L9a38

(Knotscape image)

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

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Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_12,3,13,4 X_4,9,5,10 X_14,6,15,5 X_18,14,9,13 X_16,8,17,7 X_6,16,7,15 X_8,18,1,17
Gauss code	{1, -2, 3, -4, 5, -8, 7, -9}, {4, -1, 2, -3, 6, -5, 8, -7, 9, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 3 + v 2 u 3 + v 3 u 2 -3 v 2 u 2 + 2 v u 2 + 2 v 2 u -3 v u + u + v -1$ (db)
Jones polynomial	$q^{11/2}-2 q^{9/2}+3 q^{7/2}-5 q^{5/2}+5 q^{3/2}-6 \sqrt{q}+\frac{4}{\sqrt{q}}-\frac{3}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{1}{q^{7/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + a z 5 -6 z 5 a -1 + z 5 a -3 + 4 a z 3 -12 z 3 a -1 + 4 z 3 a -3 + 4 a z -8 z a -1 + 4 z a -3 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$- z 8 a -2 - z 8 -2 a z 7 -4 z 7 a -1 -2 z 7 a -3 -2 a 2 z 6 + z 6 a -2 -2 z 6 a -4 + z 6 - a 3 z 5 + 6 a z 5 + 14 z 5 a -1 + 5 z 5 a -3 -2 z 5 a -5 + 6 a 2 z 4 + 3 z 4 a -4 - z 4 a -6 + 2 z 4 + 3 a 3 z 3 -6 a z 3 -21 z 3 a -1 -8 z 3 a -3 + 4 z 3 a -5 -3 a 2 z 2 -3 z 2 a -2 - z 2 a -4 + 2 z 2 a -6 -3 z 2 - a 3 z + 4 a z + 10 z a -1 + 4 z a -3 - z a -5 + 1- a z -1 - a -1 z -1$ (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 L9a38. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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