L11a398

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 4 - v u 4 - v 2 w u 4 + v w u 4 -3 v 2 u 3 + 5 v u 3 + 3 v 2 w u 3 -5 v w u 3 + 2 w u 3 -2 u 3 + 4 v 2 u 2 -8 v u 2 -4 v 2 w u 2 + 8 v w u 2 -4 w u 2 + 4 u 2 -2 v 2 u + 5 v u + 2 v 2 w u -5 v w u + 3 w u -3 u - v + v w - w + 1$ (db)
Jones polynomial	$- q 5 + 4 q 4 -9 q 3 + 17 q 2 -21 q + 27-25 q -1 + 23 q -2 -17 q -3 + 10 q -4 -5 q -5 + q -6$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z 8 -2 a 2 z 6 - z 6 a -2 + 5 z 6 + a 4 z 4 -6 a 2 z 4 -3 z 4 a -2 + 10 z 4 + a 4 z 2 -4 a 2 z 2 -3 z 2 a -2 + 6 z 2 - a 4 + 4 a 2 + a -2 -4- a 4 z -2 + 4 a 2 z -2 + 2 a -2 z -2 -5 z -2$ (db)
Kauffman polynomial	$2 a 2 z 10 + 2 z 10 + 7 a 3 z 9 + 15 a z 9 + 8 z 9 a -1 + 9 a 4 z 8 + 22 a 2 z 8 + 11 z 8 a -2 + 24 z 8 + 5 a 5 z 7 - a 3 z 7 -15 a z 7 - z 7 a -1 + 8 z 7 a -3 + a 6 z 6 -19 a 4 z 6 -63 a 2 z 6 -17 z 6 a -2 + 4 z 6 a -4 -64 z 6 -10 a 5 z 5 -25 a 3 z 5 -19 a z 5 -15 z 5 a -1 -10 z 5 a -3 + z 5 a -5 - a 6 z 4 + 12 a 4 z 4 + 53 a 2 z 4 + 14 z 4 a -2 -5 z 4 a -4 + 59 z 4 + 5 a 5 z 3 + 17 a 3 z 3 + 17 a z 3 + 10 z 3 a -1 + 4 z 3 a -3 - z 3 a -5 -4 a 4 z 2 -18 a 2 z 2 -5 z 2 a -2 + 2 z 2 a -4 -21 z 2 + a 5 z + 5 a 3 z + 9 a z + 5 z a -1 -2 a 2 -3 a -2 -4- a 5 z -1 -5 a 3 z -1 -9 a z -1 -5 a -1 z -1 + a 4 z -2 + 4 a 2 z -2 + 2 a -2 z -2 + 5 z -2$ (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 L11a398. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -1$	$i = 1$
$r = -6$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -3$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -2$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = -1$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 0$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{15}$
$r = 1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 2$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{11}$
$r = 3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$