L11n18

(Knotscape image)

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

Visit L11n18's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_5,12,6,13 X₈₄₉₃ X_9,16,10,17 X_11,20,12,21 X_17,22,18,5 X_21,18,22,19 X_19,10,20,11 X_2,14,3,13
Gauss code	{1, -11, 5, -3}, {-4, -1, 2, -5, -6, 10, -7, 4, 11, -2, 3, 6, -8, 9, -10, 7, -9, 8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 3 + 2 u 3 + 5 v u 2 -5 u 2 -5 v u + 5 u + 2 v -2$ (db)
Jones polynomial	$-\frac{2}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{8}{q^{9/2}}-\frac{10}{q^{11/2}}+\frac{9}{q^{13/2}}-\frac{7}{q^{15/2}}+\frac{6}{q^{17/2}}-\frac{3}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- z 3 a 9 - z a 9 - a 9 z -1 + z 5 a 7 + 2 z 3 a 7 + 2 z a 7 + a 7 z -1 + z 5 a 5 + 2 z 3 a 5 + 3 z a 5 + 2 a 5 z -1 -2 z 3 a 3 -4 z a 3 -2 a 3 z -1$ (db)
Kauffman polynomial	$- z 6 a 12 + 3 z 4 a 12 -3 z 2 a 12 + a 12 -3 z 7 a 11 + 9 z 5 a 11 -6 z 3 a 11 + z a 11 -3 z 8 a 10 + 6 z 6 a 10 + 3 z 4 a 10 -5 z 2 a 10 - z 9 a 9 -4 z 7 a 9 + 17 z 5 a 9 -12 z 3 a 9 + 2 z a 9 + a 9 z -1 -5 z 8 a 8 + 10 z 6 a 8 -3 z 4 a 8 + 2 z 2 a 8 -3 a 8 - z 9 a 7 -3 z 7 a 7 + 11 z 5 a 7 -11 z 3 a 7 + 2 z a 7 + a 7 z -1 -2 z 8 a 6 + 2 z 6 a 6 -3 z 4 a 6 + z 2 a 6 -2 z 7 a 5 + 3 z 5 a 5 -8 z 3 a 5 + 6 z a 5 -2 a 5 z -1 - z 6 a 4 -3 z 2 a 4 + 3 a 4 -3 z 3 a 3 + 5 z a 3 -2 a 3 z -1$ (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 =

-3 is the signature of L11n18. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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