 4 a 6 -5 z 2 a 6 + a 6 z -2 - a 6 + z 9 a 5 + 2 z 7 a 5 -12 z 5 a 5 + 7 z 3 a 5 + 2 z a 5 -2 a 5 z -1 + 2 z 8 a 4 -2 z 6 a 4 -9 z 4 a 4 + 9 z 2 a 4 + 2 a 4 z -2 -6 a 4 + 2 z 7 a 3 -5 z 5 a 3 + 4 z a 3 -2 a 3 z -1 + z 6 a 2 -4 z 4 a 2 + 6 z 2 a 2 + a 2 z -2 -4 a 2$ (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 L11n374. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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