L11a333

(Knotscape image)

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

Planar diagram presentation	X_10,1,11,2 X_12,4,13,3 X_20,5,21,6 X_16,7,17,8 X_8,9,1,10 X_18,12,19,11 X_6,15,7,16 X_4,14,5,13 X_22,18,9,17 X_2,19,3,20 X_14,22,15,21
Gauss code	{1, -10, 2, -8, 3, -7, 4, -5}, {5, -1, 6, -2, 8, -11, 7, -4, 9, -6, 10, -3, 11, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{(t(1) t(2)+1) \left(t(1)^2 t(2)^4-t(1) t(2)^4-3 t(1)^2 t(2)^3+5 t(1) t(2)^3-t(2)^3+3 t(1)^2 t(2)^2-7 t(1) t(2)^2+3 t(2)^2-t(1)^2 t(2)+5 t(1) t(2)-3 t(2)-t(1)+1\right)}{t(1)^{3/2} t(2)^{5/2}}$ (db)
Jones polynomial	$q^{9/2}-4 q^{7/2}+9 q^{5/2}-14 q^{3/2}+19 \sqrt{q}-\frac{23}{\sqrt{q}}+\frac{22}{q^{3/2}}-\frac{20}{q^{5/2}}+\frac{14}{q^{7/2}}-\frac{9}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 3 z 7 + 4 a 3 z 5 + 5 a 3 z 3 + 3 a 3 z + 2 a 3 z -1 - a z 9 -6 a z 7 + z 7 a -1 -13 a z 5 + 4 z 5 a -1 -13 a z 3 + 5 z 3 a -1 -8 a z + 3 z a -1 -3 a z -1 + a -1 z -1$ (db)
Kauffman polynomial	$-3 a 2 z 10 -3 z 10 -8 a 3 z 9 -15 a z 9 -7 z 9 a -1 -10 a 4 z 8 -9 a 2 z 8 -7 z 8 a -2 -6 z 8 -8 a 5 z 7 + 11 a 3 z 7 + 37 a z 7 + 14 z 7 a -1 -4 z 7 a -3 -4 a 6 z 6 + 18 a 4 z 6 + 32 a 2 z 6 + 16 z 6 a -2 - z 6 a -4 + 27 z 6 - a 7 z 5 + 13 a 5 z 5 -5 a 3 z 5 -37 a z 5 -9 z 5 a -1 + 9 z 5 a -3 + 5 a 6 z 4 -13 a 4 z 4 -34 a 2 z 4 -9 z 4 a -2 + 2 z 4 a -4 -27 z 4 + a 7 z 3 -6 a 5 z 3 + a 3 z 3 + 19 a z 3 + 7 z 3 a -1 -4 z 3 a -3 + 4 a 4 z 2 + 13 a 2 z 2 + 3 z 2 a -2 - z 2 a -4 + 13 z 2 + 2 a 5 z -4 a 3 z -9 a z -3 z a -1 -3 a 2 - a -2 -3 + 2 a 3 z -1 + 3 a z -1 + a -1 z -1$ (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 =

-1 is the signature of L11a333. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -6$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{4}$
$r = -3$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -2$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = -1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 0$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{12}$
$r = 1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 2$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$