L11n9

(Knotscape image)

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

Visit L11n9's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_9,14,10,15 X₈₄₉₃ X_5,11,6,10 X_11,20,12,21 X_19,22,20,5 X_13,19,14,18 X_21,12,22,13 X_2,16,3,15
Gauss code	{1, -11, 5, -3}, {-6, -1, 2, -5, -4, 6, -7, 10, -9, 4, 11, -2, 3, 9, -8, 7, -10, 8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 + u 3 + 2 v u 2 -2 u 2 -2 v u + 2 u + v -1$ (db)
Jones polynomial	$-q^{5/2}+2 q^{3/2}-3 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}}+\frac{1}{q^{13/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- a 7 z -1 + 2 z a 5 + 3 a 5 z -1 -2 z 3 a 3 -5 z a 3 -3 a 3 z -1 + z 5 a + 4 z 3 a + 5 z a + 2 a z -1 - z 3 a -1 -2 z a -1 - a -1 z -1$ (db)
Kauffman polynomial	$- a 3 z 9 - a z 9 -2 a 4 z 8 -4 a 2 z 8 -2 z 8 - a 5 z 7 + 2 a 3 z 7 + 2 a z 7 - z 7 a -1 + 10 a 4 z 6 + 20 a 2 z 6 + 10 z 6 + 4 a 5 z 5 + 8 a 3 z 5 + 9 a z 5 + 5 z 5 a -1 - a 6 z 4 -15 a 4 z 4 -27 a 2 z 4 -13 z 4 - a 7 z 3 -6 a 5 z 3 -18 a 3 z 3 -20 a z 3 -7 z 3 a -1 - a 8 z 2 + 8 a 4 z 2 + 12 a 2 z 2 + 5 z 2 + a 7 z + 6 a 5 z + 12 a 3 z + 11 a z + 4 z a -1 + a 8 + 2 a 6 -2 a 2 - a 7 z -1 -3 a 5 z -1 -3 a 3 z -1 -2 a z -1 - a -1 z -1$ (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 =

-3 is the signature of L11n9. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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