L11n357

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_11,19,12,18 X_7,16,8,17 X_15,8,16,9 X_17,15,18,22 X_13,21,14,20 X_19,13,20,12 X_21,5,22,14 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {-5, 4, -6, 3, -8, 7, -9, 6}, {10, -1, -4, 5, 11, -2, -3, 8, -7, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$u 2 v 4 - u v 4 - u 2 v 3 + u v 3 - v 3 - u 2 w v 2 + v 2 + u 2 w v - u w v + w v + u w - w$ (db)
Jones polynomial	$- q 4 + 2 q 3 -2 q 2 + 4 q -3 + 4 q -1 -3 q -2 + 3 q -3 - q -4 + q -5$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 -2 a 2 z 4 - z 4 a -2 + 5 z 4 + a 4 z 2 -8 a 2 z 2 -3 z 2 a -2 + 8 z 2 + 3 a 4 -8 a 2 - a -2 + 6 + a 4 z -2 -2 a 2 z -2 + z -2$ (db)
Kauffman polynomial	$a 3 z 9 + a z 9 + a 4 z 8 + 4 a 2 z 8 + 3 z 8 -5 a 3 z 7 -2 a z 7 + 3 z 7 a -1 -7 a 4 z 6 -24 a 2 z 6 + 2 z 6 a -2 -15 z 6 + 5 a 3 z 5 -8 a z 5 -12 z 5 a -1 + z 5 a -3 + 17 a 4 z 4 + 47 a 2 z 4 -6 z 4 a -2 + 24 z 4 + 5 a 3 z 3 + 18 a z 3 + 11 z 3 a -1 -2 z 3 a -3 -17 a 4 z 2 -38 a 2 z 2 + 3 z 2 a -2 + 2 z 2 a -4 -20 z 2 -7 a 3 z -10 a z -3 z a -1 + z a -3 + z a -5 + 7 a 4 + 14 a 2 - a -2 - a -4 + 8 + 2 a 3 z -1 + 2 a z -1 - a 4 z -2 -2 a 2 z -2 - z -2$ (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 =

2 is the signature of L11n357. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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