L9a22

(Knotscape image)

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

Visit L9a22's page at the original Knot Atlas.

Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_16,12,17,11 X_12,6,13,5 X_4,17,5,18 X_14,7,15,8 X_18,13,7,14 X_6,16,1,15
Gauss code	{1, -2, 3, -6, 5, -9}, {7, -1, 2, -3, 4, -5, 8, -7, 9, -4, 6, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + v u 4 + 2 v 2 u 3 -3 v u 3 + u 3 -2 v 2 u 2 + 3 v u 2 -2 u 2 + v 2 u -3 v u + 2 u + v -1$ (db)
Jones polynomial	$q^{3/2}-3 \sqrt{q}+\frac{4}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{7}{q^{5/2}}-\frac{8}{q^{7/2}}+\frac{7}{q^{9/2}}-\frac{5}{q^{11/2}}+\frac{3}{q^{13/2}}-\frac{1}{q^{15/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- a 3 z 7 + a 5 z 5 -5 a 3 z 5 + a z 5 + 3 a 5 z 3 -8 a 3 z 3 + 3 a z 3 + 2 a 5 z -4 a 3 z + a z + a 3 z -1 - a z -1$ (db)
Kauffman polynomial	$- z 3 a 9 -3 z 4 a 8 + z 2 a 8 -5 z 5 a 7 + 4 z 3 a 7 - z a 7 -6 z 6 a 6 + 8 z 4 a 6 -3 z 2 a 6 -5 z 7 a 5 + 8 z 5 a 5 -3 z 3 a 5 + z a 5 -2 z 8 a 4 -2 z 6 a 4 + 13 z 4 a 4 -7 z 2 a 4 -8 z 7 a 3 + 24 z 5 a 3 -19 z 3 a 3 + 4 z a 3 + a 3 z -1 -2 z 8 a 2 + 3 z 6 a 2 + 5 z 4 a 2 -5 z 2 a 2 - a 2 -3 z 7 a + 11 z 5 a -11 z 3 a + 2 z a + a z -1 - z 6 + 3 z 4 -2 z 2$ (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 L9a22. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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