L11a486

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_12,6,13,5 X₈₄₉₃ X_22,14,19,13 X_20,10,21,9 X_10,20,11,19 X_14,22,15,21 X_18,12,5,11 X_2,16,3,15
Gauss code	{1, -11, 5, -3}, {8, -7, 9, -6}, {4, -1, 2, -5, 7, -8, 10, -4, 6, -9, 11, -2, 3, -10}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 4 v u 4 -2 v w u 4 + 2 w u 4 -4 u 4 -7 v u 3 + 6 v w u 3 -6 w u 3 + 7 u 3 + 6 v u 2 -7 v w u 2 + 7 w u 2 -6 u 2 -2 v u + 4 v w u -4 w u + 2 u - v w + w$ (db)
Jones polynomial	$q 9 -4 q 8 + 10 q 7 -17 q 6 + 23 q 5 -25 q 4 + 27 q 3 -22 q 2 + 17 q -9 + 4 q -1 - q -2$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 a -2 + 2 z 6 a -4 + z 4 a -2 + 5 z 4 a -4 -3 z 4 a -6 - z 4 + z 2 a -2 + 4 z 2 a -4 -5 z 2 a -6 + z 2 a -8 - z 2 + 2 a -2 - a -4 -2 a -6 + a -8 + a -2 z -2 -2 a -4 z -2 + a -6 z -2$ (db)
Kauffman polynomial	$2 z 10 a -4 + 2 z 10 a -6 + 8 z 9 a -3 + 15 z 9 a -5 + 7 z 9 a -7 + 11 z 8 a -2 + 25 z 8 a -4 + 22 z 8 a -6 + 8 z 8 a -8 + 8 z 7 a -1 + z 7 a -3 -14 z 7 a -5 -3 z 7 a -7 + 4 z 7 a -9 -16 z 6 a -2 -67 z 6 a -4 -65 z 6 a -6 -17 z 6 a -8 + z 6 a -10 + 4 z 6 + a z 5 -10 z 5 a -1 -22 z 5 a -3 -23 z 5 a -5 -20 z 5 a -7 -8 z 5 a -9 + 12 z 4 a -2 + 63 z 4 a -4 + 62 z 4 a -6 + 14 z 4 a -8 -2 z 4 a -10 -5 z 4 - a z 3 + 5 z 3 a -1 + 19 z 3 a -3 + 27 z 3 a -5 + 20 z 3 a -7 + 6 z 3 a -9 -4 z 2 a -2 -27 z 2 a -4 -30 z 2 a -6 -8 z 2 a -8 + z 2 a -10 + 2 z 2 - z a -1 - z a -3 -5 z a -5 -7 z a -7 -2 z a -9 - a -2 + 2 a -4 + 5 a -6 + 3 a -8 -2 a -3 z -1 -2 a -5 z -1 + a -2 z -2 + 2 a -4 z -2 + a -6 z -2$ (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 L11a486. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 1$	$i = 3$
$r = -3$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 0$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{8}$
$r = 1$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 2$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = 3$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{14}$	${\mathbb Z}^{14}$
$r = 4$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{13}$
$r = 5$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 6$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 7$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 8$	${\mathbb Z}_2$	${\mathbb Z}$