L11a493

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 5 - v w u 5 -4 v u 4 + 4 v w u 4 -2 w u 4 + 2 u 4 + 6 v u 3 -6 v w u 3 + 5 w u 3 -5 u 3 -5 v u 2 + 5 v w u 2 -6 w u 2 + 6 u 2 + 2 v u -2 v w u + 4 w u -4 u - w + 1$ (db)
Jones polynomial	$q 10 -4 q 9 + 10 q 8 -16 q 7 + 20 q 6 -23 q 5 + 24 q 4 -18 q 3 + 15 q 2 -8 q + 4- q -1$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z 8 a -4 - z 6 a -2 + 5 z 6 a -4 -2 z 6 a -6 -3 z 4 a -2 + 10 z 4 a -4 -7 z 4 a -6 + z 4 a -8 -2 z 2 a -2 + 9 z 2 a -4 -9 z 2 a -6 + 2 z 2 a -8 + a -2 + 2 a -4 -5 a -6 + 2 a -8 + a -2 z -2 -2 a -4 z -2 + a -6 z -2$ (db)
Kauffman polynomial	$2 z 10 a -4 + 2 z 10 a -6 + 5 z 9 a -3 + 14 z 9 a -5 + 9 z 9 a -7 + 4 z 8 a -2 + 12 z 8 a -4 + 24 z 8 a -6 + 16 z 8 a -8 + z 7 a -1 -11 z 7 a -3 -26 z 7 a -5 + 2 z 7 a -7 + 16 z 7 a -9 -14 z 6 a -2 -56 z 6 a -4 -76 z 6 a -6 -24 z 6 a -8 + 10 z 6 a -10 -3 z 5 a -1 -2 z 5 a -3 -11 z 5 a -5 -39 z 5 a -7 -23 z 5 a -9 + 4 z 5 a -11 + 17 z 4 a -2 + 66 z 4 a -4 + 66 z 4 a -6 + 8 z 4 a -8 -8 z 4 a -10 + z 4 a -12 + 3 z 3 a -1 + 14 z 3 a -3 + 32 z 3 a -5 + 34 z 3 a -7 + 13 z 3 a -9 -8 z 2 a -2 -29 z 2 a -4 -25 z 2 a -6 + 4 z 2 a -10 - z a -1 -3 z a -3 -11 z a -5 -13 z a -7 -4 z a -9 + 4 a -4 + 5 a -6 + a -8 - a -10 -2 a -3 z -1 -2 a -5 z -1 + a -2 z -2 + 2 a -4 z -2 + a -6 z -2$ (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 =

4 is the signature of L11a493. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 3$	$i = 5$
$r = -3$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -1$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 0$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{7}$
$r = 1$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 2$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 3$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{14}$	${\mathbb Z}^{14}$
$r = 4$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{11}$
$r = 5$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 6$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 7$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 8$	${\mathbb Z}_2$	${\mathbb Z}$