L9a34

(Knotscape image)

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

Visit L9a34's page at the original Knot Atlas.

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 3 + v u 3 - v 3 u 2 + 3 v 2 u 2 -2 v u 2 + u 2 + v 3 u -2 v 2 u + 3 v u - u + v 2 - v$ (db)
Jones polynomial	$-\sqrt{q}+\frac{2}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{6}{q^{9/2}}-\frac{6}{q^{11/2}}+\frac{3}{q^{13/2}}-\frac{2}{q^{15/2}}+\frac{1}{q^{17/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- z 3 a 7 -2 z a 7 + z 5 a 5 + 3 z 3 a 5 + 3 z a 5 + a 5 z -1 + z 5 a 3 + 2 z 3 a 3 - z a 3 - a 3 z -1 - z 3 a -2 z a$ (db)
Kauffman polynomial	$- z 4 a 10 + 2 z 2 a 10 -2 z 5 a 9 + 4 z 3 a 9 -2 z a 9 -2 z 6 a 8 + 2 z 4 a 8 -2 z 7 a 7 + 3 z 5 a 7 -3 z 3 a 7 - z a 7 - z 8 a 6 - z 2 a 6 -4 z 7 a 5 + 10 z 5 a 5 -12 z 3 a 5 + 6 z a 5 - a 5 z -1 - z 8 a 4 + 2 z 4 a 4 - z 2 a 4 + a 4 -2 z 7 a 3 + 4 z 5 a 3 -2 z 3 a 3 + 3 z a 3 - a 3 z -1 -2 z 6 a 2 + 5 z 4 a 2 -2 z 2 a 2 - z 5 a + 3 z 3 a -2 z a$ (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 =

-3 is the signature of L9a34. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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