L9a53

(Knotscape image)

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

Visit L9a53's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_12,7,13,8 X_4,13,1,14 X_18,10,15,9 X₈₄₉₃ X_16,5,17,6 X_14,17,5,18 X_10,16,11,15 X_2,12,3,11
Gauss code	{1, -9, 5, -3}, {8, -6, 7, -4}, {6, -1, 2, -5, 4, -8, 9, -2, 3, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 3 - v w u 3 + w u 3 - u 3 -3 v u 2 + 3 v w u 2 -3 w u 2 + 3 u 2 + 3 v u -3 v w u + 3 w u -3 u - v + v w - w + 1$ (db)
Jones polynomial	$q 4 -4 q 3 + 8 q 2 -9 q + 12-10 q -1 + 10 q -2 -6 q -3 + 3 q -4 - q -5$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 6 + 2 a 2 z 4 + z 4 a -2 -3 z 4 - a 4 z 2 + 4 a 2 z 2 + z 2 a -2 -4 z 2 - a 4 + 3 a 2 + a -2 -3 + a 2 z -2 + a -2 z -2 -2 z -2$ (db)
Kauffman polynomial	$2 a 2 z 8 + 2 z 8 + 4 a 3 z 7 + 11 a z 7 + 7 z 7 a -1 + 3 a 4 z 6 + 7 a 2 z 6 + 8 z 6 a -2 + 12 z 6 + a 5 z 5 -5 a 3 z 5 -17 a z 5 -7 z 5 a -1 + 4 z 5 a -3 -6 a 4 z 4 -22 a 2 z 4 -11 z 4 a -2 + z 4 a -4 -28 z 4 -2 a 5 z 3 - a 3 z 3 + 3 a z 3 -2 z 3 a -3 + 5 a 4 z 2 + 17 a 2 z 2 + 5 z 2 a -2 + 17 z 2 + a 5 z + 3 a 3 z + 3 a z + z a -1 -2 a 4 -6 a 2 -2 a -2 -5-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 L9a53. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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