L11n348

(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_17,22,18,13 X_9,17,10,16 X_15,11,16,10 X_19,12,20,5 X_21,8,22,9 X₂₅₃₆ X_4,13,1,14
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 $u$ , $v$ , $w$ , ...)	$- v 2 w u 3 + v w u 3 - v 3 u 2 + v 2 u 2 + 2 v 2 w u 2 - w u 2 + v 3 u -2 v u - v w u + w u - v 2 + v$ (db)
Jones polynomial	$1-2 q -1 + 3 q -2 -2 q -3 + 2 q -4 + q -6 + 2 q -7 -2 q -8 + 2 q -9 - q -10$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$- a 10 + 2 z 2 a 8 + a 8 z -2 + 2 a 8 -2 a 6 z -2 -3 a 6 - z 6 a 4 -4 z 4 a 4 -2 z 2 a 4 + a 4 z -2 + a 4 + z 4 a 2 + 3 z 2 a 2 + a 2$ (db)
Kauffman polynomial	$z 7 a 11 -5 z 5 a 11 + 6 z 3 a 11 -2 z a 11 + 2 z 8 a 10 -11 z 6 a 10 + 16 z 4 a 10 -10 z 2 a 10 + 3 a 10 + z 9 a 9 -4 z 7 a 9 -3 z 5 a 9 + 14 z 3 a 9 -7 z a 9 + 3 z 8 a 8 -21 z 6 a 8 + 40 z 4 a 8 -29 z 2 a 8 - a 8 z -2 + 11 a 8 + z 9 a 7 -5 z 7 a 7 - z 5 a 7 + 16 z 3 a 7 -12 z a 7 + 2 a 7 z -1 + 2 z 8 a 6 -14 z 6 a 6 + 27 z 4 a 6 -24 z 2 a 6 -2 a 6 z -2 + 11 a 6 + 2 z 7 a 5 -11 z 5 a 5 + 14 z 3 a 5 -8 z a 5 + 2 a 5 z -1 + z 8 a 4 -3 z 6 a 4 - z 4 a 4 - z 2 a 4 - a 4 z -2 + 3 a 4 + 2 z 7 a 3 -8 z 5 a 3 + 6 z 3 a 3 - z a 3 + z 6 a 2 -4 z 4 a 2 + 4 z 2 a 2 - a 2$ (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 L11n348. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -5$	$i = -3$	$i = -1$
$r = -9$		${\mathbb Z}$
$r = -8$		${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -7$		${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -6$		${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2$	${\mathbb Z}^{2}$
$r = -3$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -1$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 0$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}^{2}$
$r = 1$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = 2$	${\mathbb Z}_2$	${\mathbb Z}$