L11a368

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 4 + v 2 u 4 - v 4 u 3 + 3 v 3 u 3 -4 v 2 u 3 + v u 3 + v 4 u 2 -4 v 3 u 2 + 5 v 2 u 2 -4 v u 2 + u 2 + v 3 u -4 v 2 u + 3 v u - u + v 2 - v$ (db)
Jones polynomial	$q^{3/2}-3 \sqrt{q}+\frac{5}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{12}{q^{7/2}}+\frac{11}{q^{9/2}}-\frac{10}{q^{11/2}}+\frac{7}{q^{13/2}}-\frac{4}{q^{15/2}}+\frac{2}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z 5 a 7 + 4 z 3 a 7 + 4 z a 7 + a 7 z -1 - z 7 a 5 -5 z 5 a 5 -9 z 3 a 5 -7 z a 5 - a 5 z -1 - z 7 a 3 -4 z 5 a 3 -4 z 3 a 3 - z a 3 + z 5 a + 3 z 3 a + z a$ (db)
Kauffman polynomial	$- z 5 a 11 + 3 z 3 a 11 -2 z a 11 -2 z 6 a 10 + 5 z 4 a 10 -3 z 2 a 10 -2 z 7 a 9 + 2 z 5 a 9 + 2 z 3 a 9 - z a 9 -2 z 8 a 8 + 2 z 6 a 8 - z 4 a 8 + z 2 a 8 -2 z 9 a 7 + 5 z 7 a 7 -13 z 5 a 7 + 15 z 3 a 7 -7 z a 7 + a 7 z -1 - z 10 a 6 + 3 z 6 a 6 -10 z 4 a 6 + 7 z 2 a 6 - a 6 -5 z 9 a 5 + 18 z 7 a 5 -31 z 5 a 5 + 24 z 3 a 5 -9 z a 5 + a 5 z -1 - z 10 a 4 -2 z 8 a 4 + 13 z 6 a 4 -17 z 4 a 4 + 6 z 2 a 4 -3 z 9 a 3 + 8 z 7 a 3 -5 z 5 a 3 + z 3 a 3 -4 z 8 a 2 + 13 z 6 a 2 -10 z 4 a 2 + 2 z 2 a 2 -3 z 7 a + 10 z 5 a -7 z 3 a + z a - z 6 + 3 z 4 - z 2$ (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 L11a368. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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