L11a9

(Knotscape image)

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

Visit L11a9's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_12,6,13,5 X₈₄₉₃ X_22,14,5,13 X_14,22,15,21 X_18,12,19,11 X_20,10,21,9 X_10,20,11,19 X_2,16,3,15
Gauss code	{1, -11, 5, -3}, {4, -1, 2, -5, 9, -10, 8, -4, 6, -7, 11, -2, 3, -8, 10, -9, 7, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{(u-1) (v-1) \left(4 v^2-7 v+4\right)}{\sqrt{u} v^{3/2}}$ (db)
Jones polynomial	$-16 q^{9/2}+18 q^{7/2}-20 q^{5/2}+\frac{1}{q^{5/2}}+17 q^{3/2}-\frac{4}{q^{3/2}}-q^{17/2}+3 q^{15/2}-6 q^{13/2}+12 q^{11/2}-14 \sqrt{q}+\frac{8}{\sqrt{q}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 3 a -7 - z a -7 + z 5 a -5 + z 3 a -5 + 2 z a -5 + a -5 z -1 + 2 z 5 a -3 + 2 z 3 a -3 -2 z a -3 -3 a -3 z -1 + z 5 a -1 - a z 3 + z a -1 + 2 a -1 z -1$ (db)
Kauffman polynomial	$z 7 a -9 -4 z 5 a -9 + 4 z 3 a -9 + 3 z 8 a -8 -12 z 6 a -8 + 16 z 4 a -8 -8 z 2 a -8 + 4 z 9 a -7 -13 z 7 a -7 + 13 z 5 a -7 -6 z 3 a -7 + 2 z a -7 + 2 z 10 a -6 + 3 z 8 a -6 -26 z 6 a -6 + 34 z 4 a -6 -17 z 2 a -6 + a -6 + 11 z 9 a -5 -29 z 7 a -5 + 20 z 5 a -5 -7 z 3 a -5 + 4 z a -5 - a -5 z -1 + 2 z 10 a -4 + 12 z 8 a -4 -42 z 6 a -4 + 36 z 4 a -4 -13 z 2 a -4 + 3 a -4 + 7 z 9 a -3 -3 z 7 a -3 -18 z 5 a -3 + 13 z 3 a -3 + 3 z a -3 -3 a -3 z -1 + 12 z 8 a -2 -20 z 6 a -2 + a 2 z 4 + 11 z 4 a -2 -4 z 2 a -2 + 3 a -2 + 12 z 7 a -1 + 4 a z 5 -17 z 5 a -1 -2 a z 3 + 8 z 3 a -1 + z a -1 -2 a -1 z -1 + 8 z 6 -6 z 4$ (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 L11a9. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -3$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -1$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 0$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{7}$
$r = 1$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 3$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 4$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 5$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 6$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 7$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 8$	${\mathbb Z}_2$	${\mathbb Z}$