L11a240

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_12,3,13,4 X_22,10,7,9 X_10,14,11,13 X_18,5,19,6 X_16,22,17,21 X_20,16,21,15 X_14,20,15,19 X₂₇₃₈ X_4,11,5,12 X_6,17,1,18
Gauss code	{1, -9, 2, -10, 5, -11}, {9, -1, 3, -4, 10, -2, 4, -8, 7, -6, 11, -5, 8, -7, 6, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + 3 v u 4 -3 u 4 + 3 v 2 u 3 -7 v u 3 + 4 u 3 -4 v 2 u 2 + 7 v u 2 -4 u 2 + 4 v 2 u -7 v u + 3 u -3 v 2 + 3 v -1$ (db)
Jones polynomial	$q^{11/2}-4 q^{9/2}+8 q^{7/2}-13 q^{5/2}+17 q^{3/2}-18 \sqrt{q}+\frac{18}{\sqrt{q}}-\frac{15}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + 3 a z 5 -4 z 5 a -1 + z 5 a -3 -3 a 3 z 3 + 9 a z 3 -7 z 3 a -1 + 2 z 3 a -3 + a 5 z -7 a 3 z + 7 a z -5 z a -1 + z a -3 + 2 a 5 z -1 -3 a 3 z -1 + a z -1$ (db)
Kauffman polynomial	$- a 2 z 10 - z 10 -3 a 3 z 9 -8 a z 9 -5 z 9 a -1 -2 a 4 z 8 -8 a 2 z 8 -10 z 8 a -2 -16 z 8 - a 5 z 7 + 8 a 3 z 7 + 14 a z 7 -6 z 7 a -1 -11 z 7 a -3 + 6 a 4 z 6 + 34 a 2 z 6 + 12 z 6 a -2 -8 z 6 a -4 + 48 z 6 + 5 a 5 z 5 -4 a 3 z 5 + 5 a z 5 + 32 z 5 a -1 + 14 z 5 a -3 -4 z 5 a -5 -3 a 4 z 4 -35 a 2 z 4 - z 4 a -2 + 7 z 4 a -4 - z 4 a -6 -41 z 4 -9 a 5 z 3 -3 a 3 z 3 -9 a z 3 -23 z 3 a -1 -6 z 3 a -3 + 2 z 3 a -5 -4 a 4 z 2 + 8 a 2 z 2 - z 2 a -2 -2 z 2 a -4 + 13 z 2 + 7 a 5 z + 5 a 3 z + 3 z a -1 + z a -3 + 3 a 4 + 3 a 2 + 1-2 a 5 z -1 -3 a 3 z -1 - a z -1$ (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 L11a240. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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