L11a376

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 4 + v 2 u 4 - v 4 u 3 + 3 v 3 u 3 -5 v 2 u 3 + 2 v u 3 + v 4 u 2 -5 v 3 u 2 + 7 v 2 u 2 -5 v u 2 + u 2 + 2 v 3 u -5 v 2 u + 3 v u - u + v 2 - v$ (db)
Jones polynomial	$q^{3/2}-3 \sqrt{q}+\frac{6}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{12}{q^{5/2}}-\frac{15}{q^{7/2}}+\frac{14}{q^{9/2}}-\frac{12}{q^{11/2}}+\frac{9}{q^{13/2}}-\frac{5}{q^{15/2}}+\frac{2}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z 5 a 7 + 4 z 3 a 7 + 5 z a 7 + a 7 z -1 - z 7 a 5 -5 z 5 a 5 -10 z 3 a 5 -8 z a 5 - a 5 z -1 - z 7 a 3 -4 z 5 a 3 -5 z 3 a 3 -2 z a 3 + z 5 a + 3 z 3 a + 2 z a$ (db)
Kauffman polynomial	$- z 5 a 11 + 3 z 3 a 11 -2 z a 11 -2 z 6 a 10 + 4 z 4 a 10 - z 2 a 10 -3 z 7 a 9 + 5 z 5 a 9 -2 z 3 a 9 + z a 9 -4 z 8 a 8 + 10 z 6 a 8 -15 z 4 a 8 + 9 z 2 a 8 -3 z 9 a 7 + 5 z 7 a 7 -6 z 5 a 7 + 3 z 3 a 7 -4 z a 7 + a 7 z -1 - z 10 a 6 -5 z 8 a 6 + 19 z 6 a 6 -29 z 4 a 6 + 14 z 2 a 6 - a 6 -6 z 9 a 5 + 15 z 7 a 5 -18 z 5 a 5 + 13 z 3 a 5 -8 z a 5 + a 5 z -1 - z 10 a 4 -5 z 8 a 4 + 18 z 6 a 4 -18 z 4 a 4 + 7 z 2 a 4 -3 z 9 a 3 + 4 z 7 a 3 + 3 z 5 a 3 -2 z 3 a 3 + z a 3 -4 z 8 a 2 + 10 z 6 a 2 -5 z 4 a 2 + z 2 a 2 -3 z 7 a + 9 z 5 a -7 z 3 a + 2 z a - z 6 + 3 z 4 -2 z 2$ (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 L11a376. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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