L11a399

(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_4,15,1,16 X_18,22,19,21 X_20,9,21,10 X_10,19,5,20 X_16,12,17,11 X_22,18,11,17
Gauss code	{1, -4, 3, -6}, {2, -1, 5, -3, 8, -9}, {10, -2, 4, -5, 6, -10, 11, -7, 9, -8, 7, -11}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$2 v 2 u 2 -4 v u 2 -2 v 2 w u 2 + 4 v w u 2 -2 w u 2 + 2 u 2 -4 v 2 u + 8 v u + 4 v 2 w u -8 v w u + 4 w u -4 u + 2 v 2 -4 v -2 v 2 w + 4 v w -2 w + 2$ (db)
Jones polynomial	$q 6 -4 q 5 + 8 q 4 -13 q 3 + 18 q 2 -20 q + 21-17 q -1 + 14 q -2 -7 q -3 + 4 q -4 - q -5$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 6 a -2 - z 6 + 2 a 2 z 4 -2 z 4 a -2 + z 4 a -4 - z 4 - a 4 z 2 + 2 a 2 z 2 -2 z 2 a -2 + z 2 a -4 - a 2 + 1 + a 4 z -2 -2 a 2 z -2 + z -2$ (db)
Kauffman polynomial	$2 z 10 a -2 + 2 z 10 + 5 a z 9 + 11 z 9 a -1 + 6 z 9 a -3 + 6 a 2 z 8 + 9 z 8 a -2 + 7 z 8 a -4 + 8 z 8 + 6 a 3 z 7 - a z 7 -22 z 7 a -1 -11 z 7 a -3 + 4 z 7 a -5 + 4 a 4 z 6 - a 2 z 6 -32 z 6 a -2 -19 z 6 a -4 + z 6 a -6 -17 z 6 + a 5 z 5 -7 a 3 z 5 -3 a z 5 + 17 z 5 a -1 + 2 z 5 a -3 -10 z 5 a -5 -7 a 4 z 4 -12 a 2 z 4 + 30 z 4 a -2 + 15 z 4 a -4 -2 z 4 a -6 + 8 z 4 - a 5 z 3 + a 3 z 3 -2 a z 3 -6 z 3 a -1 + 3 z 3 a -3 + 5 z 3 a -5 + 4 a 4 z 2 + 8 a 2 z 2 -8 z 2 a -2 -4 z 2 a -4 - a 3 z - a z + a 4 + a 2 + 1 + 2 a 3 z -1 + 2 a z -1 - a 4 z -2 -2 a 2 z -2 - z -2$ (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 =

0 is the signature of L11a399. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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