L11a499

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_16,8,17,7 X_22,17,19,18 X_20,12,21,11 X_10,20,11,19 X_18,21,5,22 X_14,10,15,9 X_8,16,9,15 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {6, -5, 7, -4}, {10, -1, 3, -9, 8, -6, 5, -2, 11, -8, 9, -3, 4, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$2 v u 3 -2 v w u 3 + 2 w u 3 -2 u 3 -5 v u 2 + 5 v w u 2 -5 w u 2 + 5 u 2 + 5 v u -5 v w u + 5 w u -5 u -2 v + 2 v w -2 w + 2$ (db)
Jones polynomial	$- q 8 + 3 q 7 -7 q 6 + 12 q 5 -16 q 4 + 19 q 3 -17 q 2 + 16 q -11 + 7 q -1 -2 q -2 + q -3$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 a -2 + z 6 a -4 + z 4 a -2 + 3 z 4 a -4 - z 4 a -6 -2 z 4 + a 2 z 2 - z 2 a -2 + 6 z 2 a -4 -2 z 2 a -6 -4 z 2 + 2 a 2 + 4 a -4 -2 a -6 -4 + a 2 z -2 + a -2 z -2 -2 z -2$ (db)
Kauffman polynomial	$z 10 a -2 + z 10 a -4 + 3 z 9 a -1 + 8 z 9 a -3 + 5 z 9 a -5 + 9 z 8 a -2 + 14 z 8 a -4 + 8 z 8 a -6 + 3 z 8 + 2 a z 7 -2 z 7 a -1 -14 z 7 a -3 -4 z 7 a -5 + 6 z 7 a -7 + a 2 z 6 -26 z 6 a -2 -46 z 6 a -4 -21 z 6 a -6 + 3 z 6 a -8 -3 z 6 -4 a z 5 + 9 z 5 a -3 -9 z 5 a -5 -13 z 5 a -7 + z 5 a -9 -4 a 2 z 4 + 27 z 4 a -2 + 62 z 4 a -4 + 27 z 4 a -6 -5 z 4 a -8 -7 z 4 -11 z 3 a -1 -3 z 3 a -3 + 21 z 3 a -5 + 11 z 3 a -7 -2 z 3 a -9 + 6 a 2 z 2 -16 z 2 a -2 -36 z 2 a -4 -14 z 2 a -6 + 12 z 2 + 4 a z + 8 z a -1 -8 z a -5 -4 z a -7 -4 a 2 + 8 a -4 + 4 a -6 -7-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 =

2 is the signature of L11a499. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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