L11a330

(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_6,9,7,10 X_18,12,19,11 X_16,8,17,7 X_4,14,5,13 X_22,16,9,15 X_8,18,1,17 X_2,19,3,20 X_14,22,15,21
Gauss code	{1, -10, 2, -7, 3, -4, 6, -9}, {4, -1, 5, -2, 7, -11, 8, -6, 9, -5, 10, -3, 11, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 5 + 2 v 2 u 5 - v u 5 + 2 v 3 u 4 -6 v 2 u 4 + 4 v u 4 -2 v 3 u 3 + 8 v 2 u 3 -7 v u 3 + u 3 + v 3 u 2 -7 v 2 u 2 + 8 v u 2 -2 u 2 + 4 v 2 u -6 v u + 2 u - v 2 + 2 v -1$ (db)
Jones polynomial	$q^{17/2}-4 q^{15/2}+9 q^{13/2}-14 q^{11/2}+19 q^{9/2}-22 q^{7/2}+21 q^{5/2}-19 q^{3/2}+13 \sqrt{q}-\frac{9}{\sqrt{q}}+\frac{4}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$- z 9 a -3 + z 7 a -1 -6 z 7 a -3 + z 7 a -5 + 4 z 5 a -1 -13 z 5 a -3 + 4 z 5 a -5 + 5 z 3 a -1 -13 z 3 a -3 + 5 z 3 a -5 + 4 z a -1 -7 z a -3 + 3 z a -5 + 2 a -1 z -1 -3 a -3 z -1 + a -5 z -1$ (db)
Kauffman polynomial	$-3 z 10 a -2 -3 z 10 a -4 -6 z 9 a -1 -15 z 9 a -3 -9 z 9 a -5 -3 z 8 a -2 -12 z 8 a -4 -13 z 8 a -6 -4 z 8 - a z 7 + 18 z 7 a -1 + 40 z 7 a -3 + 8 z 7 a -5 -13 z 7 a -7 + 31 z 6 a -2 + 45 z 6 a -4 + 18 z 6 a -6 -9 z 6 a -8 + 13 z 6 + 3 a z 5 -13 z 5 a -1 -28 z 5 a -3 + 9 z 5 a -5 + 17 z 5 a -7 -4 z 5 a -9 -33 z 4 a -2 -36 z 4 a -4 -6 z 4 a -6 + 8 z 4 a -8 - z 4 a -10 -12 z 4 -3 a z 3 + z 3 a -1 + 10 z 3 a -3 -3 z 3 a -5 -8 z 3 a -7 + z 3 a -9 + 10 z 2 a -2 + 13 z 2 a -4 + 3 z 2 a -6 -3 z 2 a -8 + 3 z 2 + a z -3 z a -1 -7 z a -3 -2 z a -5 + z a -7 -3 a -2 -3 a -4 - a -6 + 2 a -1 z -1 + 3 a -3 z -1 + a -5 z -1$ (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 =

3 is the signature of L11a330. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 2$	$i = 4$
$r = -4$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -1$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 0$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{9}$
$r = 1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 2$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 3$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 4$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 5$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 6$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 7$	${\mathbb Z}_2$	${\mathbb Z}$