L11a519

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_16,6,17,5 X_22,20,13,19 X_18,10,19,9 X_10,21,11,22 X_20,11,21,12 X_2,13,3,14 X_14,3,15,4 X₄₇₅₈ X_12,16,7,15 X_6,18,1,17
Gauss code	{1, -7, 8, -9, 2, -11}, {9, -1, 4, -5, 6, -10}, {7, -8, 10, -2, 11, -4, 3, -6, 5, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 3 - v 2 w 2 u 3 + v w 2 u 3 + v u 3 + 2 v 2 w u 3 -2 v w u 3 + 2 v 2 u 2 + 2 v 2 w 2 u 2 -4 v w 2 u 2 + w 2 u 2 -4 v u 2 -4 v 2 w u 2 + 8 v w u 2 -2 w u 2 + u 2 - v 2 u - v 2 w 2 u + 4 v w 2 u -2 w 2 u + 4 v u + 2 v 2 w u -8 v w u + 4 w u -2 u - v w 2 + w 2 - v + 2 v w -2 w + 1$ (db)
Jones polynomial	$- q 5 + 4 q 4 -9 q 3 + 15 q 2 -20 q + 24-22 q -1 + 21 q -2 -14 q -3 + 9 q -4 -4 q -5 + q -6$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z 8 -2 a 2 z 6 - z 6 a -2 + 5 z 6 + a 4 z 4 -7 a 2 z 4 -3 z 4 a -2 + 10 z 4 + 2 a 4 z 2 -8 a 2 z 2 -3 z 2 a -2 + 9 z 2 + a 4 -4 a 2 - a -2 + 4 + a 4 z -2 -2 a 2 z -2 + z -2$ (db)
Kauffman polynomial	$2 a 2 z 10 + 2 z 10 + 6 a 3 z 9 + 13 a z 9 + 7 z 9 a -1 + 7 a 4 z 8 + 14 a 2 z 8 + 10 z 8 a -2 + 17 z 8 + 4 a 5 z 7 -6 a 3 z 7 -20 a z 7 -2 z 7 a -1 + 8 z 7 a -3 + a 6 z 6 -16 a 4 z 6 -44 a 2 z 6 -16 z 6 a -2 + 4 z 6 a -4 -47 z 6 -9 a 5 z 5 -7 a 3 z 5 + 3 a z 5 -12 z 5 a -1 -12 z 5 a -3 + z 5 a -5 -2 a 6 z 4 + 12 a 4 z 4 + 45 a 2 z 4 + 12 z 4 a -2 -5 z 4 a -4 + 48 z 4 + 5 a 5 z 3 + 6 a 3 z 3 + 6 a z 3 + 12 z 3 a -1 + 6 z 3 a -3 - z 3 a -5 + a 6 z 2 -6 a 4 z 2 -25 a 2 z 2 -6 z 2 a -2 + z 2 a -4 -25 z 2 -2 a 3 z -4 a z -3 z a -1 - z a -3 + 3 a 4 + 7 a 2 + 2 a -2 + 7 + 2 a 3 z -1 + 2 a z -1 - a 4 z -2 -2 a 2 z -2 - z -2$ (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 L11a519. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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