L11a304

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 5 + 2 v 3 u 4 -2 v 2 u 4 -2 v 3 u 3 + 4 v 2 u 3 -3 v u 3 -3 v 2 u 2 + 4 v u 2 -2 u 2 -2 v u + 2 u -1$ (db)
Jones polynomial	$\sqrt{q}-\frac{2}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{6}{q^{5/2}}+\frac{7}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{8}{q^{13/2}}+\frac{6}{q^{15/2}}-\frac{4}{q^{17/2}}+\frac{2}{q^{19/2}}-\frac{1}{q^{21/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- a 5 z 9 + a 7 z 7 -8 a 5 z 7 + a 3 z 7 + 6 a 7 z 5 -24 a 5 z 5 + 6 a 3 z 5 + 12 a 7 z 3 -33 a 5 z 3 + 11 a 3 z 3 + 9 a 7 z -19 a 5 z + 6 a 3 z + 2 a 7 z -1 -3 a 5 z -1 + a 3 z -1$ (db)
Kauffman polynomial	$- z 3 a 13 + z a 13 -2 z 4 a 12 + z 2 a 12 -3 z 5 a 11 + 2 z 3 a 11 - z a 11 -4 z 6 a 10 + 5 z 4 a 10 -2 z 2 a 10 -5 z 7 a 9 + 12 z 5 a 9 -10 z 3 a 9 + 3 z a 9 -4 z 8 a 8 + 9 z 6 a 8 -2 z 4 a 8 - z 2 a 8 -3 z 9 a 7 + 9 z 7 a 7 -9 z 5 a 7 + 13 z 3 a 7 -10 z a 7 + 2 a 7 z -1 - z 10 a 6 - z 8 a 6 + 16 z 6 a 6 -23 z 4 a 6 + 13 z 2 a 6 -3 a 6 -5 z 9 a 5 + 26 z 7 a 5 -48 z 5 a 5 + 45 z 3 a 5 -21 z a 5 + 3 a 5 z -1 - z 10 a 4 + 2 z 8 a 4 + 9 z 6 a 4 -25 z 4 a 4 + 17 z 2 a 4 -3 a 4 -2 z 9 a 3 + 12 z 7 a 3 -24 z 5 a 3 + 19 z 3 a 3 -6 z a 3 + a 3 z -1 - z 8 a 2 + 6 z 6 a 2 -11 z 4 a 2 + 6 z 2 a 2 - a 2$ (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 =

-5 is the signature of L11a304. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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