L11n440

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X₂₅₃₆ X_18,11,19,12 X_3,11,4,10 X_9,1,10,4 X_7,15,8,14 X_13,5,14,8 X_19,13,20,22 X_15,21,16,20 X_21,17,22,16 X_12,17,9,18
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, 3, -11}, {-7, 6, -9, 10, 11, -3, -8, 9, -10, 8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 + x u 3 + 2 v w u 2 - w u 2 + v x u 2 - v w x u 2 -2 x u 2 + u 2 -2 v w u + w u - v x u + v w x u + 2 x u - u + v w - w x$ (db)
Jones polynomial	$q^{15/2}-2 q^{13/2}+5 q^{11/2}-6 q^{9/2}+5 q^{7/2}-7 q^{5/2}+3 q^{3/2}-5 \sqrt{q}-\frac{1}{\sqrt{q}}-\frac{1}{q^{5/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 5 a -1 + 2 z 5 a -3 + a z 3 -8 z 3 a -1 + 10 z 3 a -3 -3 z 3 a -5 + 4 a z -17 z a -1 + 21 z a -3 -9 z a -5 + z a -7 + 5 a z -1 -17 a -1 z -1 + 21 a -3 z -1 -11 a -5 z -1 + 2 a -7 z -1 + 2 a z -3 -7 a -1 z -3 + 9 a -3 z -3 -5 a -5 z -3 + a -7 z -3$ (db)
Kauffman polynomial	$- z 8 a -4 - z 8 a -6 - a z 7 - z 7 a -1 -5 z 7 a -3 -7 z 7 a -5 -2 z 7 a -7 -5 z 6 a -2 -7 z 6 a -4 -4 z 6 a -6 - z 6 a -8 - z 6 + 7 a z 5 + 12 z 5 a -1 + 23 z 5 a -3 + 24 z 5 a -5 + 6 z 5 a -7 + 32 z 4 a -2 + 43 z 4 a -4 + 26 z 4 a -6 + 4 z 4 a -8 + 11 z 4 -15 a z 3 -37 z 3 a -1 -41 z 3 a -3 -23 z 3 a -5 -4 z 3 a -7 -73 z 2 a -2 -75 z 2 a -4 -34 z 2 a -6 -6 z 2 a -8 -26 z 2 + 16 a z + 39 z a -1 + 37 z a -3 + 16 z a -5 + 2 z a -7 + 60 a -2 + 58 a -4 + 24 a -6 + 4 a -8 + 23-9 a z -1 -23 a -1 z -1 -24 a -3 z -1 -12 a -5 z -1 -2 a -7 z -1 -19 a -2 z -2 -18 a -4 z -2 -7 a -6 z -2 - a -8 z -2 -7 z -2 + 2 a z -3 + 7 a -1 z -3 + 9 a -3 z -3 + 5 a -5 z -3 + a -7 z -3$ (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 L11n440. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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