L11a505

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 3 - v 2 w 2 u 3 + v w 2 u 3 + v u 3 + 2 v 2 w u 3 -2 v w u 3 + 2 v 2 u 2 + 3 v 2 w 2 u 2 -4 v w 2 u 2 + w 2 u 2 -4 v u 2 -5 v 2 w u 2 + 9 v w u 2 -3 w u 2 + 2 u 2 - v 2 u -2 v 2 w 2 u + 4 v w 2 u -2 w 2 u + 4 v u + 3 v 2 w u -9 v w u + 5 w u -3 u - v w 2 + w 2 - v + 2 v w -2 w + 1$ (db)
Jones polynomial	$- q 8 + 4 q 7 -9 q 6 + 17 q 5 -22 q 4 + 27 q 3 -26 q 2 + 24 q -17 + 11 q -1 -5 q -2 + q -3$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$- z 8 a -2 -4 z 6 a -2 + 2 z 6 a -4 + z 6 -5 z 4 a -2 + 6 z 4 a -4 - z 4 a -6 + 2 z 4 - z 2 a -2 + 4 z 2 a -4 -2 z 2 a -6 + 2 a -2 -2 a -4 + a -2 z -2 -2 a -4 z -2 + a -6 z -2$ (db)
Kauffman polynomial	$3 z 10 a -2 + 3 z 10 a -4 + 9 z 9 a -1 + 18 z 9 a -3 + 9 z 9 a -5 + 18 z 8 a -2 + 19 z 8 a -4 + 11 z 8 a -6 + 10 z 8 + 5 a z 7 -11 z 7 a -1 -30 z 7 a -3 -6 z 7 a -5 + 8 z 7 a -7 + a 2 z 6 -57 z 6 a -2 -54 z 6 a -4 -16 z 6 a -6 + 4 z 6 a -8 -22 z 6 -9 a z 5 -8 z 5 a -1 + 6 z 5 a -3 -6 z 5 a -5 -10 z 5 a -7 + z 5 a -9 - a 2 z 4 + 46 z 4 a -2 + 48 z 4 a -4 + 11 z 4 a -6 -5 z 4 a -8 + 13 z 4 + 3 a z 3 + 8 z 3 a -1 + 5 z 3 a -3 + 5 z 3 a -5 + 4 z 3 a -7 - z 3 a -9 -11 z 2 a -2 -14 z 2 a -4 -3 z 2 a -6 + 2 z 2 a -8 -2 z 2 + 2 z a -3 + 2 z a -5 -2 a -2 -3 a -4 -2 a -6 -2 a -3 z -1 -2 a -5 z -1 + a -2 z -2 + 2 a -4 z -2 + a -6 z -2$ (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 =

2 is the signature of L11a505. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 1$	$i = 3$
$r = -4$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -1$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 0$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{11}$
$r = 1$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = 2$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{15}$
$r = 3$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 4$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{11}$
$r = 5$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 6$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 7$	${\mathbb Z}_2$	${\mathbb Z}$