L9a54

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 3 - v w u 3 + w u 3 - u 3 -2 v u 2 + 2 v w u 2 -2 w u 2 + 2 u 2 + 2 v u -2 v w u + 2 w u -2 u - v + v w - w + 1$ (db)
Jones polynomial	$- q 6 + 3 q 5 -6 q 4 + 8 q 3 -7 q 2 + 9 q -6 + 5 q -1 -2 q -2 + q -3$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 a -2 + 4 z 4 a -2 - z 4 a -4 -2 z 4 + a 2 z 2 + 7 z 2 a -2 -2 z 2 a -4 -6 z 2 + 2 a 2 + 6 a -2 -2 a -4 -6 + a 2 z -2 + a -2 z -2 -2 z -2$ (db)
Kauffman polynomial	$z 8 a -2 + z 8 + 2 a z 7 + 7 z 7 a -1 + 5 z 7 a -3 + a 2 z 6 + 11 z 6 a -2 + 8 z 6 a -4 + 4 z 6 -6 a z 5 -16 z 5 a -1 -4 z 5 a -3 + 6 z 5 a -5 -4 a 2 z 4 -35 z 4 a -2 -14 z 4 a -4 + 3 z 4 a -6 -22 z 4 + 4 a z 3 + 3 z 3 a -1 -7 z 3 a -3 -5 z 3 a -5 + z 3 a -7 + 6 a 2 z 2 + 30 z 2 a -2 + 10 z 2 a -4 + 26 z 2 + 2 a z + 6 z a -1 + 6 z a -3 + 2 z a -5 -4 a 2 -12 a -2 -4 a -4 -11-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 L9a54. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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