L11n33

(Knotscape image)

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

Visit L11n33's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_20,7,21,8 X_4,21,1,22 X_9,14,10,15 X₈₄₉₃ X_5,13,6,12 X_13,5,14,22 X_15,18,16,19 X_11,17,12,16 X_17,11,18,10 X_2,20,3,19
Gauss code	{1, -11, 5, -3}, {-6, -1, 2, -5, -4, 10, -9, 6, -7, 4, -8, 9, -10, 8, 11, -2, 3, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u + u + v -1$ (db)
Jones polynomial	$-q^{11/2}+2 q^{9/2}-3 q^{7/2}+3 q^{5/2}-2 q^{3/2}+\sqrt{q}-\frac{1}{\sqrt{q}}-\frac{1}{q^{3/2}}+\frac{1}{q^{5/2}}-\frac{2}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 5 - z 5 a -1 - a 3 z 3 + 5 a z 3 -5 z 3 a -1 + 2 z 3 a -3 -2 a 3 z + 6 a z -8 z a -1 + 5 z a -3 - z a -5 + 2 a z -1 -4 a -1 z -1 + 3 a -3 z -1 - a -5 z -1$ (db)
Kauffman polynomial	$- z 9 a -1 - z 9 a -3 - a 2 z 8 -4 z 8 a -2 -2 z 8 a -4 -3 z 8 -2 a 3 z 7 -3 a z 7 + 2 z 7 a -1 + 2 z 7 a -3 - z 7 a -5 - a 4 z 6 + 5 a 2 z 6 + 23 z 6 a -2 + 10 z 6 a -4 + 19 z 6 + 10 a 3 z 5 + 21 a z 5 + 17 z 5 a -1 + 11 z 5 a -3 + 5 z 5 a -5 + 4 a 4 z 4 -3 a 2 z 4 -34 z 4 a -2 -13 z 4 a -4 -28 z 4 -11 a 3 z 3 -34 a z 3 -40 z 3 a -1 -24 z 3 a -3 -7 z 3 a -5 -2 a 4 z 2 + 17 z 2 a -2 + 7 z 2 a -4 + 12 z 2 + 5 a 3 z + 17 a z + 24 z a -1 + 16 z a -3 + 4 z a -5 - a 2 -3 a -2 - a -4 -2-2 a z -1 -4 a -1 z -1 -3 a -3 z -1 - a -5 z -1$ (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 =

-1 is the signature of L11n33. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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