L11a378

(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_4,11,5,12 X_16,8,17,7 X_18,6,19,5 X_22,16,11,15 X_6,18,7,17 X_8,22,9,21 X_20,10,21,9 X_10,20,1,19
Gauss code	{1, -2, 3, -4, 6, -8, 5, -9, 10, -11}, {4, -1, 2, -3, 7, -5, 8, -6, 11, -10, 9, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{u^4 v^2-u^4 v+2 u^3 v^3-4 u^3 v^2+3 u^3 v-u^3+u^2 v^4-4 u^2 v^3+5 u^2 v^2-4 u^2 v+u^2-u v^4+3 u v^3-4 u v^2+2 u v-v^3+v^2}{u^2 v^2}$ (db)
Jones polynomial	$q^{13/2}-2 q^{11/2}+4 q^{9/2}-8 q^{7/2}+9 q^{5/2}-12 q^{3/2}+12 \sqrt{q}-\frac{11}{\sqrt{q}}+\frac{9}{q^{3/2}}-\frac{6}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{1}{q^{9/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- a z 5 -2 z 5 a -1 - z 5 a -3 + a 3 z 3 - a z 3 -5 z 3 a -1 -2 z 3 a -3 + z 3 a -5 + a 3 z + a z -2 z a -1 - z a -3 + 2 z a -5 + a -1 z -1 - a -3 z -1$ (db)
Kauffman polynomial	$- z 10 a -2 - z 10 a -4 -4 z 9 a -1 -6 z 9 a -3 -2 z 9 a -5 -7 z 8 a -2 - z 8 a -6 -8 z 8 -10 a z 7 + 3 z 7 a -1 + 24 z 7 a -3 + 11 z 7 a -5 -9 a 2 z 6 + 36 z 6 a -2 + 16 z 6 a -4 + 6 z 6 a -6 + 17 z 6 -6 a 3 z 5 + 20 a z 5 + 19 z 5 a -1 -28 z 5 a -3 -21 z 5 a -5 -3 a 4 z 4 + 13 a 2 z 4 -40 z 4 a -2 -31 z 4 a -4 -12 z 4 a -6 -5 z 4 - a 5 z 3 + 4 a 3 z 3 -10 a z 3 -18 z 3 a -1 + 14 z 3 a -3 + 17 z 3 a -5 -5 a 2 z 2 + 13 z 2 a -2 + 15 z 2 a -4 + 8 z 2 a -6 + z 2 - a 3 z + 3 a z + 2 z a -1 -8 z a -3 -6 z a -5 - a -2 + a -1 z -1 + a -3 z -1$ (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 L11a378. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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