L11n347

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_3,15,4,14 X_11,20,12,21 X_7,18,8,19 X_17,22,18,13 X_16,9,17,10 X_10,15,11,16 X_19,12,20,5 X_21,8,22,9 X₂₅₃₆ X_13,1,14,4
Gauss code	{1, -10, -2, 11}, {10, -1, -4, 9, 6, -7, -3, 8}, {-11, 2, 7, -6, -5, 4, -8, 3, -9, 5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 3 - v u 3 - v 2 w u 3 - v 2 u 2 + v u 2 + v 2 w u 2 - w u 2 + v 3 u - v u - v 2 w u + v w u + v + v 2 w - v w$ (db)
Jones polynomial	$1-2 q -1 + 4 q -2 -3 q -3 + 4 q -4 -3 q -5 + 3 q -6 - q -7 + q -9$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$z 2 a 8 + a 8 z -2 + 2 a 8 -2 z 2 a 6 -2 a 6 z -2 -5 a 6 - z 6 a 4 -4 z 4 a 4 -3 z 2 a 4 + a 4 z -2 + a 4 + z 4 a 2 + 3 z 2 a 2 + 2 a 2$ (db)
Kauffman polynomial	$z 6 a 10 -6 z 4 a 10 + 8 z 2 a 10 -2 a 10 - z 5 a 9 + 2 z 3 a 9 + z a 9 - z 6 a 8 + 4 z 4 a 8 -3 z 2 a 8 - a 8 z -2 + 3 a 8 + z 7 a 7 -5 z 5 a 7 + 10 z 3 a 7 -8 z a 7 + 2 a 7 z -1 + z 8 a 6 -4 z 6 a 6 + 8 z 4 a 6 -13 z 2 a 6 -2 a 6 z -2 + 9 a 6 + 3 z 7 a 5 -11 z 5 a 5 + 12 z 3 a 5 -8 z a 5 + 2 a 5 z -1 + z 8 a 4 - z 6 a 4 -6 z 4 a 4 + 3 z 2 a 4 - a 4 z -2 + 3 a 4 + 2 z 7 a 3 -7 z 5 a 3 + 4 z 3 a 3 + z a 3 + z 6 a 2 -4 z 4 a 2 + 5 z 2 a 2 -2 a 2$ (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 =

-4 is the signature of L11n347. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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