L11a492

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_22,16,19,15 X_20,9,21,10 X_8,19,9,20 X_4,17,1,18 X_12,6,13,5 X_10,4,11,3 X_18,12,5,11 X_16,22,17,21 X_2,14,3,13
Gauss code	{1, -11, 8, -6}, {5, -4, 10, -3}, {7, -1, 2, -5, 4, -8, 9, -7, 11, -2, 3, -10, 6, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 5 - v w u 5 + w u 5 - u 5 -3 v u 4 + 3 v w u 4 -3 w u 4 + 3 u 4 + 5 v u 3 -5 v w u 3 + 5 w u 3 -5 u 3 -5 v u 2 + 5 v w u 2 -5 w u 2 + 5 u 2 + 3 v u -3 v w u + 3 w u -3 u - v + v w - w + 1$ (db)
Jones polynomial	$q 6 -4 q 5 + 9 q 4 -15 q 3 + 20 q 2 -22 q + 25-19 q -1 + 15 q -2 -9 q -3 + 4 q -4 - q -5$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z 8 - a 2 z 6 -2 z 6 a -2 + 5 z 6 -3 a 2 z 4 -7 z 4 a -2 + z 4 a -4 + 10 z 4 -3 a 2 z 2 -8 z 2 a -2 + 2 z 2 a -4 + 9 z 2 - a 2 -3 a -2 + a -4 + 3 + a 2 z -2 + a -2 z -2 -2 z -2$ (db)
Kauffman polynomial	$2 z 10 a -2 + 2 z 10 + 8 a z 9 + 14 z 9 a -1 + 6 z 9 a -3 + 11 a 2 z 8 + 16 z 8 a -2 + 7 z 8 a -4 + 20 z 8 + 8 a 3 z 7 -8 a z 7 -24 z 7 a -1 -4 z 7 a -3 + 4 z 7 a -5 + 4 a 4 z 6 -23 a 2 z 6 -55 z 6 a -2 -15 z 6 a -4 + z 6 a -6 -66 z 6 + a 5 z 5 -12 a 3 z 5 -5 a z 5 -17 z 5 a -3 -9 z 5 a -5 -5 a 4 z 4 + 24 a 2 z 4 + 60 z 4 a -2 + 9 z 4 a -4 -2 z 4 a -6 + 78 z 4 - a 5 z 3 + 4 a 3 z 3 + 13 a z 3 + 21 z 3 a -1 + 19 z 3 a -3 + 6 z 3 a -5 -13 a 2 z 2 -30 z 2 a -2 -6 z 2 a -4 + z 2 a -6 -36 z 2 - a 3 z -5 a z -9 z a -1 -7 z a -3 -2 z a -5 + 2 a 2 + 6 a -2 + 2 a -4 + 7-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 L11a492. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -1$	$i = 1$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -1$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 0$	${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{14}$
$r = 1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 2$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$