L11n444

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X₂₅₃₆ X_11,19,12,18 X_10,3,11,4 X_4,9,1,10 X_16,7,17,8 X_8,15,5,16 X_13,20,14,21 X_19,15,20,22 X_21,12,22,13 X_17,9,18,14
Gauss code	{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, -3, 10, -8, 11}, {7, -6, -11, 3, -9, 8, -10, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 2 - v u 2 - v 2 w u 2 + v w u 2 - v 2 x u 2 + 2 v x u 2 - x u 2 - v 2 u + 2 v 2 w u -2 v w u + w u + v 2 x u -2 v x u - w x u + 2 x u - v 2 w + 2 v w - w + v x - v w x + w x - x$ (db)
Jones polynomial	$2 \sqrt{q}-\frac{5}{\sqrt{q}}+\frac{6}{q^{3/2}}-\frac{10}{q^{5/2}}+\frac{8}{q^{7/2}}-\frac{10}{q^{9/2}}+\frac{6}{q^{11/2}}-\frac{6}{q^{13/2}}+\frac{2}{q^{15/2}}-\frac{1}{q^{17/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 9 z -1 + a 9 z -3 -4 z a 7 -7 a 7 z -1 -3 a 7 z -3 + 5 z 3 a 5 + 14 z a 5 + 12 a 5 z -1 + 3 a 5 z -3 -2 z 5 a 3 -8 z 3 a 3 -13 z a 3 -7 a 3 z -1 - a 3 z -3 + 2 z 3 a + 3 z a + a z -1$ (db)
Kauffman polynomial	$- z 7 a 9 + 5 z 5 a 9 -10 z 3 a 9 + 10 z a 9 -5 a 9 z -1 + a 9 z -3 -2 z 8 a 8 + 7 z 6 a 8 -4 z 4 a 8 -7 z 2 a 8 -3 a 8 z -2 + 9 a 8 - z 9 a 7 -4 z 7 a 7 + 31 z 5 a 7 -50 z 3 a 7 + 35 z a 7 -14 a 7 z -1 + 3 a 7 z -3 -7 z 8 a 6 + 21 z 6 a 6 -4 z 4 a 6 -24 z 2 a 6 -6 a 6 z -2 + 21 a 6 - z 9 a 5 -11 z 7 a 5 + 54 z 5 a 5 -74 z 3 a 5 + 49 z a 5 -18 a 5 z -1 + 3 a 5 z -3 -5 z 8 a 4 + 9 z 6 a 4 + 11 z 4 a 4 -28 z 2 a 4 -3 a 4 z -2 + 18 a 4 -8 z 7 a 3 + 27 z 5 a 3 -39 z 3 a 3 + 30 z a 3 -11 a 3 z -1 + a 3 z -3 -5 z 6 a 2 + 11 z 4 a 2 -14 z 2 a 2 + 6 a 2 - z 5 a -5 z 3 a + 6 z a -2 a z -1 -3 z 2 + 1$ (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 L11n444. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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