L9a1

(Knotscape image)

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

Visit L9a1's page at the original Knot Atlas.

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 3 + 2 u 3 + 5 v u 2 -5 u 2 -5 v u + 5 u + 2 v -2$ (db)
Jones polynomial	$-q^{13/2}+3 q^{11/2}-5 q^{9/2}+8 q^{7/2}-10 q^{5/2}+9 q^{3/2}-9 \sqrt{q}+\frac{6}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{1}{q^{5/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z 5 a -1 + z 5 a -3 - a z 3 + z 3 a -1 + 2 z 3 a -3 - z 3 a -5 - z a -1 + 2 z a -3 - z a -5 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$-2 z 8 a -2 -2 z 8 a -4 -5 z 7 a -1 -9 z 7 a -3 -4 z 7 a -5 -5 z 6 a -2 -2 z 6 a -4 -3 z 6 a -6 -6 z 6 -4 a z 5 + 4 z 5 a -1 + 18 z 5 a -3 + 9 z 5 a -5 - z 5 a -7 - a 2 z 4 + 11 z 4 a -2 + 10 z 4 a -4 + 7 z 4 a -6 + 7 z 4 + 4 a z 3 -13 z 3 a -3 -7 z 3 a -5 + 2 z 3 a -7 -4 z 2 a -2 -7 z 2 a -4 -4 z 2 a -6 - z 2 + 2 z a -1 + 4 z a -3 + 2 z a -5 + 1- a z -1 - a -1 z -1$ (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 L9a1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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