L11a357

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + v u 4 -2 v 3 u 3 + 5 v 2 u 3 -3 v u 3 + u 3 - v 4 u 2 + 5 v 3 u 2 -7 v 2 u 2 + 5 v u 2 - u 2 + v 4 u -3 v 3 u + 5 v 2 u -2 v u + v 3 - v 2$ (db)
Jones polynomial	$-\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{13}{q^{7/2}}+\frac{15}{q^{9/2}}-\frac{14}{q^{11/2}}+\frac{11}{q^{13/2}}-\frac{9}{q^{15/2}}+\frac{4}{q^{17/2}}-\frac{2}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- z 3 a 9 -2 z a 9 + z 5 a 7 + 2 z 3 a 7 + 2 z a 7 + a 7 z -1 + 2 z 5 a 5 + 4 z 3 a 5 + z a 5 - a 5 z -1 + z 5 a 3 -3 z a 3 - z 3 a - z a$ (db)
Kauffman polynomial	$- z 6 a 12 + 4 z 4 a 12 -4 z 2 a 12 -2 z 7 a 11 + 7 z 5 a 11 -8 z 3 a 11 + 4 z a 11 -2 z 8 a 10 + 3 z 6 a 10 + 3 z 4 a 10 -4 z 2 a 10 -2 z 9 a 9 + 3 z 7 a 9 - z 5 a 9 -2 z 3 a 9 + 6 z a 9 - z 10 a 8 - z 8 a 8 + z 6 a 8 + 4 z 4 a 8 -4 z 2 a 8 -5 z 9 a 7 + 11 z 7 a 7 -15 z 5 a 7 + 11 z 3 a 7 -4 z a 7 + a 7 z -1 - z 10 a 6 -4 z 8 a 6 + 8 z 6 a 6 -5 z 4 a 6 - z 2 a 6 - a 6 -3 z 9 a 5 + z 7 a 5 + 3 z 5 a 5 -2 z 3 a 5 -2 z a 5 + a 5 z -1 -5 z 8 a 4 + 8 z 6 a 4 -5 z 4 a 4 + 2 z 2 a 4 -5 z 7 a 3 + 9 z 5 a 3 -5 z 3 a 3 + 3 z a 3 -3 z 6 a 2 + 5 z 4 a 2 - z 2 a 2 - z 5 a + 2 z 3 a - z a$ (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 L11a357. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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