L11a508

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_18,12,19,11 X_10,4,11,3 X_2,20,3,19 X_16,8,17,7 X_20,9,21,10 X_12,18,7,17 X_4,21,5,22 X_14,6,15,5 X_22,15,13,16 X_6,14,1,13
Gauss code	{1, -4, 3, -8, 9, -11}, {5, -1, 6, -3, 2, -7}, {11, -9, 10, -5, 7, -2, 4, -6, 8, -10}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 3 -2 v u 3 - v 2 w u 3 + 2 v w u 3 - w u 3 + u 3 -2 v 2 u 2 - v 2 w 2 u 2 + 2 v w 2 u 2 - w 2 u 2 + 4 v u 2 + 3 v 2 w u 2 -5 v w u 2 + 3 w u 2 -2 u 2 + v 2 u + 2 v 2 w 2 u -4 v w 2 u + 2 w 2 u -2 v u -3 v 2 w u + 5 v w u -3 w u + u - v 2 w 2 + 2 v w 2 - w 2 + v 2 w -2 v w + w$ (db)
Jones polynomial	$- q 7 + 4 q 6 -7 q 5 + 13 q 4 -16 q 3 + 20 q 2 -19 q + 18-13 q -1 + 8 q -2 -4 q -3 + q -4$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 6 a -2 - z 6 + a 2 z 4 - z 4 a -2 + 2 z 4 a -4 -2 z 4 + a 2 z 2 + z 2 a -2 + 2 z 2 a -4 - z 2 a -6 -2 z 2 + 2 a -2 -2 a -4 + a -2 z -2 -2 a -4 z -2 + a -6 z -2$ (db)
Kauffman polynomial	$2 z 10 a -2 + 2 z 10 a -4 + 7 z 9 a -1 + 12 z 9 a -3 + 5 z 9 a -5 + 14 z 8 a -2 + 7 z 8 a -4 + 4 z 8 a -6 + 11 z 8 + 11 a z 7 -3 z 7 a -1 -30 z 7 a -3 -15 z 7 a -5 + z 7 a -7 + 8 a 2 z 6 -49 z 6 a -2 -41 z 6 a -4 -15 z 6 a -6 -15 z 6 + 4 a 3 z 5 -13 a z 5 -16 z 5 a -1 + 14 z 5 a -3 + 10 z 5 a -5 -3 z 5 a -7 + a 4 z 4 -7 a 2 z 4 + 42 z 4 a -2 + 47 z 4 a -4 + 17 z 4 a -6 + 4 z 4 -2 a 3 z 3 + 4 a z 3 + 10 z 3 a -1 + 2 z 3 a -3 + 2 z 3 a -7 + 2 a 2 z 2 -11 z 2 a -2 -14 z 2 a -4 -5 z 2 a -6 + 2 z a -3 + 2 z a -5 -2 a -2 -3 a -4 -2 a -6 -2 a -3 z -1 -2 a -5 z -1 + a -2 z -2 + 2 a -4 z -2 + a -6 z -2$ (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 =

0 is the signature of L11a508. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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