L11a497

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 5 - v w u 5 -3 v u 4 + 3 v w u 4 -2 w u 4 + 2 u 4 + 4 v u 3 -4 v w u 3 + 4 w u 3 -4 u 3 -4 v u 2 + 4 v w u 2 -4 w u 2 + 4 u 2 + 2 v u -2 v w u + 3 w u -3 u - w + 1$ (db)
Jones polynomial	$- q 5 + 3 q 4 -7 q 3 + 11 q 2 -15 q + 19-17 q -1 + 16 q -2 -11 q -3 + 8 q -4 -3 q -5 + q -6$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z 8 -2 a 2 z 6 - z 6 a -2 + 6 z 6 + a 4 z 4 -9 a 2 z 4 -4 z 4 a -2 + 15 z 4 + 3 a 4 z 2 -16 a 2 z 2 -6 z 2 a -2 + 19 z 2 + 4 a 4 -14 a 2 -4 a -2 + 14 + 2 a 4 z -2 -5 a 2 z -2 - a -2 z -2 + 4 z -2$ (db)
Kauffman polynomial	$a 2 z 10 + z 10 + 4 a 3 z 9 + 8 a z 9 + 4 z 9 a -1 + 5 a 4 z 8 + 13 a 2 z 8 + 6 z 8 a -2 + 14 z 8 + 3 a 5 z 7 -3 a 3 z 7 -10 a z 7 + z 7 a -1 + 5 z 7 a -3 + a 6 z 6 -12 a 4 z 6 -49 a 2 z 6 -11 z 6 a -2 + 3 z 6 a -4 -50 z 6 -7 a 5 z 5 -14 a 3 z 5 -19 a z 5 -21 z 5 a -1 -8 z 5 a -3 + z 5 a -5 -3 a 6 z 4 + 10 a 4 z 4 + 68 a 2 z 4 + 12 z 4 a -2 -5 z 4 a -4 + 72 z 4 + 3 a 5 z 3 + 22 a 3 z 3 + 50 a z 3 + 37 z 3 a -1 + 4 z 3 a -3 -2 z 3 a -5 + 3 a 6 z 2 -13 a 4 z 2 -50 a 2 z 2 -11 z 2 a -2 + z 2 a -4 -46 z 2 -17 a 3 z -35 a z -23 z a -1 -4 z a -3 + z a -5 - a 6 + 8 a 4 + 22 a 2 + 5 a -2 + 19 + 5 a 3 z -1 + 9 a z -1 + 5 a -1 z -1 + a -3 z -1 -2 a 4 z -2 -5 a 2 z -2 - a -2 z -2 -4 z -2$ (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 L11a497. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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