L9a27

(Knotscape image)

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

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Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_18,10,7,9 X₂₇₃₈ X_16,13,17,14 X_6,12,1,11 X_4,16,5,15 X_14,6,15,5 X_12,17,13,18
Gauss code	{1, -4, 2, -7, 8, -6}, {4, -1, 3, -2, 6, -9, 5, -8, 7, -5, 9, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 2 + 3 v u 2 -2 u 2 + 3 v 2 u -7 v u + 3 u -2 v 2 + 3 v -1$ (db)
Jones polynomial	$q^{11/2}-3 q^{9/2}+5 q^{7/2}-8 q^{5/2}+8 q^{3/2}-9 \sqrt{q}+\frac{7}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{1}{q^{7/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z 5 a -1 -2 a z 3 + 2 z 3 a -1 -2 z 3 a -3 + a 3 z -2 a z + 3 z a -1 -2 z a -3 + z a -5 + a -1 z -1 - a -3 z -1$ (db)
Kauffman polynomial	$- z 8 a -2 - z 8 -3 a z 7 -6 z 7 a -1 -3 z 7 a -3 -3 a 2 z 6 -7 z 6 a -2 -4 z 6 a -4 -6 z 6 - a 3 z 5 + 4 a z 5 + 7 z 5 a -1 - z 5 a -3 -3 z 5 a -5 + 7 a 2 z 4 + 13 z 4 a -2 + 4 z 4 a -4 - z 4 a -6 + 15 z 4 + 2 a 3 z 3 + a z 3 + z 3 a -1 + 6 z 3 a -3 + 4 z 3 a -5 -4 a 2 z 2 -5 z 2 a -2 - z 2 a -4 + z 2 a -6 -7 z 2 - a 3 z - a z -2 z a -1 -4 z a -3 -2 z a -5 - a -2 + a -1 z -1 + a -3 z -1$ (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 L9a27. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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