L11n397

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_10,4,11,3 X_12,8,13,7 X_16,10,17,9 X_17,22,18,19 X_13,20,14,21 X_19,14,20,15 X_21,18,22,5 X_8,16,9,15 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -10, 2, -11}, {-7, 6, -8, 5}, {10, -1, 3, -9, 4, -2, 11, -3, -6, 7, 9, -4, -5, 8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + v u 4 - u 4 + v u 3 - v w u 3 + w u 3 - u 3 - v u 2 + v w u 2 - w u 2 + u 2 + v w u - w u - v w + w$ (db)
Jones polynomial	$- q 5 + 2 q 4 -2 q 3 + q 2 - q + 1 + 2 q -1 - q -2 + 3 q -3 - q -4 + q -5$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z 6 -2 a 2 z 4 + 6 z 4 + a 4 z 2 -9 a 2 z 2 -2 z 2 a -2 - z 2 a -4 + 11 z 2 + 3 a 4 -11 a 2 -3 a -2 + 11 + 2 a 4 z -2 -5 a 2 z -2 - a -2 z -2 + 4 z -2$ (db)
Kauffman polynomial	$a 3 z 9 + a z 9 + a 4 z 8 + 4 a 2 z 8 + 3 z 8 -5 a 3 z 7 -4 a z 7 + 2 z 7 a -1 + z 7 a -3 -7 a 4 z 6 -27 a 2 z 6 + 2 z 6 a -4 -22 z 6 + 3 a 3 z 5 -7 a z 5 -15 z 5 a -1 -4 z 5 a -3 + z 5 a -5 + 17 a 4 z 4 + 56 a 2 z 4 -7 z 4 a -4 + 46 z 4 + 11 a 3 z 3 + 32 a z 3 + 27 z 3 a -1 + 3 z 3 a -3 -3 z 3 a -5 -19 a 4 z 2 -49 a 2 z 2 -5 z 2 a -2 + 3 z 2 a -4 -38 z 2 -15 a 3 z -29 a z -17 z a -1 -2 z a -3 + z a -5 + 10 a 4 + 22 a 2 + 4 a -2 + 17 + 5 a 3 z -1 + 9 a z -1 + 5 a -1 z -1 + a -3 z -1 -2 a 4 z -2 -5 a 2 z -2 - a -2 z -2 -4 z -2$ (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 =

1 is the signature of L11n397. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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