L11n350

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,4,15,3 X_11,21,12,20 X_7,18,8,19 X_9,16,10,17 X_17,8,18,9 X_19,13,20,22 X_13,10,14,11 X_21,5,22,12 X₂₅₃₆ X_4,16,1,15
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 6, -5, 8, -3, 9}, {-8, -2, 11, 5, -6, 4, -7, 3, -9, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v w u 3 - w u 3 + v 2 u 2 - v u 2 - v 2 w u 2 + v w u 2 + w u 2 - v 3 u - v 2 u + v u + v 2 w u - v w u + v 3 - v 2$ (db)
Jones polynomial	$- q 5 + 2 q 4 -2 q 3 + 2 q 2 - q + 2 + q -1 - q -2 + 2 q -3 - q -4 + q -5$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z 2 a 4 + a 4 z -2 + 2 a 4 - z 4 a 2 -4 z 2 a 2 -2 a 2 z -2 -5 a 2 + z -2 + 2 + z 4 a -2 + 3 z 2 a -2 + 2 a -2 - z 2 a -4 - a -4$ (db)
Kauffman polynomial	$a 3 z 9 + a z 9 + a 4 z 8 + 3 a 2 z 8 + 2 z 8 -6 a 3 z 7 -6 a z 7 + z 7 a -1 + z 7 a -3 -7 a 4 z 6 -21 a 2 z 6 + z 6 a -2 + 2 z 6 a -4 -15 z 6 + 9 a 3 z 5 + 6 a z 5 -7 z 5 a -1 -3 z 5 a -3 + z 5 a -5 + 16 a 4 z 4 + 44 a 2 z 4 -4 z 4 a -2 -7 z 4 a -4 + 31 z 4 - a 3 z 3 + 6 a z 3 + 10 z 3 a -1 -3 z 3 a -5 -15 a 4 z 2 -36 a 2 z 2 + 2 z 2 a -2 + 4 z 2 a -4 -23 z 2 -5 a 3 z -8 a z -3 z a -1 + z a -3 + z a -5 + 6 a 4 + 13 a 2 - a -4 + 9 + 2 a 3 z -1 + 2 a z -1 - a 4 z -2 -2 a 2 z -2 - z -2$ (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 =

0 is the signature of L11n350. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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