L11n303

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_13,21,14,20 X_19,11,20,22 X_15,5,16,10 X_17,9,18,8 X_7,17,8,16 X_9,19,10,18 X_21,15,22,14 X₂₅₃₆ X_4,11,1,12
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 $u$ , $v$ , $w$ , ...)	$v 2 u 4 - v u 3 - v 2 w u 3 + v w u 3 + 2 v u 2 + v 2 w u 2 -2 v w u 2 - u 2 - v u + v w u + u - w$ (db)
Jones polynomial	$- q 8 + 2 q 7 -3 q 6 + 4 q 5 -4 q 4 + 4 q 3 -2 q 2 + 2 q + 1 + q -2$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$- z 6 a -2 + z 6 a -4 -8 z 4 a -2 + 6 z 4 a -4 - z 4 a -6 + z 4 -19 z 2 a -2 + 14 z 2 a -4 -3 z 2 a -6 + 5 z 2 -17 a -2 + 13 a -4 -3 a -6 + 7-5 a -2 z -2 + 4 a -4 z -2 - a -6 z -2 + 2 z -2$ (db)
Kauffman polynomial	$z 8 a -2 + z 8 a -4 + z 8 a -6 + z 8 + z 7 a -1 + 2 z 7 a -3 + 3 z 7 a -5 + 2 z 7 a -7 -10 z 6 a -2 -6 z 6 a -4 -2 z 6 a -6 + 2 z 6 a -8 -8 z 6 -9 z 5 a -1 -16 z 5 a -3 -14 z 5 a -5 -6 z 5 a -7 + z 5 a -9 + 31 z 4 a -2 + 15 z 4 a -4 - z 4 a -6 -6 z 4 a -8 + 21 z 4 + 22 z 3 a -1 + 41 z 3 a -3 + 26 z 3 a -5 + 4 z 3 a -7 -3 z 3 a -9 -39 z 2 a -2 -16 z 2 a -4 + 3 z 2 a -6 + 3 z 2 a -8 -23 z 2 -19 z a -1 -35 z a -3 -19 z a -5 -2 z a -7 + z a -9 + 22 a -2 + 13 a -4 - a -8 + 11 + 5 a -1 z -1 + 9 a -3 z -1 + 5 a -5 z -1 + a -7 z -1 -5 a -2 z -2 -4 a -4 z -2 - a -6 z -2 -2 z -2$ (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 L11n303. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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