L11a316

(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_6,9,7,10 X_16,7,17,8 X_8,15,1,16 X_20,13,21,14 X_18,6,19,5 X_4,20,5,19 X_22,17,9,18 X_14,21,15,22
Gauss code	{1, -2, 3, -9, 8, -4, 5, -6}, {4, -1, 2, -3, 7, -11, 6, -5, 10, -8, 9, -7, 11, -10}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 5 + v u 5 - v 3 u 4 + 4 v 2 u 4 -3 v u 4 + u 4 + 2 v 3 u 3 -5 v 2 u 3 + 5 v u 3 -2 u 3 -2 v 3 u 2 + 5 v 2 u 2 -5 v u 2 + 2 u 2 + v 3 u -3 v 2 u + 4 v u - u + v 2 - v$ (db)
Jones polynomial	$-\frac{1}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{7}{q^{5/2}}+\frac{10}{q^{7/2}}-\frac{14}{q^{9/2}}+\frac{15}{q^{11/2}}-\frac{16}{q^{13/2}}+\frac{14}{q^{15/2}}-\frac{10}{q^{17/2}}+\frac{6}{q^{19/2}}-\frac{3}{q^{21/2}}+\frac{1}{q^{23/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- z 5 a 9 -3 z 3 a 9 -2 z a 9 - a 9 z -1 + z 7 a 7 + 4 z 5 a 7 + 6 z 3 a 7 + 6 z a 7 + 3 a 7 z -1 + z 7 a 5 + 3 z 5 a 5 -4 z a 5 -2 a 5 z -1 - z 5 a 3 -3 z 3 a 3 -2 z a 3$ (db)
Kauffman polynomial	$- z 4 a 14 + z 2 a 14 -3 z 5 a 13 + 3 z 3 a 13 -5 z 6 a 12 + 5 z 4 a 12 - z 2 a 12 -7 z 7 a 11 + 11 z 5 a 11 -8 z 3 a 11 + 2 z a 11 -8 z 8 a 10 + 18 z 6 a 10 -18 z 4 a 10 + 6 z 2 a 10 - a 10 -6 z 9 a 9 + 12 z 7 a 9 -6 z 5 a 9 - z 3 a 9 - z a 9 + a 9 z -1 -2 z 10 a 8 -6 z 8 a 8 + 33 z 6 a 8 -39 z 4 a 8 + 18 z 2 a 8 -3 a 8 -10 z 9 a 7 + 33 z 7 a 7 -35 z 5 a 7 + 22 z 3 a 7 -11 z a 7 + 3 a 7 z -1 -2 z 10 a 6 - z 8 a 6 + 21 z 6 a 6 -26 z 4 a 6 + 13 z 2 a 6 -3 a 6 -4 z 9 a 5 + 13 z 7 a 5 -11 z 5 a 5 + 7 z 3 a 5 -6 z a 5 + 2 a 5 z -1 -3 z 8 a 4 + 11 z 6 a 4 -11 z 4 a 4 + 3 z 2 a 4 - z 7 a 3 + 4 z 5 a 3 -5 z 3 a 3 + 2 z a 3$ (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 L11a316. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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