L11n342

(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_7,14,8,15 X_13,8,14,9 X_15,21,16,20 X_17,13,18,22 X_21,17,22,16 X_19,5,20,12 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, -3, 9}, {-5, 4, -6, 8, -7, 3, -9, 6, -8, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 3 + v 2 u 3 + v w u 3 - w u 3 + v 3 u 2 - v u 2 - v 2 w u 2 + w u 2 - v 3 u + v u + v 2 w u - w u + v 3 - v 2 - v w + w$ (db)
Jones polynomial	$- q 5 + q 4 - q 2 + 2 q -2 + 4 q -1 -3 q -2 + 4 q -3 - q -4 + q -5$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 -2 a 2 z 4 + 5 z 4 + a 4 z 2 -8 a 2 z 2 - z 2 a -4 + 8 z 2 + 3 a 4 -10 a 2 - a -4 + 8 + 2 a 4 z -2 -5 a 2 z -2 - a -2 z -2 + 4 z -2$ (db)
Kauffman polynomial	$a 3 z 9 + a z 9 + a 4 z 8 + 5 a 2 z 8 + 4 z 8 -4 a 3 z 7 + 4 z 7 a -1 -7 a 4 z 6 -30 a 2 z 6 + z 6 a -4 -24 z 6 - a 3 z 5 -24 a z 5 -25 z 5 a -1 - z 5 a -3 + z 5 a -5 + 18 a 4 z 4 + 58 a 2 z 4 -3 z 4 a -2 -5 z 4 a -4 + 42 z 4 + 17 a 3 z 3 + 52 a z 3 + 40 z 3 a -1 + z 3 a -3 -4 z 3 a -5 -21 a 4 z 2 -51 a 2 z 2 + z 2 a -2 + 4 z 2 a -4 -33 z 2 -18 a 3 z -37 a z -23 z a -1 -2 z a -3 + 2 z a -5 + 11 a 4 + 24 a 2 + 2 a -2 - a -4 + 17 + 5 a 3 z -1 + 9 a z -1 + 5 a -1 z -1 + a -3 z -1 -2 a 4 z -2 -5 a 2 z -2 - a -2 z -2 -4 z -2$ (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 =

2 is the signature of L11n342. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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