L9a55

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u + 2 v w u -2 w u + 2 v x u - v w x u + w x u - x u + u + v - v w + 2 w -2 v x + v w x -2 w x + 2 x -1$ (db)
Jones polynomial	$q^{5/2}-4 q^{3/2}+6 \sqrt{q}-\frac{8}{\sqrt{q}}+\frac{7}{q^{3/2}}-\frac{10}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{1}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 7 z -1 + a 7 z -3 -3 z a 5 -5 a 5 z -1 -3 a 5 z -3 + 3 z 3 a 3 + 6 z a 3 + 7 a 3 z -1 + 3 a 3 z -3 - z 5 a -2 z 3 a -3 z a -3 a z -1 - a z -3 + z 3 a -1$ (db)
Kauffman polynomial	$- a 4 z 8 - a 2 z 8 - a 5 z 7 -5 a 3 z 7 -4 a z 7 - a 6 z 6 - a 4 z 6 -6 a 2 z 6 -6 z 6 - a 7 z 5 -3 a 5 z 5 + 4 a 3 z 5 + 2 a z 5 -4 z 5 a -1 -2 a 4 z 4 + 7 a 2 z 4 - z 4 a -2 + 8 z 4 + 4 a 7 z 3 + 12 a 5 z 3 + 8 a 3 z 3 + 4 a z 3 + 4 z 3 a -1 + 6 a 6 z 2 + 12 a 4 z 2 + 6 a 2 z 2 -6 a 7 z -14 a 5 z -14 a 3 z -6 a z -8 a 6 -15 a 4 -8 a 2 + 4 a 7 z -1 + 9 a 5 z -1 + 9 a 3 z -1 + 4 a z -1 + 3 a 6 z -2 + 6 a 4 z -2 + 3 a 2 z -2 - a 7 z -3 -3 a 5 z -3 -3 a 3 z -3 - a z -3$ (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 =

-1 is the signature of L9a55. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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