L9a5

(Knotscape image)

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

Visit L9a5's page at the original Knot Atlas.

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 + 3 u 3 + 4 v u 2 -5 u 2 -5 v u + 4 u + 3 v -1$ (db)
Jones polynomial	$q^{11/2}-3 q^{9/2}+6 q^{7/2}-8 q^{5/2}+9 q^{3/2}-9 \sqrt{q}+\frac{7}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{1}{q^{7/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z 5 a -1 -2 a z 3 + z 3 a -1 -2 z 3 a -3 + a 3 z -2 a z - z a -1 - z a -3 + z a -5 + a 3 z -1 -2 a -1 z -1 + a -3 z -1$ (db)
Kauffman polynomial	$- z 8 a -2 - z 8 -3 a z 7 -7 z 7 a -1 -4 z 7 a -3 -2 a 2 z 6 -10 z 6 a -2 -5 z 6 a -4 -7 z 6 - a 3 z 5 + 5 a z 5 + 11 z 5 a -1 + 2 z 5 a -3 -3 z 5 a -5 + 3 a 2 z 4 + 24 z 4 a -2 + 7 z 4 a -4 - z 4 a -6 + 19 z 4 + 3 a 3 z 3 -3 a z 3 -7 z 3 a -1 + 2 z 3 a -3 + 3 z 3 a -5 -19 z 2 a -2 -5 z 2 a -4 + z 2 a -6 -13 z 2 -3 a 3 z + 2 a z + 6 z a -1 - z a -5 - a 2 + 5 a -2 + 2 a -4 + 3 + a 3 z -1 -2 a -1 z -1 - a -3 z -1$ (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 L9a5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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