L11n352

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,3,15,4 X_11,20,12,21 X_7,18,8,19 X_9,13,10,22 X_21,17,22,16 X_17,8,18,9 X_15,11,16,10 X_19,12,20,5 X₂₅₃₆ X_4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 7, -5, 8, -3, 9}, {11, -2, -8, 6, -7, 4, -9, 3, -6, 5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 w u 3 + v w u 3 - v 3 u 2 - v 2 u 2 + 2 v u 2 + 2 v 2 w u 2 -2 v w u 2 - w u 2 + v 3 u + 2 v 2 u -2 v u -2 v 2 w u + v w u + w u - v 2 + v$ (db)
Jones polynomial	$- q 2 + 3 q -5 + 6 q -1 -5 q -2 + 6 q -3 -3 q -4 + 2 q -5 + q -6 - q -7 + q -8$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a 8 z -2 + a 8 -2 z 2 a 6 -2 a 6 z -2 -5 a 6 + 2 z 2 a 4 + a 4 z -2 + 3 a 4 + z 6 a 2 + 4 z 4 a 2 + 5 z 2 a 2 + 2 a 2 - z 4 -2 z 2 -1$ (db)
Kauffman polynomial	$a 7 z 9 + a 5 z 9 + a 8 z 8 + 2 a 6 z 8 + 2 a 4 z 8 + a 2 z 8 -7 a 7 z 7 -7 a 5 z 7 + 3 a 3 z 7 + 3 a z 7 -7 a 8 z 6 -18 a 6 z 6 -12 a 4 z 6 + 2 a 2 z 6 + 3 z 6 + 12 a 7 z 5 + 11 a 5 z 5 -7 a 3 z 5 -5 a z 5 + z 5 a -1 + 15 a 8 z 4 + 45 a 6 z 4 + 28 a 4 z 4 -9 a 2 z 4 -7 z 4 -3 a 7 z 3 + 5 a 5 z 3 + 7 a 3 z 3 -3 a z 3 -2 z 3 a -1 -14 a 8 z 2 -38 a 6 z 2 -23 a 4 z 2 + 4 a 2 z 2 + 3 z 2 -5 a 7 z -8 a 5 z -3 a 3 z + a z + z a -1 + 6 a 8 + 13 a 6 + 9 a 4 -1 + 2 a 7 z -1 + 2 a 5 z -1 - a 8 z -2 -2 a 6 z -2 - a 4 z -2$ (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 =

-2 is the signature of L11n352. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -3$	$i = -1$	$i = 1$
$r = -8$		${\mathbb Z}$
$r = -7$		${\mathbb Z}_2$	${\mathbb Z}$
$r = -6$		${\mathbb Z}$
$r = -5$		${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{4}$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}^{2}$
$r = -2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -1$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 0$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{3}$
$r = 1$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 2$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 3$	${\mathbb Z}_2$	${\mathbb Z}$