L11n302

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_3,13,4,12 X_20,13,21,14 X_22,19,11,20 X_15,5,16,10 X_17,9,18,8 X_7,17,8,16 X_9,19,10,18 X_14,21,15,22 X₂₅₃₆ X_11,1,12,4
Gauss code	{1, -10, -2, 11}, {10, -1, -7, 6, -8, 5}, {-11, 2, 3, -9, -5, 7, -6, 8, 4, -3, 9, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$u v 4 - u 2 v 3 - u v 3 + v 3 + u 2 v 2 - w v 2 - u 2 w v + u w v + w v - u w$ (db)
Jones polynomial	$q 6 - q 5 + 2 q 4 - q 3 + 2 q 2 + 1 + q -1 - q -2 + q -3 - q -4$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 6 a -2 + z 6 - a 2 z 4 -7 z 4 a -2 + z 4 a -4 + 7 z 4 -4 a 2 z 2 -17 z 2 a -2 + 4 z 2 a -4 + 16 z 2 -4 a 2 -16 a -2 + 5 a -4 + 15- a 2 z -2 -5 a -2 z -2 + 2 a -4 z -2 + 4 z -2$ (db)
Kauffman polynomial	$a 2 z 8 + z 8 a -2 + z 8 a -4 + z 8 + a 3 z 7 + 2 a z 7 + 2 z 7 a -1 + 2 z 7 a -3 + z 7 a -5 -6 a 2 z 6 -8 z 6 a -2 -5 z 6 a -4 + z 6 a -6 -8 z 6 -6 a 3 z 5 -14 a z 5 -16 z 5 a -1 -12 z 5 a -3 -4 z 5 a -5 + 10 a 2 z 4 + 22 z 4 a -2 + 7 z 4 a -4 -5 z 4 a -6 + 20 z 4 + 10 a 3 z 3 + 28 a z 3 + 37 z 3 a -1 + 21 z 3 a -3 + 2 z 3 a -5 -7 a 2 z 2 -31 z 2 a -2 -7 z 2 a -4 + 6 z 2 a -6 -25 z 2 -5 a 3 z -21 a z -33 z a -1 -16 z a -3 + z a -5 + 4 a 2 + 20 a -2 + 6 a -4 -2 a -6 + 17 + a 3 z -1 + 5 a z -1 + 9 a -1 z -1 + 5 a -3 z -1 - a 2 z -2 -5 a -2 z -2 -2 a -4 z -2 -4 z -2$ (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 =

0 is the signature of L11n302. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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